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in 1998 , bender and boettcher introduced the non - hermitian @xmath0-symmetric quantum systems @xcite .
this letter ignited a very active field .
hundreds of papers have been published on various aspects of this new quantum mechanics . for review , please see ref .
@xcite and references therein .
most of the studies have been focused on one - dimensional systems , very few have touched two or higher dimensions . in 2001 , bender _ et al _  studied the complex cubic potential @xmath1 and @xmath3 . using perturbation theory , they found real eigenvalues for the three lowest levels .
they used the wkb method to confirm the reality of the ground state energy .
their approach also applied to the complex hnon - heiles potential @xmath2 @xcite . in 2002 , nanayakkara and abayaratne revisited the @xmath4 interaction with a non - degenerated setup .
they computed energy eigenvalues up to @xmath5 . with their choice of parameters , all eigenvalues appear to be real @xcite . later in the same year
, nanayakkara obtained the analytic results for @xmath6 interaction and numerical results in @xmath7 and @xmath8 interactions .
for all three systems , the author found real spectrum for small enough coupling constant and complex one for large coupling .
the interaction of @xmath9 is also studied @xcite . in 2005 , nanayakkara observed avoided level crossings in both the real cubic interaction @xmath10 and the complex cubic interaction @xmath4 @xcite .
the results has been challenged in a comment by bla _
et al _  in ref .
@xcite . they pointed out that the quantum system with the real cubic potential is ill defined and the complex cubic potential has no avoided level crossings .    in this paper , we compute the eigenvalues of two - dimensional systems to high levels with high precision .
we study the hamiltonians associated with the complex cubic potential @xmath11 @xmath12 and with the complex hnon - heiles potential @xmath13 @xmath14 both hamiltonians are not hermitian .
rather , they respect the @xmath0 symmetry and the @xmath15 parity : @xmath16=0,&\qquad & [ h,{\cal p}_y]=0,\end{aligned}\ ] ] where the total parity @xmath17 , the time reversal @xmath18 , and the @xmath15 parity @xmath19 are defined as @xmath20 although the hamiltonians are complex , the secular equations are real due to the @xmath0 symmetry
. they can only depend on @xmath21 .
therefore , both systems are invariant under @xmath22 .
since the @xmath15 parity is an unbroken symmetry , we can diagonalize @xmath23 and @xmath19 simultaneously .
all the eigenfunctions of @xmath23 are also eigenfunctions of @xmath19 with eigenvalues @xmath24 .
we use different methods to compute the eigenvalues of each system .
we found complex conjugate pairs when the levels cross between same @xmath15-parity states . at higher level
, the level crossing happens at smaller coupling constant .
this indicates that no matter how small the coupling constant is , there are always complex eigenvalues at high enough levels .
it shows the rich structure in the two - dimensional @xmath0-symmetric quantum systems .
one must be very careful when generalizing the results in one - dimensional non - hermitian quantum mechanics to higher dimensions .
the paper is organized as the following : the methods we used are briefly reviewed in sec .
[ sec : method ] ; the results of different systems are presented in sec .
[ sec : xy2 ] and sec .
[ sec : hh ] ; we conclude in sec .
[ sec : conclusion ] .
we use three different methods to study the spectra of hamiltonians in ( [ eqn : h12 ] ) and ( [ eqn : hhh ] ) .    *
the first method is the perturbation theory . because the energy levels of unperturbed hamiltonian , @xmath25 has degeneracy
, we must apply the degenerate perturbation theory . after getting the eigenvalues as power series of the coupling constant @xmath26
, we use pad expansions to extract the information from the divergent series .
the results for two systems are shown as lines in fig .
[ fig : xy2_re ] and fig .
[ fig : hh_re ] , respectively . *
the second method is numerical computation by the finite - element method ( fem ) .
the fem results for eigenvalues with @xmath11 potential are shown as crosses in fig .
[ fig : xy2_re ] and fig .
[ fig : xy2_im ] . *
the third method is based on the expansion in two - dimensional harmonic oscillator ( ho ) basis @xcite .
we first analytically compute the non - vanishing matrix elements of the full hamiltonian on the two - dimensional ho eigenfunctions basis . depending on the precision requirement
, we truncate the sparse matrix to a finite size . and the eigenvalues of the full hamiltonian can be obtained by numerically diagonalizing this finite matrix .
this method is a mixture of analytic and numerical techniques .
it turns out that this is the best way to compute the eigenvalues of the two - dimensional complex hamiltonians .
the results are shown as dots in fig .
[ fig : xy2_re ] and fig .
[ fig : xy2_im ] and crosses in fig .
[ fig : hh_re ] and fig .
[ fig : hh_im ] .    from fig .
[ fig : xy2_re ] we can clearly to see that all three methods provide consistent informations about the real part of the eigenvalues . and fig .
[ fig : xy2_im ] shows the two numerical methods are consistent on the imaginary part of the eigenvalues .
using perturbation theory , we calculate the eigenvalues as power series of the coupling constant @xmath26 . here are the first few terms for the first 10 eigenvalues : @xmath27 we actually continue this expansion up to the order of @xmath28 , then use @xmath29 pad to extract the information from the divergent series .
the results are shown as lines in fig .
