The weakly coupled fractional one-dimensional Schr\"{o}dinger operator with index $\bf 1<\alpha \leq 2$

TL;DR

This paper investigates the properties of a fractional one-dimensional Schr"odinger operator, analyzing its Green's function, spectral characteristics, and conditions for bound states, especially in the weak coupling regime.

Contribution

It provides a detailed spectral analysis of the fractional Schr"odinger operator, including Green's function asymptotics and criteria for bound states at small coupling.

Findings

Asymptotic behavior of the Green's function is characterized.

Finite-rank approximation of the Birman-Schwinger operator is derived.

Necessary and sufficient conditions for bound states at weak coupling are established.

Abstract

We study fundamental properties of the fractional, one-dimensional Weyl operator $\hat{\mathcal{P}}^{\alpha}$ densely defined on the Hilbert space $\mathcal{H}=L^2({\mathbb R},dx)$ and determine the asymptotic behaviour of both the free Green's function and its variation with respect to energy for bound states. In the sequel we specify the Birman-Schwinger representation for the Schr\"{o}dinger operator $K_{\alpha}\hat{\mathcal{P}}^{\alpha}-g|\hat{V}|$ and extract the finite-rank portion which is essential for the asymptotic expansion of the ground state. Finally, we determine necessary and sufficient conditions for there to be a bound state for small coupling constant $g$.