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Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains

Benjamin Hinrichs, Oliver Matte

Abstract

We prove Feynman-Kac formulas for the semigroups generated by selfadjoint operators in a class containing Fröhlich Hamiltonians known from solid state physics. The latter model multi-polarons, i.e., a fixed number of quantum mechanical electrons moving in a polarizable crystal and interacting with the quantized phonon field generated by the crystal's vibrational modes. Both the electrons and phonons can be confined to suitable open subsets of Euclidean space. We also include possibly very singular magnetic vector potentials and electrostatic potentials. Our Feynman-Kac formulas comprise Fock space operator-valued multiplicative functionals and can be applied to every vector in the underlying Hilbert space. In comparison to the renormalized Nelson model, for which analogous Feynman-Kac formulas are known, the analysis of the creation and annihilation terms in the multiplicative functionals requires novel ideas to overcome difficulties caused by the phonon dispersion relation being constant. Getting these terms under control and generalizing other construction steps so as to cover confined systems are the main achievements of this article.

Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains

Abstract

We prove Feynman-Kac formulas for the semigroups generated by selfadjoint operators in a class containing Fröhlich Hamiltonians known from solid state physics. The latter model multi-polarons, i.e., a fixed number of quantum mechanical electrons moving in a polarizable crystal and interacting with the quantized phonon field generated by the crystal's vibrational modes. Both the electrons and phonons can be confined to suitable open subsets of Euclidean space. We also include possibly very singular magnetic vector potentials and electrostatic potentials. Our Feynman-Kac formulas comprise Fock space operator-valued multiplicative functionals and can be applied to every vector in the underlying Hilbert space. In comparison to the renormalized Nelson model, for which analogous Feynman-Kac formulas are known, the analysis of the creation and annihilation terms in the multiplicative functionals requires novel ideas to overcome difficulties caused by the phonon dispersion relation being constant. Getting these terms under control and generalizing other construction steps so as to cover confined systems are the main achievements of this article.
Paper Structure (39 sections, 34 theorems, 192 equations)

This paper contains 39 sections, 34 theorems, 192 equations.

Table of Contents

  1. Introduction and main results
  2. General introduction
  3. Brief description of the main result
  4. Remarks on closely related previous work
  5. Remarks on mathematical novelties
  6. Standing assumptions and construction of polaron Hamiltonians
  7. Fock space calculus
  8. Standing hypotheses
  9. Assumptions on coupling functions
  10. Assumptions on electrostatic and magnetic vector potentials
  11. Definition of the Hamiltonian via quadratic forms
  12. Presentation of the Feynman--Kac formulas
  13. Brownian motions and their time-reversals
  14. Path integrals involving $A$ and $V$
  15. Processes appearing in the interaction terms
  16. ...and 24 more sections

Key Result

Theorem 2.4

For every $\Psi\in\mathcal{D}(\mathfrak{q}^{\min})$, the iterated integral is well-defined. The so-obtained quadratic form $\mathfrak{w}(v)$ is infinitesimally $\mathfrak{q}^{\min}$-bounded. In fact, for all $E\geqslant1$, where $L_E(v)\xrightarrow{E\to\infty}0$ by assumption. Finally,

Theorems & Definitions (89)

  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • proof
  • Definition 2.6
  • Corollary 2.7
  • proof
  • Lemma 3.1
  • ...and 79 more