On the Birman-Schwinger principle applied to (-Delta + m^2)^(1/2) - m

TL;DR

This paper applies the Birman-Schwinger principle to analyze the spectral properties of the operator (-Delta + m^2)^(1/2) - m, deriving conditions for zero to be an eigenvalue and developing a series expansion for the inverse coupling constant.

Contribution

It extends the Birman-Schwinger framework to a relativistic operator and derives a second-order series expansion for the inverse coupling constant near zero energy.

Findings

Derived conditions for zero eigenvalues of the operator.

Developed a second-order series expansion for (l^{-1})(a).

Applied the Birman-Schwinger principle to a relativistic operator.

Abstract

The condition for E = 0 to be an eigenvalue of the operator (-Delta + m^2)^(1/2) -m + l V is obtained through the use of the Birman-Schwinger principle. By setting E=-a^2 and using the analyticity of the corresponding Birman-Schwinger kernel, the series development of (l^(-1))(a) is obtained up to second order on a.