text:
Adei Ad
Photo: Tazpit News Agency
Israel places two radical right-wing suspects in administrative detention
Police raid West Bank outposts in crackdown on 'hilltop youth,' as part of recent efforts to curb Jewish violence against Palestinians in wake of the deadly arson attack on the Dawabsheh family last month.
Itay Blumenthal, Yoav Zitun|Last update: 08.09.15 , 14:02
Defense Minister Moshe Ya'alon issued on Sunday administrative detention orders for two radical right-wing suspects, Meir Ettinger and Evyatar Slonim.
Their remand was scheduled to be extended on Sunday, but following Ya'alon's edict they could remain in custory for up to six months without being brought before a judge.
The two, who are in their early 20s, were arrested last week. Another suspected Jewish extremist, Mordechai Mayer, was placed under six-month administrative detention last week.
Last week, the Security Cabinet approved the use of harsh measures to combat the trend, including administrative detention, which allows Israel to hold suspects for lengthy periods without charge. The measure has been mainly used against Palestinians suspected of involvement in terrorist groups, and rarely against Israelis.
Meir Ettinger (Photo: Avihu Shapira)
The Shin Bet security agency has accused Ettinger of leading an extremist Jewish movement responsible for encouraging attacks on Palestinian property and Christian holy sites, including an arson attack on a well-known church near the Sea of Galilee in northern Israel that marks the New Testament story of the miracle of the loaves and fish.
Adv. Aharon Roza of Honenu said on behalf of his clients that "it is unacceptable that Ettinger and Slonim's administrative detention will substitute their court hearing today. It seems like the judicial process is just a charade." The Defense Ministry said that the decree was issued following the recommendation of Shin Bet security services.
Israeli human rights activists who advocate on behalf of Palestinians, as well as lawyers for the Israeli suspects, criticized the use of administrative detention, portraying it as a draconian measure intended to appease an Israeli public shocked at the firebomb attack.
"It is carried out based on an administrative order only, without indictment or trial, and the detainee cannot defend himself against the allegations as the evidence is classified," a statement by human rights group B'Tselem said.
"This measure is dangerous...for the entire legal system and for democracy," said Aharon Rozeh, a lawyer for Ettinger and Slonim, who said his clients were innocent.
Jewish Radicalism
West Bank inches closer to boiling point / Oded Shalom and Elior Levy
The rising tensions are evident everywhere. In Meir Ettinger's fanaticism; in Palestinian villages where night watchmen groups are formed; in Givat Ronen outpost where 'the most hotheaded guys are drawn,' and in the police, where investigators are trying to solve the arson in Duma and prepare for possible Jewish retaliation.
Israeli security forces raided the settlement of Kokhav HaShahar and the surrounding area in the West Bank late Saturday night and made a series of arrests, settlers said Sunday morning.
Israel Police's nationalist crimes unit also operated at the nearby Adei Ad outpost in the Binyamin area in the wake of the arson attack on a Palestinian home in the village of Duma that claimed the lives of a baby, Ali Dawabsheh, and his father, Saed Dawabsheh, and left the 27-year-old mother and her second child, aged 4, in serious condition. A gag order has been imposed on the investigation.
Settlers claimed that some of the search warrants were issued in the middle of last week, while others were issued a week ago.
The Shiloah Valley area, which has seen repeated clashes between Jews and Palestinians, is considered a main stomping ground for the so-called hilltop youth, a radical faction of the settler movement.
Police cars making arrests in Adei Ad, early Sunday morning (Photo: Tazpit News Agency)
A statement issued by Honenu, an organization offering legal advice to Jewish hate crime suspects, said that at least 15 "hilltop youths" have been arrested across the Binyamin region and taken for questioning. Some have already been released after being questioned by the Shin Bet.
Defense lawyer Itamar Ben-Gvir said that the arrests are "meant to assuage politicians' anxieties. I hope that next time they'll use a tranquilizer instead of sending police to impinge on people's liberties."
"I only hope that Arab terror will produce a similar public backlash that will lead police to arrest terrorists preemptively," he added.
On Saturday, Saed Dawabsheh, who was injured in the Duma arson attack last month, was laid to rest after succumbing to his wounds. Hundreds attended his funeral, which was followed by clashes with Israel Defense Forces troops.
Not a single Palestinian Authority official attended, as opposed to dozens of Hamas operatives who were conspicuously present.
In a statement published following the news of Dawabsheh's death, the Islamic organization said: "Dawabsheh's death highlights the crimes perpetrated by the Zionists. Resistance has become our right, our duty and the only way to defends ourselves in the West Bank. The only way to deter the settlers is to engage proactively against them, rather than wait until they attack our villages and homes. Our people in the West Bank are left with no choice but to launch an all-out attack against Israel, without further ado.
The Associated Press and Reuters contributed to this report.
See all talkbacks "Israel places two radical right-wing suspects in administrative detention"


text:
Conesus Lake
Invasive species are non-native, introduced species that cause harm to the environment, human health, and/or economy. International trade and travel has increased the rate at which species migrate to new environments where they can have unintended, ill effects. Invasive Species possess certain traits that allow them to out compete and displace native species in their new environment. In the absence of natural predators, an invasive population can grow relatively unchecked and disrupt the native ecosystem until the environment slowly begins to evolve and adapt on its own.
Environmental:
Reductions in biodiversity and localized extinctions of native species
Losses in ecosystem services such as erosion and flood control
Replacement of forage food for native species
Direct predation on native species including sport fishes
Reductions in dissolved oxygen content in the water, which can lead to fish kills
Changes in water clarity
Changes in macrophyte abundance
Recreational:
Reductions in game fish populations
Creation of weed beds impassable to boaters and swimmers
Damage to boat engines, steering equipment and other parts
Negative aesthetic impacts
Sharp shells deter swimmers and divers
Fouling of fishing equipment
Human Health:
The sap of Giant Hogweed (Heracleum mantegazzianum), in combination with moisture and sunlight, can cause severe skin and eye irritation, painful blistering, permanent scarring and blindness.
The sharp shells of Zebra Mussels (Dreissena polymorpha) can potentially cut or scrape swimmers and divers.
Feral swine (Sus scrofa) have razor sharp tusks and can be aggressive toward humans and their pets and can carry and transmit several diseases including swine brucellosis, E. coli, trichinosis, and pseudorabies to humans and/or livestock.
Feral swine rooting and wallowing habits can disturb water quality in streams, and their resulting waste can foul water supplies.
Economic:
Direct invasive species control and management costs
Costs attributed to losses in ecosystem services such as erosion control and fisheries
Economic losses to local businesses and tourism relating to impaired uses such as fishing, boating, swimming, navigation, ecological biodiversity, aesthetics, and other recreation uses
Annual economic losses due to invasive species in the United States are estimated at over $120 billion per year (Pimental et al 2005)
In the Great Lakes region, direct costs attributed to aquatic invasive species accounts for over $100,000,000 a year (Rosaen et al, 2012).
Prevention is the most cost effective management strategy for the control of invasive species. Increasing awareness levels of invasive species issues and educating the public on prevention best management practices are key. Once a species is established in an ecosystem, direct costs associated with management and indirect economic losses can spiral out of control.
Best Management Practices for Invasive Species Prevention
Clean, drain, and dry all boating, angling, and recreational gear before moving between waterways.
Never transport uncertified baitfish between waters.
Always dispose of unwanted bait in the trash, not the ground.
Never release unwanted species from home aquariums.
Never release live seafood into the wild.
Conesus Lake Watershed Council Invasive Species Prevention and Response Plan
The Conesus Lake Watershed Council approved an Invasive Species Prevention and Response Plan in 2013. The plan identifies pathways for invasive species introduction and associated prevention projects, outlines a monitoring program, and prescribes a detailed response protocol in the event of a confirmed sighting.
To report a suspect invasive species sighting, please contact the Conesus Lake Watershed Manager at (585) 243-7550.
Watershed Council
Watershed Management Plan
About the Watershed
About the Lake
Watershed Inspection Program
Reports, Publications & Educational Resources

redpajama_set_name: RedPajamaCommonCrawl


Deforestation is a bigger cause of climate change than all the world's cars and trucks combined. Some of the damage is done underwater, where entire forests of kelp and seaweed are vanishing.
One fix may come from a Japanese company in an industry better known for its carbon-dioxide emissions than for environmentalism. Nippon Steel & Sumitomo Metal Corp. says it's found a way to use some of its waste to stop underwater erosion and as a fertilizer that rejuvenates dying seaweed.
Skeptics might be tempted to dismiss this as slick PR or as a cover-story for dumping. But fishermen say the technology has helped restore fish populations off Hokkaido. And Shinji Sato, a coastal engineering expert at the University of Tokyo, says it could help absorb greenhouse gases. "This is definitely a worthwhile effort," he says.
The idea that waste can be put to productive work goes back at least as far as the Kalundborg industrial park in Denmark, where since the 1970s dozens of factories have been saving money by trading byproducts. A residue collected by one plant's smoke-stack scrubbers, for example, becomes an ingredient for construction materials made by another; surplus heat goes to warm local homes.
Mixing steel slag and dredged soil on the coast of Himeji city in western Japan.Source: Nippon Steel & Sumitomo Metal Corp.
Nippon Steel's furnaces generate 17 million tons of molten slag each year. Most of it is technically not waste because it's turned into sealant for road asphalt or used as a cement additive. But there is still a large remainder, which is only good for making things like gravel — not a money maker.
Finding a better use for that material is why company engineers around 2000 came up with the idea of using the iron-rich residue as an underwater fertilizer. Japan has lost kelp and eelgrass along long stretches of its coastline, partly because river dams have starved coastal plants of nutrients that would otherwise have run off into the sea, according to Sato, the University of Tokyo professor.
Installing Nippon Steel's iron supply units.Source: Nippon Steel & Sumitomo Metal Corp.
So, with the permission of local fisherman, Nippon Steel in 2004 launched a small test off the coast of Hokkaido, seeding the seabed with bails of slag and fermented wood pulp.
Within a year, kelp staged a spectacular recovery, growing 220 times more than in a nearby, untreated area. A larger study started in 2014 is also showing results, though somewhat less dramatic, according to Hiroyuki Aiuchi, who represents a fishing cooperative in the nearby town of Mashike. He says catches of abalone and sea urchins, delicacies coveted by sushi chefs, have improved.
"It seems to be working,"he said by phone.
Eiji Kiso, the Nippon Steel staffer who oversees marketing for the project, says the company hopes eventually to sell one million tons of slag each year to fisheries and local governments.
He says steel slag can also be used as a kind of glue to stop erosion around places like ports, where dredging has turned the seabed into fallow sand. Under a swath of sea the size of 20 football fields, near one of its iron mills on the edge of Tokyo bay, Nippon Steel has used some of the material to build a sort of underwater planter's bed. Seaweed growth has taken off, attracting fish and drawing scores of sports fishermen on weekends, Kiso says.
Ecologist Tomohiro Kuwae is a fan of these experiments. A scientist at the Port and Airport Research Institute, a government-funded think tank near Tokyo, he says as much of 20 percent of the slag generated by Japanese metal producers could be used to restore ocean ecosystems.
Revival of kelp forests after placement of iron supply units at Mashike town.Source: Nippon Steel & Sumitomo Metal Corp.
The big payoff, he says, has less to do with sushi than climate change. Healthier plant life along Japan's coastline could help absorb an extra nine million tons of carbon-dioxide annually over the next decade or so, according to calculations by him and other scientists. Kuwae belongs to a group of researchers pushing to have seaweed counted as a carbon sink in international agreements.
If that happens, Nippon Steel's inventions could help Japan meet its goals for cutting greenhouse emissions by a quarter by 2030.
First, though, the company will have to win over more fishermen.


TMZ reports that actor Edward Herrmann has passed away from cancer at age 71.
In St. Elsewhere's finest episode(s), the fourth season two-parter "Time Heals", we meet the hospital's founder, an Irish Catholic priest from Southie named Father Joseph McCabe, played by Edward Herrmann. McCabe re-appeared in the fifth season opener, "Where There's Hope, There's Crosby", returning from Arizona (with Herrmann in old-age makeup) suffering from ALS. For that appearance, Herrmann was nominated for an Emmy award for Outstanding Performance by a Guest Actor.
According to a report from TMZ, Herrmann had been battling brain cancer and had been in the ICU for the past three and half weeks. He was taken off life support and passed away this morning at the age of 71.
Herrmann was more recently known for his role on Gilmore Girls, and he assembled a long list of credits dating back to the early seventies. To 80s kids like me, he was the King of the Vampires in The Lost Boys.
Follow this link to a YouTube playlist to enjoy Edward Herrmann's performance in "Time Heals".
Edward Herrmann will also be remembered for his uncanny portrayal of President Franklin Delano Roosevelt in two made for TV movies from the 1970s - "Eleanor and Franklin" and "Eleanor and Franklin: The White House Years" (both of which earned him Best Actor Emmy Award nominations), And as recently as last year, Herrmann was the voice of FDR in Ken Burns' documentary series "The Roosevelts: An Intimate History," I always thought that Hermann reminded me a little of FDR when he played Father McCabe in those legendary St. Elsewhere episodes, especially the flashback scenes that he's in from the 1930s and 1940s.
I've been think that this opening scene is how I always remember Edward Hermann. it was nice to find it again.
When he played the old man in the fifth season opener, he was again Fr. Joseph MCabe (back from Arizona where the character had been transferred according to events described in 'Time Heals.'). He was such a wonderful presence on St. Elsewhere that the character was often referenced in later seasons - even adding another 'McCabe' as a young chaplain in the final season.
Thanks for your comment; I have updated the wording so it's more clear what I meant!


text:
Q: Linux Kernel compile: File format not recognized I compile the latest kernel version 4.15.15. My current kernel is 3.13 on Ubuntu 12.04, gcc version is 7.3.0.  When it executes the command make, it has many errors, such as
objdump: mm/.tmp_oom_kill.o: File format not recognized

How to fix the problems?

