## Semiclassical expansions of eigenvalues of Schrödinger operators

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Considering Schrödinger operators $$H(\hbar) = \hbar \Delta + V$$ where $V$ is some potential, perturbation theory tells that the eigenvalues of $H(\hbar)$ are holomorphic on some region containing the positive real axis, but don't necessarily have an analytic continuation to the origin.

However, one tries to find asymptotic expansions in $\hbar$ in $0$, thus hopes that $$\lambda(\hbar) \sim \lambda_0 + \hbar \lambda_1 + \hbar^2 \lambda_2 + \dots.$$ Usually one finds those expansions by WKB-methods, belonging to eigenfunctions that concentrate at the minimums of $V$.

Now, there are a lot of theorems about the behavior of this and statements that tell you that the expansions you get by WKB-methods actually belong to a "true" eigenvalue, but to me it seems that nobody ever tries to answer the question, if every eigenvalue of $H(\hbar)$ (considered as holomorphic function in $\hbar$ on some region $U$ with $0 \in \partial U$) actually admits an asymptotic expansion at $0$. Instead, it seems, that this is always automatically assumed.

But this is per se is not clear to me at all; it means for example that $0$ is not a branch point of $\lambda(\hbar)$.

Do you know about this question? I can't even have a guess if it is true that every eigenvalue admits an asymptotic expansion in the general case or if that is true only if $V$ is "well-behaved", like for example having no nondegenerate minima (and always assuming that it is such that the spectrum is discrete). Neither do I have an idea how one would prove something like that so far.

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## 2 Answers

If your model is the anharmonic oscillator $$V=x^2+x^4$$ one can do a simple unitary transformation based on scaling $x\rightarrow \lambda x$, $\frac{d}{dx}\rightarrow \lambda^{-1} \frac{d}{dx}$ to reduce your problem to that of the analyticity of $\sqrt{\hbar}E_n(\sqrt{\hbar})$ where $E_n(\beta)$ are the eigenvalues of the Hamiltonian $$-\Delta+x^2+\beta x^4\ .$$ This has been studied for instance in B. Simon, "Coupling constant analyticity for the anharmonic oscillator", Ann. Phys. 58 (1970), 76-136. You might find the kind of methods you need in this article or in the ones referring to it, say on Google Scholar.

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There is no specific potential given. Just possibly some condition that it is morse, i.e. that all critical points are nondegenerate. – Kofi Feb 23 2012 at 22:32
@Kofi: sure. But whatever general theorem you are looking for, it should apply to this example. – Abdelmalek Abdesselam Feb 24 2012 at 15:38

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If you look at:

Microlocal WKB expansions by A. Martinez (available through cite seer), you will see that this is a fairly popular subject, where many results have been obtained by Sjostrand et al.

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I didn't now that specific paper, but the older works by Helffer and Sjöstrand. As far as I could see is, that Martinez "only" proves similar results for a wider class of operators. They always start with formal "local" eigenvectors and eigenvalues and then prove that this is the asymptotic expansion of an actual eigenvector. What I am wondering about is the question, if it is not possible to prove by abstract methods that all eigenvalus admit an asymptotic expansion at zero, just by looking at the operator itself. – Kofi Dec 28 2011 at 22:36
Ah, I see... Interesting question... – Igor Rivin Dec 28 2011 at 23:29
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