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

Abstract

For perturbed Stark operators $Hu=-u^{\prime \prime }-xu+qu$, the author has proved that ${\mathrm{limsup}}_{x\to \infty }{x}^{\frac{1}{2}}|q(x)|$ must be larger than $\frac{1}{\sqrt{2}}{N}^{\frac{1}{2}}$ in order to create $N$ linearly independent eigensolutions in $L^2({\mathbb{R}}^{+})$. 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 $\frac{1}{\sqrt{2}}{N}^{\frac{1}{2}}$ is sharp.

References in corpus (9)

• Criteria for embedded eigenvalues for discrete Schrödinger operators
• Asymptotics of resonances for 1d Stark operators
• Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators
• Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators
• Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian
• Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique
• Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach
• Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von Neumann approach
• The asymptotical behaviour of embedded eigenvalues for perturbed periodic operators