A: Perhaps this tmp file is left over from a failed build. I suggest removing the object file and re-trying the build.
You can use the file command to find out more about the file, e.g.:
file mm/.tmp_oom_kill.o

maybe that will explain why you are getting that error.

redpajama_set_name: RedPajamaStackExchange


The following block of PHP codes (line 15 - 32) at the top of osCommerce homepage (index.php) determine what the contents of the Main Content (Body text) will be displayed. The secret is the $category_depth variable.
The picture below summarize how the contents of Main Content of osCommerce homepage will be displayed according to the value of $category_depth passing to the main program.
Now, we know how the $category_depth variable control the contents display on the Main Content of osCommerce homepage.


text:
De figuur van de priester-arbeider was een initiatief in de Katholieke Kerk in Frankrijk dat tot doel had de missie van de Kerk onder de arbeiders te intensiveren. De gedachte erachter was dat priesters zouden gaan werken in fabrieken, om daar de arbeiderspopulatie te evangeliseren en te bekeren tot, dan wel te behouden voor de katholieke Kerk.

Het initiatief tot de beweging van de priester-arbeiders werd genomen door Jacques Loew. Loew, een dominicaan, die in 1941 als arbeider begon in de haven van Marseille. Aanvankelijk was het niet eens de bedoeling dat hij als arbeider aan de slag zou gaan. Zijn opdracht was vooral om de omstandigheden waaronder arbeiders werkten en leefden te bestuderen. Loews initiatief had al spoedig navolging in andere Franse industriesteden, waar het werd gezien als een uitgelezen middel om het vertrouwen van de arbeidersklasse, die de Kerk massaal de rug had toegekeerd, te herwinnen. De Franse theoloog Marie-Dominique Chenu, eveneens dominicaan, zette zich in om het pastoraat van priester-arbeiders theologisch te funderen.

Gaandeweg werden sommige priester-arbeiders echter lid van (linkse) vakbonden en politieke partijen. Dit ging het Vaticaan te ver. In 1953 vaardigde paus Pius XII een apostolische exhortatie uit, waarin de beweging werd verboden.

Literatuur
 Bonn, Robert L., and Doyle, Ruth T. 1974. "Secularly Employed Clergymen: A Study in Occupational Role Recomposition". Journal for the Scientific Study of Religion, 13(3): 325-343.
 Flower, J.E. 1967. "Forerunners of the Worker-Priests". Journal of Contemporary History, 2(4): 183-199.
 Siefer, Gregor. 1964. The Church and Industrial Society. London: Darton, Longman, and Todd.

Bron
 Worker-Priest op en-wikipedia

Katholicisme in Frankrijk
Sociale geschiedenis

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}
Giant-dipole states are an exotic species of Rydberg atoms in crossed electric and magnetic fields. They have been first explored theoretically \cite{Baye1992,Dzyaloshinskii1992,Dippel1994,Schmelcher1998,Schmelcher2001} and experimentally \cite{Fauth1987,Raithel1993} firstly in the early 1990s. The giant-dipole states are of decentered character and possess, in contrast to the usual Rydberg states, a huge electric dipole moment.
It has been shown that the total potential of the electronic motion possesses a gauge invariant term which leads to an outer potential well \cite{Schmelcher1993a}. This potential well supports weakly bound states possessing a large spatial separation of the Rydberg electron and the ionic core. The mathematical origin of this effect is the non-separability of the center of mass and electronic motions in the presence of the external fields \cite{Avron1978,Herold1981,Johnson1983,Ackermann1997,Shertzer1998}. More precisely, translation symmetry and conservation of the total momentum in field-free space is replaced by the conservation of a novel quantity, the so-called total pseudomomentum. Recently, the existence of giant-dipole Wannier excitons in crossed electric and magnetic fields have also been predicted inside a $\textrm{Cu}_2\textrm{O}$ semiconductor environment \cite{Kurz2017}. Furthermore, giant-dipole molecular states consisting of a highly excited atom in crossed fields and a neutral ground state atom have been shown to exist and the corresponding electronic configurations as well as their rovibrational properties have been analyzed in detail \cite{Kurz12}.

In the last years, emphasis has been laid on diatomic ultra-long range molecules consisting of a Rydberg atom whose electron binds a neutral ground state atom as a result low-energy scattering off the ground state species. These particular molecules were predicted and experimentally verified to exist in ultracold traps \cite{Greene2001,Hamilton2002,Stanojevic2006,Bendkowsky2009}. Since then a number of observations of Rydberg molecules  with a variety of electronic and rovibrational structures were made \cite{Tallant2012,Krupp2014,Gonzalez2014,Anderson2014,Booth2015,DeSalvo2015,Sassmannshausen2015,Sassmannshausen2016,Niederpruem2016,Niederpruem2016b,Camargo2016,Gonzalez2016}. The standard approach to describe these ultra-long range molecules is the Fermi-pseudopotential approach \cite{Fermi1934}. However, an alternative approach based on a Green's function method has been established as well \cite{Fabrikant2002,Greene2006,Bendkowsky2010,Fey2015,Fey2016}. In a recent work, C.\ Fey and coworkers pointed out the interconnections and limitations of both approaches \cite{Fey2015}. In particular, it was shown that the standard Fermi-pseudopotential approach modeled by a bare delta scattering potential leads to non-converging results. These difficulties can be overcome by considering a regularized $\delta$-function potential which is equivalent to the application of a regularized Green's function approach.

In this work, we apply the analysis performed in Ref.\ \cite{Fey2015} to the giant-dipole systems in general. After a short introduction, we present in Sec.\ 2 some general properties of the Green's function for the case of decoupled subsystems. In Sec.\ 3 we derive the Green's function for magnetically field-dressed systems including the giant-dipole system. The result of this analysis is then used in Sec.\ 4 to calculate a spectroscopic response function, namely the dynamical polarizability of giant-dipole systems. In Sec.\ 5 we analyze the adiabatic potential energy surfaces (PES) where we show a divergent behavior whenever a bare Fermi-pseudopotential approach is used to model the electron-perturber interaction. Using the result for the giant-dipole Green's function, we verify this divergent behavior for the giant-dipole system. In Sec.\ 6 we use the giant-dipole Green's function approach to calculate the PES in different levels of approximations. Finally, we present the result for the regularized giant-dipole Green's function.
\section{Green's function of decoupled systems \label{gd_intro}}
We first consider the Green's function of a system $H$ consisting of an arbitrary number $N$ of decoupled subsystems $h_i$. The related Green's function $G(\bfmath{r}, \bfmath{r}';E)$ of the total system $H$ in energy space is given by
\begin{eqnarray}
(E-H)G=\bfmath{1}\ \Rightarrow \ (E-H(\bfmath{r}))G(\bfmath{r}, \bfmath{r}';E)=\delta(\bfmath{r}-\bfmath{r}')\equiv \prod\limits^{N}_{i=1}\delta(\bfmath{r}_i-\bfmath{r}^{'}_{i}),\label{GFtotal}
\end{eqnarray}
where $H(\bfmath{r})=\sum_{i}h_i(\bfmath{r}_i),\ \bfmath{r}= (\bfmath{r}_1,...,\bfmath{r}_N)$. Using  the eigenfunctions $\phi_i$ of the subsystems $h_i$
\begin{eqnarray}
h_i \phi^{(i)}_{k}({\bf r}_i)=\varepsilon^{(i)}_{k}\phi^{(i)}_{k}({\bf r}_i),\ \ \ i=2,...,N,\label{eigen}
\end{eqnarray}
we can expand the Green's function $G$ of the total system as
\begin{eqnarray}
G(\bfmath{r},\bfmath{r}';E)=\sum_{k_2...k_N}G_{1}(\bfmath{r}_1,\bfmath{r}_1';E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_i}) \prod^{N}_{n=2} \phi^{(n)*}_{k_n}(\bfmath{r}^{\prime}_n)\phi^{(n)}_{k_n}(\bfmath{r}_{n}) \label{GFtotalexp}
\end{eqnarray}
whereby $G_1$ is the Green's function related to the Hamiltonian $h_1$, i.e.\
\begin{eqnarray}
(E-h_1)G_{1}(\bfmath{r}_1,\bfmath{r}_1';E)=\delta(\bfmath{r}_1-\bfmath{r}^{'}_{1}).\label{delta}
\end{eqnarray}
Using the completeness relation of the $\phi^{(i)}_k$-functions
\begin{eqnarray}
\delta(\bfmath{r}_i-\bfmath{r}^{'}_{i})=\sum_{k}\phi^{(i)*}_{k}(\bfmath{r}_i) \phi^{(i)}_{k}(\bfmath{r}^{'}_i)
\end{eqnarray}
one can directly verify that Eq.\ (\ref{GFtotalexp}) solves Eq.\ (\ref{GFtotal}) (see Appendix A). Furthermore, from Eq.\ (\ref{GFtotal}) one obtains the known spectral representation of an arbitrary Green's function $G$
\begin{eqnarray}
G(\bfmath{r},\bfmath{r}';E)=\sum_i \frac{\psi_i(\bfmath{r})\psi^{*}_{i}(\bfmath{r}^\prime)}{E-\varepsilon_i}, \label{GFeigenexp}
\end{eqnarray}
where $\psi_i$, $\varepsilon_i$ denote the eigenfunctions and eigenvalues of $H$.

