The Zeta Function for the Triangular Potential

TL;DR

This paper investigates the zeta functions associated with the Schrödinger equation for a triangular potential, employing mathematical techniques to compute and analyze their properties and implications for quantum Hamiltonians.

Contribution

It introduces a comprehensive analysis of the zeta functions for the triangular potential, including new methods for their computation and exploration of their analytic properties.

Findings

Consistent zeta function values from two different methods.

Computed values at zero and negative integers.

Analyzed pole structure and residues.

Abstract

The zeta functions for the Schr\"odinger equation with a triangular potential are investigated. Values of the zeta functions are computed using both the Weierstrass factorization theorem and analytic continuation via contour integration. The results were found to be consistent where the domains of the two methods overlap. Analytic continuation is used to compute values of the zeta functions at zero and the negative integers, explore the pole structure (and residue values), as well as the value of the slopes at the origin. Those results are used for the computation of the trace and determinant of the associated Hamiltonians.