Semiclassical expansions of eigenvalues of Schrödinger operators

Question

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.

Answer 1

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.

Answer 2

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.