Monotonicity of the polaron energy

TL;DR

This paper uses self-dual cone analysis to prove that the polaron energy decreases monotonically as the ultraviolet cutoff increases in the Fr"ohlich Hamiltonian, providing new mathematical insight into polaron models.

Contribution

It introduces a novel application of self-dual cone analysis to establish the monotonicity of polaron energy in the Fr"ohlich model.

Findings

Proves $E_{\Lambda} > E_{\Lambda'}$ for $\Lambda < \Lambda'$

Clarifies the monotonic behavior of polaron energy with respect to the ultraviolet cutoff

Provides a rigorous mathematical foundation for energy monotonicity in polaron models

Abstract

In condensed matter physics, the polaron has been fascinating subject. It is described by the Hamiltonian of H. Fr\"ohlich. In this paper, the Fr\"ohlich Hamiltonian is investigated from a viewpoint of the self-dual cone analysis proposed in Miyao (2011). This point of view clarifies the monotonicity of polaron energy, i.e., denoting the lowest energy of the Fr\"ohlich Hamiltonian with the ultraviolet cutoff $\Lambda$ by $E_{\Lambda}$, we prove $E_{\Lambda}>E_{\Lambda'}$ for $\Lambda<\Lambda'$.