All-coupling solution for the continuous polaron problem in the Schr\"{o}dinger representation

TL;DR

This paper presents an all-coupling solution for the large-radius Fröhlich polaron in the Schrödinger representation, providing a unified approach across all coupling strengths and addressing longstanding methodological issues.

Contribution

It introduces a novel algebraic method to solve the polaron problem in the Schrödinger representation applicable to all coupling regimes, bridging gaps between different theoretical approaches.

Findings

Calculated polaron energy and effective mass across coupling constants

Method aligns with Feynman's variational results

Addresses inequalities in functional and operator approaches

Abstract

The solution for the large-radius Fr\"{o}hlich polaron in the Schr\"{o}dinger representation of the quantum theory is constructed in the entire range of variation of the coupling constant. The energy and the effective mass of the polaron are calculated by simple algebraic transformations and are analogous to the results found by Feynman on the basis of the variational principle for the path-integrals of this system. It allows us to solve the long-lived problem of the inequalities of the functional and operator approaches for the polaron problem. The developed method is important for other models of particle-field interaction including those ones for which the standard perturbation theory is divergent.