The discrete spectrum in the singular Friedrichs model

TL;DR

This paper investigates the negative spectrum of a Friedrichs model operator with a specific potential, revealing conditions under which it has finitely or infinitely many negative eigenvalues and explicitly calculating their number.

Contribution

It provides explicit criteria for the finiteness and infiniteness of negative eigenvalues in the Friedrichs model and computes their exact count for various parameter regimes.

Findings

Infinite negative eigenvalues for certain parameter values.

Finite number of negative eigenvalues for others, independent of coupling strength.

Explicit formulas for the number of negative eigenvalues.

Abstract

A typical result of the paper is the following. Let $H_\gamma=H_0 +\gamma V$ where $H_0$ is multiplication by $|x|^{2l}$ and $V$ is an integral operator with kernel $\cos< x,y\rang le$ in the space $L_2(R^d)$. If $l=d/2+ 2k$ for some $k= 0,1,...$, then the operator $H_\gamma$ has infinite number of negative eigenvalues for any coupling constant $\gamma\neq 0$. For other values of $l$, the negative spectrum of $H_\gamma$ is infinite for $|\gamma|> \sigma_l$ where $\sigma_l$ is some explicit positive constant. In the case $\pm \gamma\in (0,\sigma_l]$, the number $N^{(\pm)}_l$ of negative eigenvalues of $H_\gamma$ is finite and does not depend on $\gamma$. We calculate $N^{(\pm)}_l$.