Half-Line non-self-adjoint Schrödinger operators with polynomial potentials: Asymptotics of eigenvalues

Abstract

For integers $m\ge 3$, we study the non-self-adjoint eigenvalue problems $-u^{\prime \prime }(x)+(x^m+P(x))u(x)=Eu(x)$, $0\le x<+\infty$, with the boundary conditions $u(+\infty )=0$ and $\alpha u(0)+\beta u^{\prime }(0)=0$ for some $α, β\in\C$ with $|\alpha |+|\beta |\ne 0$, where $P(x)=a_1{x}^{m-1}+a_2{x}^{m-2}+...+a_{m-1}x$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $E_n$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues.

15 pages, 1 figure

References in corpus (1)

• Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues