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Spectral Analysis of Large Particle Systems

by Jonas Dahlbæk
PhD Dissertations | August 2017

The Fröhlich polaron model is defined as a quadratic form, and its discrete spectrum is studied for each fixed total momentum $\xi \in\mathbb{R}^d$ in the weak coupling regime. Criteria are determined by means of which the number of discrete eigenvalues may be deduced. The analysis is based on relating the spectral analysis of the Fröhlich polaron model to an equivalent problem in terms of a family of generalized Friedrichs models. This is possible by employing a combination of the Birman-Schwinger principle and the Haynsworth inertia additivity formula. The number of discrete eigenvalues of a generalized Friedrichs model is analyzed explicitly. In order to determine the family generalized Friedrichs models induced by the Fröhlich polaron model, it is necessary to compute a certain Feshbach operator.

A method for computing Feshbach operators in bosonic Fock space is developed. The focus is on obtaining a framework which unifies and generalizes frameworks that have appeared previously in the literature. The end result is a calculus for creation/annihilation symbols, where Wick's theorem provides a formula for the product of finitely many symbols. The framework is then applied to the Fröhlich polaron model. The framework is also applied to the spin boson model. The application to the spin boson model is based on the spectral renormalization group.

It is shown that the spectral renormalization group scheme can be naturally posed as an iterated Grushin problem. While it is already known that Schur complements, Feshbach maps and Grushin problems are three sides of the same coin, it seems to be a new observation that the smooth Feshbach method can also be formulated as a Grushin problem. Based on this, an abstract account of the spectral renormalization group is given.

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Thesis advisor: Jacob Schach Møller