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Gaussian integrals depending by a quantum parameter in finite dimension

Simone Camosso

Abstract

A common theme in mathematics is the evaluation of Gauss integrals. This, coupled with the fact that they are used in different branches of science, makes the topic always actual and interesting. In these notes we shall analyze a particular class of Gaussian integrals that depends by the quantum parameter $\hbar$. Starting from classical results, we will present an overview on methods, examples and analogies regarding the practice of solving quantum Gaussian integrals.

Gaussian integrals depending by a quantum parameter in finite dimension

Abstract

A common theme in mathematics is the evaluation of Gauss integrals. This, coupled with the fact that they are used in different branches of science, makes the topic always actual and interesting. In these notes we shall analyze a particular class of Gaussian integrals that depends by the quantum parameter . Starting from classical results, we will present an overview on methods, examples and analogies regarding the practice of solving quantum Gaussian integrals.

Paper Structure

This paper contains 16 sections, 13 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. The Gaussian integral in $1$--dimension
  3. Ways to write the Gaussian integral
  4. Gaussian integral or Gauss theorem?
  5. The Gaussian integral in $n$--dimensions
  6. Gaussian integrals and homogeneous functions
  7. Gaussian matrix integrals
  8. The symplectic structure
  9. The quantum postulates
  10. Gaussian integrals depending by a quantum parameter
  11. Quantum integrals
  12. On a Gaussian integral in one dimension and the Lemniscate
  13. The Berezin--Gaussian integral: a generalization of the Gaussian integral
  14. On the Laplace method and a "mixed Gaussian Integral"
  15. Quantum Gaussian integrals with a perturbative term
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $m$ be a positive integer and $k$ be a real positive constant, then where $P_{m}(\xi)\,=\,\xi^m+\sum_{j\geq 1}p_{mj}\xi^{m-2j}$ is a monic polynomial in $\xi$ of degree $m$ and parity $(-1)^m$.

Theorems & Definitions (14)

  • Proposition 2.1
  • Theorem 3.1: Hörmander
  • Theorem 3.2
  • Theorem 4.2
  • Proposition 4.3
  • Example 7.1: Schrödinger quantization
  • Proposition 8.1
  • Proposition 8.2
  • Proposition 8.3
  • Proposition 10.1
  • ...and 4 more