Obviously, the expansion Eq.\ (\ref{GFtotalexp}) is not unique as any subsystem $h_i$ can be used to construct the Green's function of the total system. Furthermore, it is easy to verify that in case of an unitary transformation $\mathcal{U}$ of the Hamilton $H$, the corresponding Green's function transforms analogously, i.e.\ $\tilde{G}(\bfmath{r},\bfmath{r}';E)=\mathcal{U}G(\bfmath{r},\bfmath{r}';E)\mathcal{U}^{-1}$.
\section{Green's functions of magnetically field-dressed systems}
We first derive the Green's function of various magnetically field-dressed systems including the considered giant-dipole system.
\subsection{Harmonically trapped systems}
As a first example we consider a particle with charge $q$ of mass $\mu$ in a homogeneous magnetic field \cite{atomicunits} along the $z$-direction ($\bfmath{ B}=B\bfmath{e}_z$), confined to an additional harmonic potential in $z$-direction
\begin{equation}
H=H_{xy}+H_z,\ \ H_{xy}=\frac{1}{2 \mu }\left(\bfmath{p}^{2}_{x}+\bfmath{p}^{2}_{y} \right)-\frac{qB}{2 \mu}L_{z}+\frac{q^2B^2}{8 \mu^2}(x^2+y^2),\ \ H_z=\frac{p^{2}_{z}}{2 \mu}+\frac{\mu}{2}\omega^{2}_{z}z^2.\label{Hxy}
\end{equation}
In order to determine the Green's function related to this system we use the Green's function $G_{xy}$ of a magnetically field dressed charge in two dimensions. This is given by \cite{Bychov1960,Ueta1992}
\begin{eqnarray}
G_{xy}(x,y,x',y';E)=e^{i(x'y-xy')\mu \omega_c/2}G_{0}(\eta;E),\ \ \ \omega_c=\frac{qB}{\mu},
\end{eqnarray}
where we have introduced $\eta=((x-x')^2+(y-y')^2)\mu \omega_c/2$ and
\begin{eqnarray}
G_{0}(\eta;E)=e^{-\eta /2} \frac{\mu \omega_c}{2\pi}\sum^{\infty}_{n=0}\frac{L_{n}(\eta)}{E-\omega_c (n+\frac{1}{2})}.
\end{eqnarray}
Here the expression $L_{n}(x)$ denotes the Laguerre polynomials \cite{Abramowitz1972}. Using both the eigenenergies $\varepsilon_n=\omega_z(n+1/2)$ and eigenfunction $\phi_n(z)=(\mu \omega_z/\pi)^{-1/4}(2^n n!)^{-1/2}$ $H_{n}(\sqrt{\mu \omega_z}z)\exp(-\mu \omega_zz^2/2)$ of the harmonic oscillator, we easily obtain the Green's function $G_{xy}$ of this system as
\begin{eqnarray}
G_{xy}(\bfmath{r},\bfmath{r}';E)=e^{i(x'y-xy')\mu \omega_c/2-\eta /2}\frac{\mu \omega_c}{2\pi}\sum^{\infty}_{n,m=0}\frac{ \phi_m(z)\phi_{m}(z')}{E-\omega_z(m+1/2)-\omega_c(n+\frac{1}{2})}L_{n}(\eta).
\end{eqnarray}
Next, we consider a situation with an additional anisotropic harmonic confinement both in $x$- and $y$-directions, whereby the harmonic confinement remains untouched. For such a configuration the Hamilton $H_{xy}$ from Eq.\ (\ref{Hxy}) is rewritten and given by
\begin{eqnarray}
H_{xy}=\frac{1}{2 \mu }\left( \bfmath{p}^{2}_{x}+\bfmath{p}^{2}_{y} \right)-\frac{qB}{2 \mu}L_{z}+\frac{q^2 B^2}{8 \mu^2}(x^2+y^2)+\frac{\mu}{2}\omega^{2}_{x}x^2+\frac{\mu}{2}\omega^{2}_{y}y^2,\ \ \omega_x \not= \omega_y.\label{gdxy}
\end{eqnarray}
In contrast to Eq.\ (\ref{Hxy}) this Hamiltonian has lost its azimuthal symmetry. However, in Ref.\ \cite{Dippel1994} it has been shown that such a Hamiltonian can be diagonalized via an unitary transformation
\begin{eqnarray}
U=\exp(i \alpha xy)\exp(i \beta p_x p_y),\ \alpha ,\beta \in \mathbb{R}
\end{eqnarray}
which transforms $H_{xy}$ into the sum of two decoupled harmonic oscillators with rescaled masses $M_{1,2}$ and frequencies $\omega_{1,2}$ with
\begin{eqnarray}
M_{1,2}&=&\frac{\sqrt{(\omega^{2}_{x}+\omega^{2}_{y}+\omega^{2}_{c})^2-4\omega^{2}_{x}\omega^{2}_{y}}}{\textrm{sgn}(\omega^{2}_{x}-\omega^{2}_{y})(\omega^{2}_{x}-\omega^{2}_{y} \pm \omega^{2}_{c})+\sqrt{(\omega^{2}_{x}+\omega^{2}_{y}+\omega^{2}_{c})^2-4\omega^{2}_{x}\omega^{2}_{y}}},\\
\omega_{1,2}&=&\frac{1}{\sqrt{2}}[\omega^{2}_{x}+\omega^{2}_{y}+\omega^{2}_{c} \pm \textrm{sgn}(\omega^{2}_{x}-\omega^{2}_{y})\sqrt{(\omega^{2}_{x}+\omega^{2}_{y}+\omega^{2}_{c})^2-4\omega^{2}_{x}\omega^{2}_{y}}]^{1/2}.
\end{eqnarray}
Thus, the  eigenenergies $\varepsilon_{n_1 n_2}$ and eigenfunctions $\psi_{n_1 n_2}(x,y)$ of Eq.\ (\ref{gdxy}) are given by
\begin{eqnarray*}
\varepsilon_{n_1 n_2}=(n_1+1/2)\omega_1+(n_2+1/2)\omega_2,\ \ \psi_{n_1 n_2}(x,y)=U(\phi_{n1}(x)\phi_{n2}(y)),\ n_{i} \in \mathbb{N}_{0}.\label{harmonic_oscillator_prop}
\end{eqnarray*}
The functions $\phi_{n_1}(x),\phi_{n_2}(y)$ denote the eigenfunctions of one-dimensional harmonic oscillator with masses $M_i$ and frequencies $\omega_i$. The specific form of the eigenfunction $\psi_{n_1n_2}(x,y)$ depends on $U$ and has been derived explicitly in Ref.\ \cite{Dippel1994}. These eigenenergies and eigenfunctions can be used to expand the Green's function related to the Hamiltonian $H=H_{xy}+H_z$ from Eq.\ (\ref{Hxy}) by using the Green's function of the one-dimensional harmonic oscillator which is given by \cite{Bakhrakh1970}
\begin{eqnarray}
G^{\rm (1d)}_\text{har}(z, z'; E) = - \sqrt{\frac{\mu}{\pi \omega_z}} \, \Gamma\left(\frac 1 2 - \frac E {\omega_z}\right) \, D_{E / \omega_z - 1 / 2}(-\sqrt{2 \mu E} \, z_<) \, D_{E / \omega_z - 1 / 2}(\sqrt{2 \mu E} \, z_>).
\end{eqnarray}
Here $D_{E}$ denotes a parabolic cylinder function \cite{Abramowitz1972} and $z_{<},\ z_{>}$ are the lesser and greater of the spatial variables $z$ and $z'$, $z_{<} \equiv \textrm{min}(z,z'),\ \ z_{>} \equiv \textrm{max}(z,z')$. Using Eq.\ (\ref{GFtotalexp}) we easily obtain
\begin{eqnarray}
\mathcal{G}_{\rm gd}(\bfmath{r},\bfmath{r}';E)=\sum_{n_1 n_2}G^{\rm (1d)}_{\textrm{har}}(z,z',E-\varepsilon_{n_1n_2})\psi_{n_1n_2}(x,y)\psi^{*}_{n_1n_2}(x',y').\label{gdgf}
\end{eqnarray}
The poles of the Green's function (\ref{gdgf}), which are related to the poles of the Gamma function $\Gamma$, determine the eigenenergies of the total system $H$. As $\Gamma(x)$ possesses poles for $x=-n,\ n \in \mathbb{N}_0$, the eigenenergies of $H$ are given by $E_{n_1n_2n}=\varepsilon_{n_1n_2}+\omega_z(n+1/2)$. This is the same result as it is expected from the structure of the transformed Hamiltonian $\tilde{H}=UHU^{-1}$.
\subsection{Giant-dipole Green's function \label{gd_gf}}
The main focus this work are a special kind of atomic species, the so-called giant-dipole atoms which consists of a highly excited atom in crossed homogeneous electric and magnetic fields. In particular, we consider the case that a highly excited single valence electron orbits around a positively charged ionic core of coordinate $\bfmath{r}_c$ and mass $m_c$. If we approximate the ionic core as a charged point particle, we find the following Hamiltonian
\begin{eqnarray}
H=\frac{(\bfmath{p}+\bfmath{A}(\bfmath{r}_e))^2}{2m_e} + \frac{(\bfmath{p}-\bfmath{A}(\bfmath{r}_c))^2}{2m_c}+\bfmath{E}\cdot(\bfmath{r}_e-\bfmath{r}_c)-\frac{1}{|\bfmath{r}_e-\bfmath{r}_c|},\ \ \bfmath{A}(\bfmath{r})=\frac{1}{2}\bfmath{B}\times \bfmath{r},\ \ \bfmath{B} \perp \bfmath{E}.\label{laber}
\end{eqnarray}
This system has been studied in detail in Refs.~\cite{Dippel1994,Schmelcher1998,Schmelcher1993a}. In systems of crossed electric and magnetic fields the total momentum $\bfmath{P}$ is not a conserved quantity, but the so-called pseudomomentum $\bfmath{K}\equiv\bfmath{P}-\bfmath{B}\times \bfmath{r}/2$ \cite{Dippel1994}. For this reason, Eq.\ (\ref{laber}) can be transformed into a single particle representation where the corresponding Hamiltonian is given by
\begin{eqnarray}
H_\text{gd}=\frac{(\bfmath{p}-q\bfmath{A}(\bfmath{r}))^2}{2\mu}+V_{\rm gd}(\bfmath{r}),\ \ V_{\rm gd}(\bfmath{r})=\bfmath{E}\cdot \bfmath{r}-\frac{1}{r}+\frac{1}{2M}(\bfmath{K}+\bfmath{B}\times  \bfmath{r})^2
\end{eqnarray}
with $\bfmath{r}=\bfmath{r}_e-\bfmath{r}_c$, $M=m_e+m_c$, $\mu=m_em_c/M$ and $q=(m_e-m_c)/M$. Obviously, this is the problem of a particle of mass $\mu$ and charged $q$ in an external magnetic field and potential $V_{\rm gd}$. Using the transformation $\bfmath{K}\rightarrow \bfmath{K}-M(\bfmath{E}\times \bfmath{B})/B^2$, the electric field can be incorporated into the pseudomomentum. Hence we can set $\bfmath{E}=0$ in the following and discuss the properties of the potential $V_{\rm gd}$ via $\bfmath{K}$ only. Note that $K=|\bfmath{K}|=1\ \rm a.u.$ corresponds to an electric field strength of $2.8 \cdot 10^3\ \rm V/m$ \cite{Dippel1994}.

For a specifically chosen set of parameters, this potential exhibits an outer well containing many bound electronic states. As atomic states bound in this outer potential well exhibit huge permanent electric dipole moments of around $10^5\, \textrm{Debye}$, they have become known as giant-dipole states \cite{Dippel1994}. Expanding the potential $V_{\rm gd}$ around the minimum \textbf{$\bfmath{r}_{\rm min}$} of the outer well up to second order
\begin{eqnarray}
V_{\rm gd}(\bfmath{r})=\frac{\mu}{2}\left( \omega^{2}_{x}(x-x_{\rm min})^2 + \omega^{2}_{y}(y-y_{\rm min})^2 + \omega^{2}_{z}(z-z_{\rm min})^2 \right)
\end{eqnarray}
and performing a corresponding spatial transformation $\bfmath{r}\rightarrow \bfmath{r}+\bfmath{r}_{\rm min}$ \cite{Dippel1994} we arrive at the giant-dipole Hamiltonian which represents a charge $q$ in a homogeneous magnetic field $\bfmath{B}$ and a three-dimensional harmonic potential
\begin{eqnarray}
H_{\rm gd}= \frac{(\bfmath{p}-q\bfmath{A}(\bfmath{r}))^2}{2\mu}+\frac{\mu}{2}\left( \omega^{2}_{x}x^2 + \omega^{2}_{y}y^2 + \omega^{2}_{z}z^2 \right).
\end{eqnarray}
The frequencies $\omega_{i}\equiv\omega_i(\bfmath{K},\bfmath{B})$ depend on the applied fields and characterize the anisotropy of the outer potential well. As $m_c \gg m_e$ one has $q \approx -1$ and $\mu \approx m_e$, which means that the giant-dipole system can be effectively described by an electron in a homogeneous magnetic field and the external potential $V_{\rm gd}$.

Throughout this work we consider $B=2.35\, \text{T},\ K=1\, \rm a.u.$. For arbitrary field strengths the Green's function related to the giant-dipole Hamiltonian $H_{\rm{gd}}$ is given by $\mathcal{G}_{\rm gd}$ in Eq.\ (\ref{gdgf}). However, for the fields  considered in this work one can verify that the system possesses nearly cylindrical symmetry, i.e.\ $\omega_x \approx \omega_y \equiv \omega_{\rho}$. If not stated otherwise, we adopt this azimuthal approximation throughout this work. In this case, it is straightforward to verify that the corresponding giant-dipole Green's function $G_{\rm gd}$ reads as
\begin{eqnarray}
G_\text{gd}(\bfmath{r},\bfmath{r}';E)=\frac{1}{2\pi}\sum^{\infty}_{n=0}\sum^{\infty}_{m=-\infty}G^{\rm (1d)}_{\text{har}}(z,z',E-\varepsilon_{nm})R_{nm}(\rho)R_{nm}(\rho')e^{i m(\phi-\phi')}\label{gd_azymuthal}
\end{eqnarray}
with
\begin{eqnarray}
R_{nm}(\rho)&=& \sqrt{\frac{2 \mu \Omega (n+|m|)!}{n!(|m|!)^2}}\mathrm{e}^{-\frac1 2 \mu \Omega\rho^2} \left(\sqrt{\mu \Omega } \rho\right)^{|m|} {}_1F_1(-n;|m|+1;\mu \Omega\rho^2)
\end{eqnarray}
and
\begin{eqnarray}
\varepsilon_{nm}&=&2\Omega \left( n+\frac{|m|+1}{2} \right) + \omega_c \frac m 2.
\end{eqnarray}
Here ${}_1F_1$ denotes the confluent hypergeometric function \cite{Abramowitz1972} of the first kind, while $\omega_c = -B /\mu$ is modified cyclotron frequency and $\Omega^2=\omega_c^2/4+\omega^{2}_{\rho}$.