[ fig : xy2_re ] . from the figure
, we may see that some pad expansions have poles and cross each others .    in the same figure ,
we show the real parts of eigenvalues from fem as crosses and from ho expansions as dots .
the two sets of numerical results clearly agree with each other . and
surprisingly , up to the crossing points , the pad expansions fit the numerical results very well .
although the hamiltonian @xmath30 has @xmath0 symmetry , some energy eigenvalues are complex .
the imaginary parts of the eigenvalues by two numerical methods are shown in fig .
[ fig : xy2_im ] .     with the complex cubic potential .
lines are from perturbative expansion using @xmath29 pad .
crosses are results using fem .
dots are results using the method based on two - dimensional ho basis expansions .
blue disks have even @xmath15-parity and purple squares have odd @xmath15-parity .
the system is symmetric under @xmath22 , only the part with positive @xmath26 is shown.,scaledwidth=75.0% ]     with the complex cubic potential .
crosses are results using fem .
dots are results using the method based on two - dimensional ho basis expansions .
blue disks have even @xmath15-parity and purple squares have odd @xmath15-parity .
the system is symmetric under @xmath22 , only the part with positive @xmath26 is shown.,scaledwidth=75.0% ]    from the fig .
[ fig : xy2_re ] , we observe two types of level crossings . if the crossing is between different @xmath15-parity states , the eigenvalues remain real .
one example of this type is the crossing between the third and fourth lowest states near @xmath31 .
if the crossing is between same @xmath15-parity states , then eigenvalues become complex conjugate pairs .
for example , the sixth and seventh lowest states cross near @xmath32
.    we also observed that the level crossing happens in higher levels at smaller value of coupling constant .
this indicates that no matter how small the coupling constant is , there is always complex eigenvalues at high enough levels .
there is no critical value , @xmath33 , such that for @xmath34 , the entire spectrum is real .
in the case of the complex hnon - heiles potential , a similar patten appears . the energy eigenvalues become complex when @xmath35 .
there are special features for this interaction .    * first , there are all rational numbers in every term of eigenvalues in the perturbation expansion . *
second , there are an accident degeneracy between odd and even @xmath15-parity . most of the energy levels are degenerated , but interestingly , not all of them .
for example , @xmath36 , @xmath37 , and @xmath38 , but @xmath39 .
in fact , @xmath40 and @xmath41 are the same up to the order of @xmath21 , but for the order of @xmath42 and higher , they separate .
this patten remains the same for higher exciting states : * * for even @xmath43 , @xmath44 , @xmath45 , etc .
* * for odd @xmath46 , @xmath47 , @xmath48 , etc .
* * for odd @xmath46 , @xmath49 starting from the order of @xmath42 . * third , @xmath50 appears to be non - alternating in signs in the expansion . and
this is a unique case up to @xmath51 .
here are the first few terms for the first 21 eigenvalues : @xmath52 as in the previous section , we actually continue this expansion up to the order of @xmath28 , then use @xmath29 pad to extract the information from the divergent series .
the results are shown as lines in fig .
[ fig : hh_re ] .
the results from ho expansions are shown as crosses in the same figure . once again , we found the pad expansions fit the numerical results up to the crossing points .
the @xmath0 symmetry is also broken in this hamiltonian .
some eigenvalues cross and form complex conjugate pairs .
the imaginary parts of the eigenvalues by ho expansions are shown in fig .
[ fig : hh_im ] .     with the complex hnon - heiles potential .
lines are from perturbative expansion using @xmath29 pad .
crosses are from ho expansions .
even @xmath15-parity states are using `` @xmath53 '' and odd @xmath15-parity states are using `` @xmath54''.,scaledwidth=75.0% ]     with the complex hnon - heiles potential . only the results from ho expansions
are shown .
even @xmath15-parity states are using `` @xmath53 '' and odd @xmath15-parity states are using `` @xmath54''.,scaledwidth=75.0% ]
we concluded that both hamiltonians have complex spectra .
the origin of this broken @xmath0 symmetry is unclear
. it could be due to the interaction between hermitian and non - hermitian potentials as in ref .
it could also be due to the level contraction instead of level repulsion in the non - hermitian quantum mechanics .
the results indicate that there is no critical value of the coupling constant to have reality in this degenerated setup . in the analogue with the @xmath6 interaction studied in ref .
@xcite , it is possible that small enough coupling constant exhibits entire real spectrum in the non - degenerated setup .
this would be confirmed by a future study .
h.  bla , m.  tater , m.  znojil , `` _ comment on : ` comparison of quantal and classical behavior of @xmath0-symmetric systems at avoided crossings ' [ phys .  lett .  a 334 ( 2005 ) 144 ] _ , '' phys .
a * 351 * , 452 - 456 ( 2006 )
two - dimensional @xmath0-symmetric quantum - mechanical systems with the complex cubic potential @xmath1 and the complex hnon - heiles potential @xmath2 are investigated . using numerical and perturbative methods , 
 energy spectra are obtained to high levels . 
 although both potentials respect the @xmath0 symmetry , the complex energy eigenvalues appear when level crossing happens between same parity eigenstates .    * pac numbers : 03.65.ge , 02.30.lt , 02.60.lj *