The Role of the Branch Cut of the Logarithm in the Definition of the Spectral Determinant for Non-Self-Adjoint Operators

Abstract

The spectral determinant is usually defined using the spectral zeta function that is meromorphically continued to zero. In this definition, the complex logarithms of the eigenvalues appear. Hence, the notion of the spectral determinant depends on the way in which one chooses the branch cut in the definition of the logarithm. We give results for the non-self-adjoint operators that specify when the determinant can and cannot be defined and how its value differs depending on the choice of the branch cut.