According to Sec.\ \ref{gd_intro} we can rewrite Eq.\ (\ref{gd_azymuthal}) by using the Green's function of the two-dimensional isotropic harmonic oscillator $G^{\rm (2d)}_{\rm har}$ is given by \cite{Bakhrakh1972}
\begin{eqnarray}
G^{(\rm 2d)}_{\rm har}(\rho,\rho',\phi-\phi^\prime;E)=-\frac{1}{2 \pi \rho \rho^\prime}\sum^{\infty}_{m=-\infty}\frac{\Gamma \left( \frac{|m|+1-E}{2} \right)}{\Gamma \left( \frac{|m|+1}{2} \right)}M_{E/2,|m|/2}(\rho^{2}_{<})W_{E/2,|m|/2}(\rho^{2}_{>})e^{im(\phi-\phi^{\prime})},
\end{eqnarray}
where $M_{a,b}$ and $W_{a,b}$ are Whittaker functions \cite{Abramowitz1972} and $\rho_{<(>)}$ are, respectively, the lesser and the greater of $\rho$ and $\rho^\prime$. This gives an alternative expression to Eq.\ (\ref{gd_azymuthal}), namely
\begin{eqnarray}
G_\text{gd}(\bfmath{r},\bfmath{r}';E)=\sum^{\infty}_{n_z=0}G^{(\rm 2d)}_{\rm har}(\rho,\rho',\phi-\phi^\prime;E-\omega_z(n_z+\frac{1}{2}))\phi_{n_z}(z)\phi_{n_z}(z').
\end{eqnarray}
\section{Dynamic polarizabilities of giant-dipole states}
As a first application of the giant-dipole Green's function, we consider the dynamic polarizability $\alpha(\omega)$ that connects the system's induced dipole moment $d_{\rm ind}$ as a response to an external time-dependent driving field with frequency $\omega$, i.e.\ $\bfmath{d}_{\rm ind} = \underline{\bfmath{\alpha}}(\omega) \bfmath{E}(\omega)$. As the dynamic polarizability tensor $\underline{\bfmath{\alpha}}(\omega)$ is represented by a symmetric $3 \times 3$ matrix, the induced dipole moment and the corresponding energy shift is determined by three independent quantities according to the principal axis theorem. In case of the giant-dipole system the dynamic polarizability depends on the entries $\alpha_{ii},\ i=x,y,z$, which are the polarizabilities parallel or perpendicular to the magnetic field direction. In particular, it can be shown that $\alpha^{(n)}_{ii}$ is related to the system's Green's function via the relation \cite{Davydkin1971}
\begin{eqnarray}
\alpha^{(nmn_z)}_{ii}(\omega)&=&-\sum_{\sigma = \pm 1}\langle \Psi_{nmn_z} |x_iG_{\rm gd}(\omega_{nmn_z} \pm \sigma \omega)x_i|\Psi_{nmn_z}\rangle \nonumber\\
&=&- \sum_{\sigma = \pm 1}\int {\bf d} \bfmath{r} \, {\bf d} {\bfmath{r}}^\prime \, \Psi^{*}_{nmn_z}(\bfmath{r}) x_i G_{\rm gd} \left( \bfmath{r}, \bfmath{r}^\prime; \omega_{nmn_z} + \sigma \omega \right) {x_i}^\prime \Psi_{nmn_z}(\bfmath{ r}^\prime).
\end{eqnarray}
where $\Psi_{nmn_z}$ and $\omega_{nmn_z}$ denote the giant-dipole eigenfunctions and eigenenergies, respectively. Expressing the confluent hypergeometric functions as generalized Laguerre polynomials \cite{Abramowitz1972} and using three term recursion formulas we find the polarizabilities parallel and perpendicular to $\bfmath{B}$
\begin{equation}
\alpha_n^\parallel(\omega) = \frac 1 \mu \frac{1}{\omega_z^2 - \omega^2},\ \
\alpha_n^\perp(\omega) = \frac{1}{2 \mu \Omega} \left(\frac{\Omega + \omega_c/2}{(\Omega + \frac 1 2 \omega_c)^2 - \omega^2} + \frac{\Omega - \omega_c/2}{(\Omega - \frac 1 2 \omega_c)^2 - \omega^2}\right) \, \text .\label{pol_dyn}
\end{equation}
For finite atomic lifetimes $\tau$, i.e.\ a decay rate $\Gamma=1/\tau$ for the giant-dipole states, one has to replace $\omega^2 \rightarrow \omega^2+i \omega \Gamma$. The imaginary part of the complex polarizability provides information about the dispersion properties. However, in this analysis we have taken the limit $\Gamma \rightarrow 0$.
\begin{figure}
\centering
\begin{minipage}[t]{0.49\textwidth}
\includegraphics[width=\textwidth]{./fig_1a.pdf}
\end{minipage}
\begin{minipage}[t]{0.49\textwidth}
\includegraphics[width=\textwidth]{./fig_1b.pdf}
\end{minipage}
\caption{(a) Frequency dependent scaled dynamical polarizabilities $\mu  \omega^{2}_{z}\alpha^{||}_{n}(\omega)$ and $\mu \Omega \omega_c \alpha^{\perp}_{n}(\omega)$ as functions of $\omega/\omega_z$ and $\omega/\omega_c$, respectively. The black line near $\omega/\omega_{c(z)}=1$ indicates the divergence of the polarizabilities. (b) Magnified plot for the interval $\omega/\omega_c \in [0,10^{-3}]$ for $\mu \Omega \omega_c \alpha^{\perp}_{n}(\omega)$. The divergence around $\omega/\omega_c=5 \cdot 10^{-4}$ is clearly visible.}\label{dyn_pol}
\end{figure}

The polarizability parallel to the magnetic field is described by a Drude-Lorentz model, where a single valence electron is considered to be confined in a one-dimensional harmonic potential. In the limiting case of $\omega_c \to 0$, the same result is obtained for the polarizability in the perpendicular $\bfmath{B}$-field direction. The polarizabilities do not depend on the specific state, a result which is a typical feature of harmonically confined particles and also obtained in the Drude-model. This result can be extended beyond the azimuthal approximation, i.e.\ for arbitrary electric and magnetic field strengths. The explicit results of this analysis is presented in Appendix B. In Fig.\ \ref{dyn_pol} we show the frequency dependence of the scaled dynamical polarizabilities $\mu \omega^{2}_{z}\alpha^{||}_{n}(\omega)$ and $\mu \Omega \omega_{c}\alpha^{\perp}_{n}(\omega)$, respectively. Due to their dependence on $\omega_z, \omega_c$ and $\Omega$ the polarizabilities are implicit functions of the applied magnetic field strength and the pseudomomentum $\bfmath{K}$. In case of the parallel to $\bfmath{B}$ polarizability we clearly see the resonances for $\omega/\omega_z=1$. For the polarizability perpendicular to $\bfmath{B}$ we find four resonances at $\omega_{1,2}/\omega_c=(\Omega/\omega_c \pm 1/2)$, with give $\omega_1 \approx 1.00054$ and $\omega_{2} \approx 5.4 \times 10^{-4}$, respectively.

Note that the quantity $\alpha(\omega=0)$ commonly denotes the system's static polarizability which relates the induced dipole moment to a static electric field. However, for giant-dipole atoms it is not reasonable to define such a quantity since an additional external electric field simply redefines the system to a new giant-dipole species.
\section{The electronic problem of ultra-long range molecules - Green's function ansatz \label{gf_div}}
The general problem of ultra-long range molecules consists of atomic species of highly excited electronic character possessing a single excited valence electron with a very low kinetic energy. The first molecular species under consideration were based on alkali systems with only one valence electron in the outermost electronic orbital \cite{Greene2001,Bendkowsky2009,Sassmannshausen2015}. However, in recent years molecules consisting of divalent constituents such as strontium \cite{DeSalvo2015,Camargo2016} have been realized as well.

The standard ansatz for the electronic Hamiltonian in adiabatic approximation is given by
\begin{eqnarray}
H_\text{el}=H_0+V_\text{en}(\bfmath{r},\bfmath{R})\ \ \text{with}\ \ H_{0}=T+V(\bfmath{ r})\label{elecham}.
\end{eqnarray}
Here, the coordinates $\bfmath{r}=(x,y,z)$ and $\bfmath{R}=(x_n,y_n,z_n)$ denote the spatial positions of the electron and the neutral ground state atom (neutral pertuber for short), respectively. $H_0$ is the Hamiltonian describing the Rydberg electron in its ionic core potential $V(\bfmath{r})$ and $V_{\rm en}(\bfmath{r},\bfmath{R})$ is the interaction between the Rydberg electron and the neutral perturber. Finally, we model the interatomic potential for the low-energy scattering between the Rydberg electron and the neutral perturber by the Fermi-pseudopotential \cite{Fermi1934,Omont1977}
\begin{eqnarray}
V_{\rm{en}}(\bfmath{r,R})=2\pi A_s[k]\delta(\bfmath{r}-\bfmath{R}).\label{fermi}
\end{eqnarray}
In Eq.\ (\ref{fermi}), the quantity $A_s[k] = -\tan ( \delta_0( k ))/ k$ denotes the energy-dependent triplet $s$-wave scattering length which is evaluated from the corresponding phase shifts $\delta_0(k)$ \cite{Fabrikant2002}. The kinetic energy $E_{\rm{kin}} = k^2/2$ of the valence electron at the collision point with the neutral perturber can be taken in a semiclassical approximation $k^2/2 =E_{\rm{el}}-V(\bfmath{r})$.

Over time, two approaches have been established to deduce the electronic eigenenergies $\varepsilon(\bfmath{R})$ of $H_{\rm el}$, which parametically depend on the perturber position and serve as an input for its dynamics. First, the electronic Hamiltonian $H_{\rm{el}}$ can be diagonalized using an arbitrary single particle basis $\{ | \phi_i \rangle \}_{i=1,...,N}$. Depending on the specific problem, quantum defect rubidium wave function \cite{Greene2001}, giant-dipole states \cite{Kurz12} and hybridized hydrogen states \cite{Eiles2016} have been applied. In an alternative approach, the Green's function $G_{0}(\bfmath{r},\bfmath{r}^\prime)$ of the unperturbed electronic problem $H_0$ satisfying
\begin{eqnarray}
(E-H_0)G_{0}(\bfmath{r},\bfmath{r}';E)=\delta(\bfmath{r}-\bfmath{r}')\label{gf_formula}
\end{eqnarray}
is used to formulate an implicit problem for the system's energy $E$. In particular, in Ref.\ \cite{Fey2015} it has been shown that this approach leads to the following self-consistent integral equation for the perturbed wave function and energies $E$, respectively
\begin{eqnarray}
\psi({\bf r})=-\int d \bfmath{r}' G_{0}(\bfmath{r},\bfmath{r}';E)V_{\rm{en}}(\bfmath{r}',\bfmath{R})\psi(\bfmath{ r}')\ \Rightarrow \ 1 - 2\pi A_s[k]G_{0}(\bfmath{R},\bfmath{R};E)=0.\label{wavefunc}
\end{eqnarray}
In previous works the Green's function approach has been applied to systems where the highly excited Rydberg electron is attached to an ionic core potential $V(r)=-1/r+V_{\rm{qd}}(r)$, where $V_{\rm{qd}}(r)$ is a short ranged quantum defect potential which takes into account deviations of the Coulomb potential due to finite size effects of the core of non-hydrogenic atomic species. It is known that for such systems the Green's function can be written as $G_{0}(\bfmath{r},\bfmath{r}';E)=G_{\rm{C}}(\bfmath{r},\bfmath{r}';E)+G_{\rm{qd}}(\bfmath{r},\bfmath{r}';E)$ whereby $G_{\rm{C}}$ denotes the Coulomb Green's function \cite{Fabrikant2002}. This Green's function diverges for $\bfmath{r} \rightarrow \bfmath{r}'$, in particular $G_{\rm{C}}(\bfmath{r},\bfmath{r}';E)\rightarrow -1/2\pi|\bfmath{r}-\bfmath{r}'|$ \cite{Fabrikant2002}. However, it can be shown that this kind of divergent behavior is not only restricted to Rydberg systems interacting via a Coulomb-like potential, but to all potentials $V(\bfmath{r})$ that are less divergent than a bare Coulomb potential.

To verify this prediction we introduce the relative position vector $\boldsymbol{\xi}=\bfmath{r}-\bfmath{r}'$, which gives $\Delta_{\bfmath{r}} \rightarrow \Delta_{\bfmath{\xi}}\equiv\partial^{2}_{\xi_1}+\partial^{2}_{\xi_2}+\partial^{2}_{\xi_3}$, $V(\bfmath{r})=V(\bfmath{\xi};\bfmath{r}^\prime)$ and $G_{0}(\bfmath{r},\bfmath{r}';E)=G_{0}(\bfmath{\xi},\bfmath{r}';E)$. For $|\boldsymbol{\xi}|=\xi \rightarrow 0$ we assume that $\xi^2 V \rightarrow 0$. In this case the kinetic energy term is dominant compared to the interaction potential $V$ in the Hamiltonian $H_0$ and $E$. From Eq.~(\ref{gf_formula}) we obtain
\begin{eqnarray}
\left( E-\frac{\bfmath{p}^2}{2}-V(\boldsymbol{\xi};\bfmath{r}^\prime) \right)G_{0}(\boldsymbol{\xi},\bfmath{r}^\prime;E)=\delta(\boldsymbol{\xi})\ \underset{\xi \rightarrow 0}{\Rightarrow} \ \frac{1}{2}\Delta_{\bfmath{\xi}} G_{0}(\boldsymbol{\xi},\bfmath{r}^\prime;E)=\delta(\boldsymbol{\xi}).\label{proof_1}
\end{eqnarray}
This problem is related to the Green's function of a free particle. $G_0$ is now a function of $\xi$ only, i.e. $G_{0}(\xi)$, and closely related to the Green's function of the three-dimensional Laplace operator which is given by $-1/(4\pi\xi)$. Thus, from Eq.\ (\ref{proof_1}) we easily derive
\begin{eqnarray}
G_{0}(\xi) \approx -\frac{1}{2 \pi \xi} = -\frac{1}{2 \pi |\bfmath{r}-\bfmath{r}^\prime|}.\label{gf_limit_behavior}
\end{eqnarray}
Obviously, this is the functional behavior known from the Coulomb problem \cite{Fabrikant2002} as it has been predicted. We note that the $1/\xi$ dependence is not a direct result of the electron potential $V(\bfmath{r})$, but stems from the kinetic energy term. As the Green's function approach is equivalent to the exact diagonalization of the electronic Hamiltonian $H_0$ in Eq.\ (\ref{elecham}), Eqs.\ (\ref{wavefunc}) and (\ref{gf_limit_behavior}) indicate a non-convergent behavior in the case a Fermi-pseudopotential is used to model the electron-perturber interaction for any kind of adiabatic electron system.

For this reason, in order to study such systems, alternative approaches such as an exact diagonalization scheme using a regularized Fermi-pseudopotential as well as Green's function methods have been considered \cite{Fey2015,Fabrikant2002}. In the case that the regularized Green's function method is employed, one has to determine the energy $E$ from the self-consistent equation
\begin{eqnarray}
1 - 2\pi A_s[k]G^{\rm (reg)}_{0}(\bfmath{R},\bfmath{R};E)=0.\label{Greg_E}
\end{eqnarray}
The function $G^{\rm (reg)}_{0}$ denotes the systems regularized Green's function. Throughout the literature a number of regularization procedures have been discussed \cite{Fey2015, Fabrikant2002}. Here we follow the procedure applied by Fabrikant {\it et al.} \cite{Fabrikant2002} and define
\begin{eqnarray}
G^{\rm (reg)}_{0}(\bfmath{r};\bfmath{r}^{\prime};E)\equiv G_{0}(\bfmath{r};\bfmath{r}^{\prime};E)+\frac{1}{2 \pi |\bfmath{r}-\bfmath{r}^\prime|}\ \ \textrm{with}\ \  G^{\rm (reg)}_{0}(\bfmath{R},\bfmath{R};E) \equiv \lim_{\bfmath{r} \rightarrow \bfmath{R}} G^{\rm (reg)}_{0}(\bfmath{r};\bfmath{R};E),
\end{eqnarray}
which means that one simply subtracts the divergent term from $G_0$. This definition slightly differs from the definition in Ref.\ \cite{Fey2015}, but gives identical results in the case of $\bfmath{r}=\bfmath{r}^\prime= \bfmath{R}$ which is required for the determination of $E$ via Eq.\ (\ref{Greg_E}) (see Appendix C).
\section{Ultra-long range giant-dipole molecules}
In this section we consider a highly excited hydrogen atom which is prepared in a giant-dipole state interacting with a neutral ground state $^{87}$Rb atom (see Sec.\ \ref{gf_div}). For this particular system, the molecular Hamiltonian reads
\begin{eqnarray}
H=\frac{\bfmath{P}^2}{2m_n}+H_{\textrm{gd}}+V_{\textrm{gd,n}}(\bfmath{r},\bfmath{R}),\label{gd_ham}
\end{eqnarray}
where $V_{\textrm{gd,n}}$ denotes the interaction of the neutral perturber atom with the giant-dipole Rydberg electron. For deeply bound states in the outer potential well the electron possesses a low kinetic energy. For this reason, it is legitimate to model this interaction potential via a Fermi-type $s$-wave pseudopotential as it has been discussed in Sec.\ \ref{gf_div}. Analogous to previous works we consider the electron-perturber interaction to be purely determined by the triplet scattering channel \cite{Kurz12}.
In order to solve the eigenvalue problem associated with Hamiltonian (\ref{gd_ham}) an adiabatic ansatz is employed for the electronic and perturber degrees of freedom, leading to the electronic Hamiltonian $H_{\textrm{el}}=H_{\textrm{gd}}+V_{\textrm{gd,n}}$ which parametically depends on the perturber position. In order to obtain the PES, an exact diagonalization scheme using the eigenstates of $H_{\textrm{gd}}$ has been adopted in a previous analysis \cite{Kurz12}. As the giant-dipole Hamiltonian $H_{\textrm{gd}}$ describes an magnetically-field dressed electron in an external three dimensional harmonic potential, its eigenfunctions are identical to those of Eq.\ (\ref{harmonic_oscillator_prop}).

However, in this work we employ the Green's function approach as it was presented in Sec.~\ref{gd_intro} and 3, respectively. In this case, we have to determine the energies $E$ which are the roots of the equation
\begin{eqnarray}
1-2\pi A_s[k]G_{\textrm{gd}}(\bfmath{R},\bfmath{R};E)=0, \label{eq_gf_E}
\end{eqnarray}
which has to be determined via a numerical root-finding routine. As discussed in Section \ref{gf_div} the Green's function $G_{\rm gd}$ of the giant-dipole system for arbitrary field strengths would be given by Eq.\ (\ref{gdgf}), i.e. $G_{\rm gd}=\mathcal{G}_{\rm gd}$.

As it has been discussed in Sec.~\ref{gf_div} the bare giant-dipole Green's functions \eqref{gd_azymuthal} possess a singularity for $\bfmath{r}=\bfmath{r}'$. This behavior can easily be verified for $\bfmath{r}'=0$:
\begin{eqnarray}
\lim\limits_{\bfmath{r}\rightarrow 0}G_{\rm gd}(\bfmath{r},0,E)\sim \lim\limits_{z\rightarrow 0} \sum^{\infty}_{n=0} G^{\rm (1d)}_{\rm har}(z,0,E-\varepsilon_{n0}) \sim -\sum^{\infty}_{n_z=0}\phi^{2}_{2n_z}(0)\sum^{\infty}_{n=0}\frac{1}{n}\rightarrow -\infty.
\end{eqnarray}
According to the analysis provided in Sec.~\ref{gf_div} this divergence is supposed to behave as $-1/(2\pi r)$. To analyze this in more detail, we introduce the truncated giant-dipole Green's function $G^{\rm (tr)}_{\rm gd}$ via its spectral representation, i.e.
\begin{equation}
G^{\rm (tr)}_\text{gd}(\bfmath{r}, \bfmath{r}^\prime; E;\bfmath{N}) = \frac{1}{2\pi}\sum_{n=0}^N \sum_{m=-M_{\rm min} }^{M_{\rm max}} \sum_{n_z = 0}^{N_z}\frac{\phi_{n_z}(z)\phi_{n_z}(z') \, R_{nm}(\rho) R_{nm}(\rho') \, \mathrm{e}^{i m \, (\phi - \phi')}}{E - \varepsilon_{nm} - \varepsilon_{n_z}}, \label{gf_trunc}
\end{equation}
with $\bfmath{N}=\{ M_{\rm min}, M_{\rm max}, N_z,N \}$. In case we restrict the summations up to a set of finite $\bfmath{N}$ we can analyze different levels of approximations of the exact function $G_{\rm gd}$. In particular, using the truncated Green's function Eq.\ (\ref{gf_trunc}) we can analyze the divergent behavior in more detail as for this approximation the divergence for $\bfmath{r}=\bfmath{r}^\prime$ is cut off.

\begin{figure}
\centering
\begin{minipage}[t]{0.455\textwidth}
\includegraphics[width=\textwidth]{./fig_2a.pdf}
\end{minipage}
\begin{minipage}[t]{0.535\textwidth}
\includegraphics[width=\textwidth]{./fig_2b.pdf}
\end{minipage}
\caption{Double logarithmic plot for the asymptotic behavior of the truncated giant-dipole Green's function $G^{\rm (tr)}_{\rm gd}(\bfmath{r},0; E)$ given by Eq.\ (\ref{gf_trunc_2}) for $N=0,200,...,1000$ in the limit $\bfmath{r} \to 0$. Fig.\ (a) shows $G^{\rm (tr)}_{\rm gd}$ as a function of $z$ for $\rho=0$, where Fig.\ (b) presents the same function dependent on $\rho$ for $z=0$. In both figures the dashed black line indicates the limiting case $-1/(2\pi r)$. The inset in figure (b) shows the oscillations of $-2 \pi \rho G^{\rm (tr)}_{\rm gd}(\rho,0; E)$ for $N=200$ and $N=1000$, respectively.}
\label{fig:GreenDivergence3D}
\end{figure}
A detailed analysis of this issue is presented in Fig.\ \ref{fig:GreenDivergence3D}(a,b), where we present the functional behavior of the truncated Green's function for different levels of approximations. In particular, we have chosen the spatial coordinates to be $\bfmath{r}^\prime=0,\ \phi=0$ and take the limit $N_{z}\rightarrow \infty$. In this case the summation over $m$ is restricted to $m=0$ and we obtain
\begin{eqnarray}
G^{\rm (tr)}_\text{gd}(\rho,z,0; E)= \frac{1}{2 \pi}\sum^{N}_{n=0}G^{\rm (1d)}_{\rm har}(z,0;E-\varepsilon_{n0})\label{gf_trunc_2}R_{n0}(\rho)R_{n0}(0).
\end{eqnarray}
In Fig.\ \ref{fig:GreenDivergence3D}(a), we depict a double logarithmic plot for $G^{\rm (tr)}_{\rm gd}(0,z,0;E=0)$ for a number of different $N=0,200,...,1000$ as a function of $z$. In this representation the limiting curve $-1/(2 \pi z)$ is simply a straight line (black dashed curve) which makes it very convenient for comparison with the calculated data. Obviously, for increasing $N$ the Green's function approaches the divergent behavior as the numerical curves more and more approach the black dashed curve in the vicinity of $z \rightarrow 0$.

In Fig.\ \ref{fig:GreenDivergence3D}(b) we present the same analysis for $z=0$ following $\rho \rightarrow 0$, i.e.\ $G^{\rm (tr)}_{\rm gd}(\rho,0,0;E=0)$. Here, the situation is more complicated as the Green's function possess as strongly oscillating structure as a function of $\rho$. For increasing $N$ the numerical curves decrease approaching the $-1/2\pi \rho$ limit (black dashed curve) for $\rho \rightarrow 0$. However, the oscillations move inwards as well, making a precise analysis of the $N \rightarrow \infty$ behavior quite challenging. To get more insight we have analyzed the oscillatory behavior separately which is presented in the inset of Fig.\ \ref{fig:GreenDivergence3D}(b). Here, we show the oscillations of $N=200$ and $N=1000$ numerical curve. Obviously, for larger $N$ the frequency of the oscillating has increased while its amplitude decreases for sufficiently large $\rho$. In Appendix D we present a more systematic analysis where we show that the oscillatory behavior is well approximated by $J_{1}(2 \sqrt{N}\rho)$ where $J_{1}(x)$ denotes the $J_1$-Bessel function \cite{Abramowitz1972}. As $J_{1}(2 \sqrt{N}\rho) \rightarrow 0$ for  $N \rightarrow \infty$ we see that the oscillations vanish in the case of the full giant-dipole Green's function possessing a $-1/(2\pi r)$ behavior.

Although the full Green's function possesses a divergent behavior, the truncated representation Eq.\ (\ref{gf_trunc}) can still be used to calculate approximate PES when it is inserted into Eq.\ (\ref{eq_gf_E}). As it has been discussed in \cite{Fey2015}, this is equivalent to the exact diagonalization approach where one uses a finite set of giant-dipole basis functions $\{ \Psi_{nmn_z}(\bfmath{r}) \}$ with $n=0,...,N$, $m=-M_{\rm min},...,M_{\rm max}$, $n_{z}=0,...,N_z$ to diagonalize the electronic Hamiltonian $H_{\rm el}$ given in Eq.~(\ref{elecham}). In the present case, we have calculated the ground state PES both via exact diagonalization and, in comparison, via the truncated Green's function approach using the same basis functions. The result of this analysis is presented in Figs.\ \ref{fig:GreenDiagCompareGD}(a,b), respectively.

In particular, Fig.\ \ref{fig:GreenDiagCompareGD}(a) shows the ground state PES for $z_n=0$ as a function of $\rho_n$ for different levels of approximations. The blue solid line labeled as ``pert.\ theory (exact diag.)'' shows the pure first order perturbation theory result for the giant-dipole state $\Psi_{000}(\bfmath{r})$, i.e.\
\begin{eqnarray}
E(\rho_n,z_n)=\varepsilon_{00}+\frac{\omega_z}{2}+A_{s}[k]\phi^{2}_{0}(z_n)R^{2}_{00}(\rho_n).
\end{eqnarray}
The same result is easily derived from Eq.\ (\ref{eq_gf_E}) for $M_{\rm min}=N_{z}=M_{\rm max}=N=0$, which is indicated by the blue dots and labeled as ``pert.\ theory (Gf-approach)''. In the same figure the solid red curve shows the exact diagonalization results  for a basis set of $M_{\rm min}=60,\ N_{z}=30$ and $M_{\rm max}=0,\ N=0$, the same parameters which had been employed for the truncated Green's function ansatz. The results of those calculation are indicated by red dots and labeled as ``finite basis (exact diag.)'', in comparison to the results of the exact diagonalization approach which is labeled as ``finite basis (Gf-approach)''. We see that both results are identical as expected from Ref.\ \cite{Fey2015}. In comparison with the perturbation theory results, the corresponding potential curves lie energetically below the latter.

Finally, Fig.\ \ref{fig:GreenDiagCompareGD}(a) shows a result provided by the Green's function approach which cannot be obtained within the exact diagonalization scheme. In particular, in the limit $N_z \rightarrow \infty$ we obtain from Eq.\ (\ref{gf_trunc})
\begin{eqnarray}
G^{\rm (tr)}_{\rm gd}(\bfmath{R},\bfmath{R};E;\bfmath{N})=\frac{1}{2 \pi}\sum^{M_{\rm max}}_{m=-M_{\rm min}}G^{\rm (1d)}_{\rm har}(z_n,z_n;E-\varepsilon_{0m})R^{2}_{0m}(\rho_n), \label{gf_trunc_3}
\end{eqnarray}
which is inserted into Eq.\ (\ref{eq_gf_E}). Because the basis set used for exact diagonalization is necessarily finite, the result provided by Eq.\ (\ref{gf_trunc_3}) can only be analyzed in the Green's function approach. In Fig.~\ref{fig:GreenDiagCompareGD}(a), the result of this particular analysis is indicated by black dots and labeled as ``Gf-approach ($N_z \rightarrow \infty$)''. One observes that this PES lies energetically below the other curves with a maximum deviation of around $11\%$ for $\rho=0$ from the finite basis set analysis. This deviation is related to the inclusion of higher $N_z$ terms, but remains finite as the summations over the $n$ and $m$ quantum numbers are truncated at $N=0$ and $(M_{\rm min},M_{\rm max})$, respectively. Performing the unlimited summations over $n$ and $m$ would lead to a divergent shift of the potential curve as in this limit we obtain the full Green's function Eq.\ (\ref{gdgf}) (see Sec.\ \ref{gf_div}).

In Fig.\ \ref{fig:GreenDiagCompareGD}(b), we present the analogous study for the ground state PES for $\rho_n=0$ as a function of $z_n$ for the same parameters as in Fig.\ \ref{fig:GreenDiagCompareGD}(a). Because of the $z \rightarrow -z$ parity symmetry of the electronic Hamilton $H_{\rm el}$, the potential curve is symmetric in $z_n$. The potential curve along the $z_n$-direction forms a Gaussian-like well. For this reason, the three-dimensional potential surface forms a single well with a minimum at $\rho_n=z_n=0$. As we have seen in the calculations for the graphs in Fig.~\ref{fig:GreenDiagCompareGD}(a) the potential curves energetically decrease for increasing basis set. Again, the limiting case $N_{z} \rightarrow \infty$ is indicated by black dots. The deviations of these calculations with respect to the finite basis set results (solid red line) are again in the range of $11\%$.
\begin{figure}
\centering
\begin{minipage}[t]{0.495\textwidth}
\includegraphics[width=\textwidth]{./fig_3a.pdf}
\end{minipage}
\begin{minipage}[t]{0.495\textwidth}
\includegraphics[width=\textwidth]{./fig_3b.pdf}
\end{minipage}
\caption{Ground state PES of a giant-dipole ultra-long range molecule with a rubidium perturber for $B = 2.35 \, \text T$, $K = 1.0$. Figure (a)((b)) show potential curves for $z_n=0$ ($\rho_n=0$). Both figures present results for different approaches: blue curve - first order perturbation theory, blue dots - perturbative Green's function, orange curve - exact diagonalization with $N_{z}=30$, $M_{\rm min} = 60$, $N=M_{\rm max}=0$, red dots - truncated Green's function approach Eq.\ (\ref{gf_trunc_2}) with the same set of functions, black dots - Green's function approach with $N_z \rightarrow \infty$ (Eq.~ (\ref{gf_trunc_3})).}\label{fig:GreenDiagCompareGD}
\end{figure}

Finally, we derive the regularized giant-dipole Green's function $G^{\rm (reg)}_{\rm gd}$ as it was discussed in Sec.\ \ref{gf_div}. For this derivation we use a the cylindrical representation of the $1/|\bfmath{r}-\bfmath{r}^\prime|$ term which is given by \cite{Abramowitz1972}
\begin{eqnarray}
\frac{1}{|\bfmath{r}-\bfmath{r}^{\prime}|}=\frac{1}{\pi \sqrt{\rho \rho^{\prime}}}\sum^{\infty}_{m=-\infty}e^{im(\phi-\phi^\prime)}Q_{m-1/2}(\chi)
\end{eqnarray}
in terms of the Legendre-Q functions $Q_{m-1/2}(x)$ \cite{Abramowitz1972} with $\chi=(\rho^2+\rho^{\prime 2}+(z-z^{\prime})^2)/(2\rho \rho^{\prime})$. Using this relation we then easily derive the following analytic expression for the regularized giant-dipole Green's function at $\bfmath{r}=\bfmath{r}^\prime$
\begin{eqnarray}
G^{\rm (reg)}_{\rm gd}(\bfmath{r},\bfmath{r};E)=\frac{1}{2 \pi}\sum^{\infty}_{m=-\infty}\left( \sum^{\infty}_{n=0}G^{\rm (1d)}_{\text{har}}(z,z;E-\varepsilon_{nm})R^{2}_{nm}(\rho)
+\frac{1}{\pi \rho}Q_{m-1/2}(1)\right).
\end{eqnarray}
\section{Conclusion}
In this work, we have presented a Green's function analysis of giant-dipole systems. In a first step, we have derived the exact Green's function representation of several magnetically field-dressed systems. In particular, we calculated the Green's function for giant-dipole systems for arbitrary field parameters as well as in the two-dimensional harmonic approximation. Using these results of the Green's function analysis, we have calculated the dynamical polarizability. This property is of special interest as it characterizes the systems response to an external time-dependent driving field, which provides insights into the dynamics of interacting atom-laser systems. Furthermore, we have applied the Green's function approach to deduce electronic properties for a specific species of ultra-long range molecules, so-called diatomic giant-dipole molecules. In particular, we have calculated the adiabatic potential energy surfaces which we compare to results that have been obtained within an exact diagonalization approach \cite{Dippel1994,Kurz12}. Beside the specific electronic potential curves of these particular molecular species, we have also shown a general divergent behavior in the potential energy calculation of ultra-long range molecules within the Fermi-pseudopotential approach, which indicates a general insufficiency of this specific ansatz. For this reason, we derived the regularized Green's function of the giant-dipole problem, as this approach has proven to avoid the insufficiencies of the pseudopotential approach in previous works \cite{Fabrikant2002,Fey2015}.

Although our study has provided novel information on the Green's function approach of giant-dipole systems, there are still open questions. For instance, a derivation of a closed representation of the giant-dipole Green's function would be useful for the implementation of numerically robust routines. Furthermore, in this work we have only considered a pure triplet pseudopotential for the electron-perturber interaction. Thus, the question arises how the inclusion of both the singlet scattering channel as well a hyperfine structure of the perturber atoms change the PES. Furthermore, the inclusion of higher-order terms in the Fermi-pseudopotential ansatz has shown to provide additional features for both single- and multi-perturber ultra-long range molecules \cite{Hamilton2002,Fey2016}. In particular, in Ref.\ \cite{Fey2016} it was shown that the Green's function ansatz is quite suitable to study multi-perturber systems, which is an open question for molecular giant-dipole systems as well. For this reason, molecular giant-dipole systems provide a plethora of interesting problems which can be addressed in future studies.
\section*{Acknowledgments}
T.\ S.\ acknowledges financial support from ``Evangelisches Studienwerk Villigst''.
\section*{Appendix A (Derivation of the Green's function expression)}
We show that Eq.\ (\ref{GFtotalexp}) solves Eq.\ (\ref{GFtotal}) by performing the corresponding calculation. We get
\begin{eqnarray}
&\ &(E-H)G(\bfmath{r},\bfmath{r}';E) \nonumber\\
&=&\sum_{k_2...k_N}(E-H)G_1(\bfmath{r},\bfmath{r}^{\prime}_{1};E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}})\prod^{N}_{n=2}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n})\ \ , \ \ \ \ \ \left( H=\sum^{N}_{i=1}h_i \right) \nonumber\\
&=&\sum_{k_2...k_N}[ E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}}-h_1+\sum^{N}_{i=2}(\varepsilon^{(i)}_{k_{i}}-h_i) ] G_1(\bfmath{r},\bfmath{r}^{\prime}_{1};E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}})\prod^{N}_{n=2}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n}) \nonumber\\
&=&\sum_{k_2...k_N}(E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}}-h_1)G_1(\bfmath{r},\bfmath{r}^{\prime}_{1};E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}})\prod^{N}_{n=2}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n}) \nonumber\\
&\ &+\sum_{k_2...k_N}\sum^{N}_{i=2}(\varepsilon^{(i)}_{k_{i}}-h_i))G_1(\bfmath{r},\bfmath{r}^{\prime}_{1};E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}})\prod^{N}_{n=2}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n}). \label{derivation}
\end{eqnarray}
Using Eqs.\ ((\ref{eigen}),(\ref{delta})) we get
\begin{eqnarray*}
(E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}}-h_1)G_1(\bfmath{r},\bfmath{r}^{\prime}_{1};E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}})=\delta(\bfmath{r}_1-\bfmath{r}^{\prime}_{1}),\ \ \textrm{and}\ \ h_i \phi^{(i)}_{k_{i}}(\bfmath{r}_i) = \varepsilon^{(i)}_{k_i}\phi^{(i)}_{k_{i}}(\bfmath{r}_{i}).
\end{eqnarray*}
For this reason the second term in Eq.\ (\ref{derivation}) cancels and the first term can be rewritten to
\begin{eqnarray*}
 \sum_{k_2...k_N}\delta(\bfmath{r}_1-\bfmath{r}^{\prime}_{1})\prod^{N}_{n=2}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n})=\prod^{N}_{n=1}\delta(\bfmath{r}_n-\bfmath{r}^{\prime}_{n})=\delta(\bfmath{r}-\bfmath{r}^{\prime}),
\end{eqnarray*}
which is the expression that is to be derived.\\
\\
Alternatively, one can derive Eq.\ (\ref{GFtotalexp}) by using Eq.\ (\ref{GFeigenexp}). Here, one has to consider the fact that the total wave function $\psi$ is a product state of the $\phi$-functions and that the total eigenenergies are sums of the single Hamiltonian eigenenergies $\varepsilon^{(i)}_{k}$. We get
\begin{eqnarray*}
G(\bfmath{r},\bfmath{r}^{\prime};E)&=&\sum_{k_1...k_N}\frac{\prod^{N}_{n=1}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n})}{E-\sum^{N}_{i=1}\varepsilon^{(i)}_{k_{i}}}\\
&=& \sum_{k_2...k_N}\sum_{k_1}\frac{\phi^{(n)*}_{k_1}(\bfmath{r}^{\prime}_{1}) \phi^{(n)}_{k_1}(\bfmath{r}_1)}{E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}}-\varepsilon_1}\prod^{N}_{n=2}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n})\\
&=&\sum_{k_2...k_N}G_{1}(\bfmath{r},\bfmath{r}^{\prime}_{1};E-\sum^{N}_{i=2}\varepsilon^{(i)}_{k_{i}})\prod^{N}_{n=2}\phi^{(n)*}_{k_{n}}(\bfmath{r}^{\prime}_{n}) \phi^{(n)}_{k_{n}}(\bfmath{r}_{n}).
\end{eqnarray*}
In the final step we used the expansion of the $G_1$ Green's function (see Eq.\ (\ref{GFeigenexp})). Obviously, any Green's function $G_i$ can be used to express the total function $G(\bfmath{r},\bfmath{r}^{\prime};E)$.
\section*{Appendix B (Calculation of giant-dipole polarizabilities)}
The giant-dipole system for arbitrary field strengths can be transformed via a unitary transformation into a systems of three decoupled harmonic oscillators with Green's function $\tilde{G}^{\rm (3d)}_{\rm har}(\bfmath{r},\bfmath{r}^\prime;E)$ \cite{Dippel1994}. Because the eigenfunctions $\Psi_n(\bfmath{r})$ and the giant-dipole Green's function transform according to
\begin{eqnarray}
| \Psi_n \rangle=U|\phi_{n_1}\phi_{n_2}\phi_{n_z}\rangle,\ \ \ \mathcal{G}_{\rm gd}(E)=U\tilde{G}^{\rm (3d)}_{\rm har}(E)U^{\dagger}
\end{eqnarray}
we get
\begin{eqnarray}
\alpha^{(n)}_{ii}(\omega)&=&-\sum_{\sigma = \pm 1}\langle \phi_{n_1}\phi_{n_2}\phi_{n_z} |U^{\dagger}x_iU\tilde{G}^{\rm (3d)}_{\rm har}(\omega_n \pm \sigma \omega)U^{\dagger}x_iU|\phi_{n_1}\phi_{n_2}\phi_{n_z}\rangle
\end{eqnarray}
The transformation of the spatial coordinates under the unitary transformation is explicitly given as \cite{Meyer1988}
\begin{eqnarray*}
 U^{\dagger}xU=x-\beta p_y,\ \ \ U^{\dagger}yU=y-\beta p_x,\ \ \ U^{\dagger}zU=z,
\end{eqnarray*}
which leads to the expressions for the dynamical polarizabilities
\begin{eqnarray*}
\alpha^{(n)}_{xx}(\omega)&=& -\sum_{\sigma = \pm 1}\langle \phi_{n_1}\phi_{n_2}\phi_{n_z} |(x-\beta p_y)\tilde{G}^{\rm (3d)}_{\rm har}(\omega_n \pm \sigma \omega)(x-\beta p_y)|\phi_{n_1}\phi_{n_2}\phi_{n_z}\rangle,\\
\alpha^{(n)}_{yy}(\omega)&=& -\sum_{\sigma = \pm 1}\langle \phi_{n_1}\phi_{n_2}\phi_{n_z} |(y-\beta p_x)\tilde{G}^{\rm (3d)}_{\rm har}(\omega_n \pm \sigma \omega)(y-\beta p_x)|\phi_{n_1}\phi_{n_2}\phi_{n_z}\rangle,\\
\alpha^{(n)}_{zz}(\omega)&=& -\sum_{\sigma = \pm 1}\langle \phi_{n_1}\phi_{n_2}\phi_{n_z} |z\tilde{G}^{\rm (3d)}_{\rm har}(\omega_n \pm \sigma \omega)z|\phi_{n_1}\phi_{n_2}\phi_{n_z}\rangle.
\end{eqnarray*}
Obviously, the result of $\alpha^{(n)}_{zz}(\omega)$ remains unchanged and is given by Eq.\ (\ref{pol_dyn}), i.e.\ $\alpha^{(n)}_{zz}(\omega)=\alpha^{||}_{n}(\omega)$.
\section*{Appendix C (Regularized Green's function)}
Analogous to Ref.\ \cite{Fey2015}, we define the regularized Green's function $\tilde{G}^{\rm (reg)}_{0}$ as
\begin{eqnarray*}
\tilde{G}^{\rm (reg)}_{0}(\bfmath{r},\bfmath{r}^\prime;E) \equiv \frac{\partial}{\partial \xi}\left(\xi G_{0}(\bfmath{r},\bfmath{r}^\prime;E) \right),\ \ \ \xi=|\bfmath{r}-\bfmath{r}^\prime|.
\end{eqnarray*}
With $G_{0}=-1/(2\pi \xi)+G^{\rm (reg)}_{0}$ we obtain
\begin{eqnarray*}
\tilde{G}^{\rm (reg)}_{0}(\bfmath{r},\bfmath{r}^\prime;E)= G_{0}(\bfmath{r},\bfmath{r}^\prime;E)+\frac{1}{2\pi \xi}+\xi \left( \frac{\partial}{\partial \xi}G^{\rm (reg)}_{0}(\bfmath{r},\bfmath{r}^\prime;E) \right).
\end{eqnarray*}
In the limit $\xi \rightarrow 0$ we then find
\begin{eqnarray*}
\tilde{G}^{\rm (reg)}_{0}(\bfmath{r},\bfmath{r};E)=\lim_{\xi \rightarrow 0}\left( G_{0}(\bfmath{r},\bfmath{r}^\prime;E)+\frac{1}{2\pi \xi} \right) + \underbrace{\lim_{\xi \rightarrow 0}\xi \left( \frac{\partial}{\partial \xi}G^{\rm (reg)}_{0}(\bfmath{r},\bfmath{r}^\prime;E) \right)}_{=0}=G^{\rm (reg)}_{0}(\bfmath{r},\bfmath{r};E).
\end{eqnarray*}
\section*{Appendix D (Oscillations of the truncated giant-dipole Green's function)}
The oscillatory part of $G^{(\rm tr)}_{\rm gd}(\rho,0;E=0)$
\begin{eqnarray*}
\textrm{osc}[-2\pi \rho G^{(\rm tr)}_{\rm gd}(\rho,0;E=0)]=2 \pi \rho \left(G^{(\rm tr)}_{\rm gd}(z=\rho,0;E=0)-G^{(\rm tr)}_{\rm gd}(\rho,0,E=0)\right) \approx J_{1}(2 \sqrt{N}\rho)
\end{eqnarray*}
for $\bfmath{r}\rightarrow 0$ is shown in Fig.\ \ref{fig:osci} dependent on the summation limit $N$. For sufficiently large $N$ it is well approximated by a $J_1$-Bessel function \cite{Abramowitz1972}, in particular $J_{1}(2 \sqrt{N}\rho)$. In the case of $N \rightarrow \infty$ it is $J_{1}(2 \sqrt{N}\rho)\rightarrow 0$ for $\rho > 0$. For this reason the oscillations vanish in the limit of the full giant-dipole Green's function.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{./fig_4.pdf}
\caption{Oscillatory part $\textrm{osc}[-2\pi \rho G^{(\rm tr)}_{\rm gd}(\rho,0;E=0)]$ (solid curves) in comparison to a Bessel function approximation given by $J_{1}(2 \sqrt{N}\rho)$ (dashed curves).}
\label{fig:osci}
\end{figure}
\section*{Bibliography}

