Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators

Abstract

For perturbed Stark operators Hu = −u ′′ −xu+qu, the author has proved that lim sup x→∞ x 1 2 |q(x)| must be larger than 1 √ 2 N 1 2 in order to create N linearly independent eigensolutions in L 2 (R + ) [25]. In this paper, we apply generalized Wigner-von Neumann type functions to construct embedded eigenvalues for a class of Schrödinger operators, including a proof that the bound 1 √ 2 N 1 2 is sharp.

Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

“…It is robust and fundamental in that it can be applied in a variety of contexts to construct embedded eigenvalues. In the forthcoming work it is adapted by one of the authors and Ong to construct eigenvalues embedded into the spectral band for perturbed periodic operator, in both continuous and discrete settings [20,23], and also to construct eigenvalues embedded into the absolutely continuous spectrum for perturbed Stark type operators [18,22].…”