Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues

Abstract

For integers $m\ge 3$ and $1\le \ell \le m-1$, we study the eigenvalue problem $-u^{\prime \prime }(z)+[(-1{)}^{\ell }(iz{)}^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi }{2}\pm \frac{(\ell +1)\pi }{m+2}$ in the complex plane, where $P(z)=a_1{z}^{m-1}+a_2{z}^{m-2}+...+a_{m-1}z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues ${\lambda }_n$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues. Also, we show for arbitrary $\mathcal{P}\mathcal{T}$-symmetric polynomial potentials of degree $m\ge 3$ and all symmetric decaying boundary conditions that the eigenvalues are all real and positive, with only finitely many exceptions.

31 pages, 1 figure

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