redpajama_set_name: RedPajamaArXiv


Whether you're a digital nomad, business professional, digital content creator or designer, see how these technologies in the new Yoga S940, Yoga A940, Yoga C730 with AMOLED and Yoga Mouse with Laser Presenter can help you get things done faster and make life easier.
The new Yoga S940 ultra-slim laptop comes with new AI features to automatically filter out ambient noise and blur out distracting backgrounds during video calls.
The minute you look away from your laptop screen, the Yoga S940's AI sensors can detect your absence and auto-lock the display to protect your data from unwanted eyes. It can also detect and alert you when a nosy neighbor is shoulder surfing from your screen. Beyond the added benefits of security and privacy, the Yoga S940 has attention-sensing AI technology, which can help increase productivity too. Look from your main laptop display to a second screen, and it can automatically track your movement and shift open windows and content to a connected external monitor. All these are part of an entire suite of Lenovo Smart Assist AI-enabled features available on the new Yoga S940.
Lenovo Smart Assist includes hands-free PC log-in through its built-in infrared camera and Windows Hello for an added layer of security, and voice-matching personalization to respond to just your voice when speaking to Amazon Alexa or Microsoft Cortana from across the room. The Yoga S940 learns from your usage patterns to optimize its performance. For instance, it'll auto-adjust its power settings for longer-lasting power so that you get every last squeeze out of the up to 15 hours of battery in FHD.
The Yoga S940 is also the first laptop with Contour Glass that wraps around its bezels – reducing the look of its already razor-thin bezels.
And it doesn't stop there. At 1.2kg (2.65lbs) light and 12.2mm (0.48in) thin,the ultra slim and portable Yoga S940 has Windows 10, up to Intel Core, up to 16GB LPDDR3 memory and up to 1TB PCIe SSD storage.
Or, binge-watch shows on its optional 4K HDR display with up to 500 nits peak brightness, Dolby Vision and Dolby Atmos Speaker System with smart audio amplifiers.
It will start at $1,499.99 and is expected to be available at Lenovo and Micro Center starting in May.
For artists and designers, the new Yoga A940 all-in-one desktop is here. Built for digital content creators, the Windows 10 Yoga A940 is a creative canvas with a large 27-inch optional 4K IPS touchscreen display with Dolby Vision. It features a rotating hinge that enables the display to tilt to a 25-degree drafting mode, for flexibility and stability when drawing, sketching or annotating with a digital pen.
The AiO comes with the Lenovo Precision Dial, a controller that's tailor made for designers, photographers and videographers to make precise selections and adjustments to their content.
The Yoga A940 has a Dolby Vision-enabled 4K wide-angle display, boosted by Dolby Atmos Speaker System. A second set of speakers under the rotating display ensures powerful sound even in drafting mode. It also comes with a range of Lenovo Smart Assist AI-enabled features. With up to 8th Generation Intel Core i7 processors, AMD Radeon RX 560 discrete graphics, up to 32GB DDR4 memory and up to 512GB PCIe SSD or 2TB SATA HDD storage, the Yoga A940 could be the ideal centerpiece in any content creator's home or workspace.
The Lenovo Yoga A940 will start at $2,199.99 and is expected to be available starting in March.
Equipped with JBL speakers and Dolby Atmos, the 15-inch Yoga C730 2-in-1 convertible laptop gets an upgrade with up to a 4K AMOLED display. The laptop features 8th Generation Intel Core i7 processors, Windows Ink for digital pen input, Windows Hello for instant log-in, and the smart convenience of Cortana and Alexa.
The 15-inch Lenovo Yoga C730 with AMOLED will start at $1,649.99 and is expected to be available starting in April.
Lenovo streamlined two accessories into one and designed the latest generation Yoga Mouse with Laser Presenter. With just one twist, it converts from a mouse with up to 1,600 DPI for high-precision tracking on nearly any surface into a flat, fully functional laser presenter (and vice versa). It will start at $69.99 and is expected to be available starting in June.
Lenovo and Google unveiled the Lenovo Smart Clock with the Google Assistant, the latest addition to Lenovo's smart living product line. Purpose-made for your bedroom, the Lenovo Smart Clock helps you unwind in the evening, kickstart your day, control your smart home and listen to your favorite music with multi-room audio grouping.
The Lenovo Smart Clock supports voice and touch controls on a 4-inch IPS screen to perform tasks that are regularly carried out before and after sleep. It blends into your bedroom decor, with a full fabric soft-touch cover and a touch screen.
The Google Assistant makes it easier to set up sleep routines, such as dimming the lights and displaying your first calendar event with just a single command, such as "Hey Google, Good Night." Don't worry about your phone though, Lenovo has included a full-size USB port for charging.
In addition, ask the Google Assistant to play some relaxing music or guided meditation, and the Lenovo Smart Clock will play it.
The Lenovo Smart Clock hides a full-range 6W speaker and two passive radiators that can fill up a large bedroom, tuned with Dolby's latest background noise reduction technology. And with multi-room functionality, you can also add the Lenovo Smart Clock to a home speaker group and play music, audiobooks, podcasts and radio across multiple devices in several rooms. The Lenovo Smart Clock also works with Chromecast, so you can stream shows and music on your TV or speakers with just your voice.
Awakening to the Lenovo Smart Clock is designed to be as pleasant as possible with the Google Assistant's gentle wake-up routine. Set your lights to progressively brighten and your alarm to gradually increase in volume starting 30 minutes before your scheduled wake time. Your customized morning alarm will then automatically trigger a morning routine, telling you about today's weather, your personal agenda and reminders, as well as displaying your commute to work. And whatever your choice of sound, music or ringtones to wake up to, just tap on the Lenovo Smart Clock to snooze or dismiss the alarm.
What's more, users can now have another convenient hub from which to control their home. Set the Lenovo Smart Clock when you wake up to automatically turn on your coffee maker in the kitchen or heater in the living room. Or check your front door security camera and the kids' room from your bed. The Lenovo Smart Clock connects with every smart device in your home, supporting over 10,000 products and 1,000 partners and brands.
Lenovo has also included a mute button for those who want the Lenovo Smart Clock to stop listening. When not in use, the screen showcases the time with multiple customizable clock faces.
The Lenovo Smart Clock is available at $79.99 starting spring 2019.


This is the one and only true Common Mode Filter Choke / RF line Isolator for 1-54 MHz on the market. PATENT pending! It is only product with over 7000 Ohms of CMI on 50MHz while maintaining a Resistive part of CMI for more than 50MHz of the frequency spectrum, actually, it has same values all the way up to 85MHz.
Perfect to cure RFI/EMI problems.
It is a cure for noisy locations, reduce the noise floor of your receiver by several S-units instantly. If you have unbalanced antennas such as OCF dipole or your antennas close to your home, adding one CMC154-3K will greatly reduce RF Interference problems on your equipment and 120/220V wiring inside your home. We do not claim "one size fits all" but actually, we design, measure and make such products. Our CMC-154K has huge Resistive values of Common Mode Choking Impedance from 1.8-54MHz, click the chart on the side for actual measurement data.
Unlike other companies, we do not claim unrealistic impedance values or frequency coverage. Most important, the value of CMC-154-3K Common Mode Choking Impedance and its Resistive (Rs) part is always higher than Reactive part (Xs) over the working bandwidth. This parameter, Rs>Xs ensure that our CMC will NOT increase common mode currents as could with inferior coaxial cable chokes.
Only chokes with Resistive (Rs>Xs) CMI- Common Mode Impedance stops and dissipate unwanted RF or noise coming to your equipment for sure.
SO-239 PTFE (Teflon®) Connectors which produce no impedance bump but a continuity of coaxial cable where the choke is inserted, it helps minimize leakage of RF in multi/contest stations setups.
PTFE (Teflon®) insulated coaxial cable used can handle a lot more than legal power (1500W) in The USA. It is rated only to 3kW because of cable limit at the 50MHz.
The enclosure is UV stable and could be used outside.
Use it as BALUN for YAGI, HEX beam, G5RV/ZS6BKW, Dipoles etc.


