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

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

2005-02-24

arxiv.org/abs/math/0502522v1

Mathematics

Spectral Theory

15 pages, 1 figure

Scientific paper

For integers $m\geq 3$, we study the non-self-adjoint eigenvalue problems $-u^{\prime\prime}(x)+(x^m+P(x))u(x)=E u(x)$, $0\leq x<+\infty$, with the boundary conditions $u(+\infty)=0$ and $\alpha u(0)+\beta u^{\prime}(0)=0$ for some $\alpha, \beta\in\C$ with $|\alpha|+|\beta|\not=0$, 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.

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