Embedded eigenvalues of generalized Schr\"odinger operators

TL;DR

This paper constructs examples of generalized Schrödinger operators with decaying potentials that possess embedded eigenvalues, linking the decay rate to the curvature of Fermi surfaces and Fourier restriction theory.

Contribution

It introduces new examples of operators with embedded eigenvalues based on potential decay and geometric properties, expanding understanding of spectral phenomena.

Findings

Operators with decaying potentials can have embedded eigenvalues.

The decay rate relates to the curvature of Fermi surfaces.

Connections are made to Fourier restriction theory counterexamples.

Abstract

We provide examples of operators $T(D)+V$ with decaying potentials that have embedded eigenvalues. The decay of the potential depends on the curvature of the Fermi surfaces of constant kinetic energy $T$. We make the connection to counterexamples in Fourier restriction theory.