There are unlimited kinds of stuff that you can see within the very wide Internet. But its vast area to search from, it could be a real time-killer to find the right materials you need for your artistic eyes and brains. Worry not as we have done the searching for your convenience.
For this article, we have gathered some of the best Test Tube Logo Designs that you can find through the net. Scroll down, check them all out below and maybe get some fresh ideas that you can use for your next design projects. Come, take a peek, and get inspired!
Always love looking at logo inspiration, great article.


Dress 50, and the only me made of the week: The Green Pineapple dress worn one evening to a local theater performance, and then again two days later because it still looked ironed.
Dress 51 – Mossimo – an oldie but a goodie. Actually, I don't know how many times I have almost given this one away. Worn for a day of puppy sitting, Walmarting, and general around the house stuff.
Dress 52 – Rue 21 with a crop top. I think this has been worn three times already but never documented. Finally, I've got it archived today.
Dress 53 – Something I found at Macy's many summers ago. This one has been in the to get rid of pile for a while now. It didn't fit the last time I wore it, but surprise surprise, it fit just fine when I put it on the other day. I am happy about that because I really didn't want to part with it. I love the somewhat 90's look of it, and the fact that it has pockets. Worn for an afternoon of quilting and hanging out with friends.
That's all that I've worn this week for my wardrobe challenge; I am happy with my progress. I have a little trick to help me keep track of what I have worn, after I wear these dresses and they get washed, I hang them up with the hangers facing the wrong way. Visually, it seems I am about half way done – woo hoo! I do have a few in other rooms of the house though that I must not forget to count. Until next time, I will continue to be wearing those forgotten dresses living in the back of my closet.


No Documents are apprearing in "Content"
In my profile, the blogs are appearing however no documents are apperaing.
can you please post links to the documents in question?
I'll have to write to your offline (eMail) to validate this request.


text:
namespace snarkfront {

////////////////////////////////////////////////////////////////////////////////
// command line handling
//

class Getopt
{
public:
    Getopt(int argc, char *argv[],
           const std::string& string_opts,
           const std::string& number_opts,
           const std::string& flag_opts);

    bool operator! () const;
    bool empty() const;

    bool valid(const std::string& string_opts,
               const std::string& number_opts,
               const std::string& flag_opts);

    std::string getString(const char c);
    std::size_t getNumber(const char c);
    bool getFlag(const char c);

    const std::vector<std::string>& getArgs() const;

private:
    std::map<int, std::string> m_string;
    std::map<int, std::size_t> m_number;
    std::set<int> m_flag, m_string_opts, m_number_opts, m_flag_opts;
    std::vector<std::string> m_args;

    bool m_error;
};

} // namespace snarkfront

#endif

redpajama_set_name: RedPajamaGithub


Multiplication Chart to 25 Fresh Multiplication Table Template Kenindle fortzone is a part of Multiplication Chart to 25 pictures gallery. To download this Multiplication Chart to 25 Fresh Multiplication Table Template Kenindle fortzone in High Resolution, right click on the image and choose "Save Image As" and then you will get this image about Multiplication Chart to 25 Fresh Multiplication Table Template Kenindle fortzone.
This digital photography of Multiplication Chart to 25 Fresh Multiplication Table Template Kenindle fortzone has dimension 390 × 507 pixels. You can see another items of this gallery of Multiplication Chart to 25 below. Get interesting article about Multiplication Chart to 25 that may help you.
, we all choices the best series using greatest resolution exclusively for you, and now this photographs is usually one among photos collections in your best images gallery concerning Multiplication Chart to 25. I'm hoping you may want it.
submitted by simply amsafall with 2018-10-29 05:36:27. To determine most graphics in Multiplication Chart to 25 pictures gallery you need to stick to this web page link.


text:
#019: Pastor Peter Pilt
by Dave Pellowe | 31 May, 2017 | Video
Peter Pilt is a political pastor. He's most widely known as an outspoken and controversial blogger, but wishes people could debate opinions without being insulting or easily offended. His "Morning News Wrap" is read by several thousand people six days a week.
After 15 years as the Senior Pastor of Nowra City Church in NSW he became the National Director of Global Care, the charity organisation funded by INC (International Network of Churches, formerly Christian Outreach Centre). He's also on the National Executive for INC, their Missions Director, and part of their Christian Heritage College's council.
Peter Pilt has a Master's Degree in Leadership specialising in Poverty Eradication & Islam from Fuller Theological Seminary (USA), and a Bachelor's Degree in Economics from Wollongong Uni with a double major in Business Management and Economics. Peter sits on the corporate board of an accountancy firm, has an investment partnership, and other corporate board positions.
"We're called to stand up and speak. Who's pushing back on the leftist agenda and the whole doctrine of tolerance? Christians need to articulate a well thought-out opinion. #DontBeALemming"
Peter Pilt takes aim at stupid facebook posts and outrage, and compares blindly repeating stuff we heard "somewhere" to cute but clueless characters from a favourite 90s PC game.
Peter Pilt provoked plenty of outrage when he asked people to send him as much Halal certified chocolates as possible. It wasn't deliberate (at first) – it was just that Cadbury Flakes are his favourite chocolate.
We discuss the credible and effective ways people can achieve positive change in our culture, including boycotting and petitioning companies that campaign against Judeo-Christian Australian values.
The Good Samaritan broke down walls and hatred by being a great model of kindness, one of the things that can change the world, by inserting himself practically inserting himself in the solution his "neighbour" needed.
God says He doesn't care about #religious activity before we learn to do good; seek justice, rebuke the oppressor, defend the fatherless and plead for the widow… social issues!
Foreign Aid has been reduced "yet again" in our federal budget. Plenty of opinions abound about who, how and if we should be help our international neighbours. Peter Pilt says we have a responsibility to help those less fortunate than ourselves as the 12th largest economy in the world. We shouldn't withhold this help from people we disagree with, and it should be given strategically, minimising potential for waste and corruption.
Are you looking for a break from the regularly regurgitated mainstream rhetoric?
Do you value news and opinions which prioritise Truth over feelings?
We have no taxpayer funding or corporate backing. It's only support from people like you: mums and dads, workers and business owners which is helping produce & publish more of what you're looking for from leading conservative voices.
Together, we can't be censored. Would you like to help?
Consider generously supporting The Good Sauce today & help us plan our growth with as little as $10 per month.
I'll help!
Get a squirt of The Good Sauce on these social media & podcast channels & via email.
Woot! Woot! HURRAH!
"Being taught to avoid talking about politics and religion has led to a lack of understanding of politics and religion.
What we should have been taught was how to have a civil conversation about a difficult topic."
© Copyright The Good Sauce. All rights reserved. | #MyGenWillEndAbortion
News & views you can trust
Conservative voices are constantly being deplatformed and shadow-banned from the most popular social media platforms. Like & follow is no guarantee you'll see the posts you need & want.
The only guaranteed way to guard against corporate censorship is to go direct to the source and bypass the strangers deciding what you shouldn't see & read.
To get a regular digest of articles, interviews & news direct to your inbox SUBSCRIBE NOW!


text:
Gwen Stacy is Taking Neoartcore by Storm!
Gwen Stacy is a fictional character created by Marvel Comics and is one of the most popular characters in the Marvel Universe. Gwen has been portrayed in various mediums, including film, television and video games.
She is best known for her role as Peter Parker's love interest and eventual wife in the original Spider-Man comic books. Gwen Stacy was created by writer Stan Lee and artist Steve Ditko, with her first appearance in The Amazing Spider-Man #31 (December 1965). She originally served as a romantic foil for Peter Parker but over time she developed into a strong, independent character of her own. Gwen was a college student at Empire State University when she was first introduced. She had a strained relationship with her father, police captain George Stacy, which intensified after she developed feelings for Peter Parker. Gwen is often associated with the neoartcore style of artwork that has become popular in recent decades. Neoartcore is an art movement that combines traditional comic book techniques with modern digital techniques to create unique interpretations of characters from Marvel Comics and other pop culture franchises. Neoartcore artwork often features vibrant colors and intricate details, making it stand out from traditional comic book artwork. Gwen Stacy's popularity has grown over the years due to her appearances in multiple mediums, including both live-action films and animated series. She has been portrayed by several actresses over the years including Bryce Dallas Howard (Spider-Man 3), Emma Stone (The Amazing Spider-Man), Hailee Steinfeld (Spider-Man: Into The Spider-Verse), Sarah Gadon (Captain America: Civil War) and Brie Larson (Avengers: Endgame). Her appearances have been met with critical acclaim from fans and critics alike. In recent years, Gwen Stacy's character has undergone significant changes with her superhero alter ego being called Ghost-Spider instead of Spider-Woman or Spider-Girl (as she was previously known). This change was made to reflect Gwen's newfound independence as an empowered individual in her own right. With new stories being written featuring her heroic journey as Ghost-Spider, Gwen Stacy continues to be a beloved character among comic book fans all around the world.


Preheat your oven to 180° C and put some baking parchment into you loaf form.
In a bowl mix the flax seed flour, coconut flour, salt, baking soda and the three tablespoons of (previously grind) flax seed and mix them well.
In the blender put the two eggs, the honey, the olive oil and all the water. Close the top and give it a short pulse just to pre-mix the liquid ingredients.
Now add the dry ingredients and blend all for half a minute. It comes out a very soft but not liquid dough.
Let it cool for a few minutes in the loaf pan and then take it out and let it cool completely on a rack.
The bread is very soft, like a real toast or sandwich bread. You can toast it as well, comes out great.


text:
Home Entertainment Chris Pratt Professes His Love For His Wife, Katherine
Chris Pratt Professes His Love For His Wife, Katherine
Chris Pratt has admitted that he is head over heels in love with his wife Katherine. The Star-Lord in Guardians of the Galaxy was on Instagram this week with a pic of the gorgeous couple who recently welcomed Lyla, their daughter last August.
In the adorable pic, Chris Pratt can be seen staring, but his wife can be seen gazing lovingly at Chris with a loving smile playing on her face.
Chris Pratt gushed over the look and challenged viewers to find someone who looks at her love similarly. He said that Katherine had given him a wonderful life and a gorgeous daughter. He says that she chews loudly, enough to make him put on his earbuds.
But that is love, he says. He says that she helps him with everything. And in return, he opens a jar of pickles for her.
Chris Pratt has a son, Jack, with Anna Faris, his ex-wife. Chris continued gushing about Katherine's pure heart. He said that it belonged to him.
He said that Katherine was his greatest treasure, next only to his baseball card of Ken Griffey Jr. Chris Pratt recently signed in a forthcoming film as the voice of Garfield.
Chris added that Kathrine's birthday was after 6 weeks. He said in jest that if he fails to get anything for her, she will always have the post to look at and love.
Katherine, the author of Gift of Forgiveness, responded to Chris's note in the comment section in equal jest. She said that she loved him, but only when she was having cereals.
Chris Pratt And Katherine First Met In 2018
Chris Pratt and Katherine were first spotted together at a picnic in June 2018. they tied the knot a year later in 2019 at a ceremony in California. Their first child, Lyla Maria, was born in August 2020.
Chris split from his first wife, Faris in August 2017 after 8 years of marriage and their divorce was finalized in November 2018.
Previous articleElizabeth Hurley Reveals More Than Her Bikini As She Shows Off Her White Chain-Link Creation
Next articleKristen Stewart Engaged To Girlfriend Dylan After A Couple Of Years Together
https://www.theunionjournal.com/
Peacocks At It Again: Stun 3rd Seed Purdue
Andrew Cuomo To Repay $5.1M From Book Profits
EXCLUSIVE! Sir Alex Ferguson opens up to Gary Neville on his incredible career in...
Rick Dennison Reluctant Towards Vaccination, Loses Job
Allison Mack Going To Be Sentenced Over NXIVM Case
Debt Consolidation Specialists Symple Lending Talk About Shortening the Path to Becoming Debt Free
We spoke with Symple Lending, a highly respected team of debt consolidation specialists, about the growing segment of American consumers who woke up to...
Erling Haaland Ensures City Remains Firms In Race For English Premier League
Manchester City was back to their winning way after a thrilling win, overcoming a two-goal deficit to beat Spurs 4-2 this Thursday. The match...
Trevor Noah Departure From The Daily Show Exactly As He Planned
Trevor Noah's bombshell resolution to depart from The Daily Show stunned fans and even his staff. In a recording, the host discloses the events...
Last Few Days To Claim Stimulus Checks
The fourth stimulus check comes with lots of rules and criteria. And one needs to follow that thoroughly in order to claim that the...
TWICE Has A Lot To Talk About
The k-pop group of nine young aspiring women TWICE has recently completed their fourth world tour. Which included US, Japan, and Korea. And their...