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Bohr-Sommerfeld quantization conditions for Schrodinger operator: the Method of Microlocal Wronskian and Gram Matrix

Abdelwaheb Ifa

Abstract

In this work, we present a new formulation of the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a Schrodinger operator within the algebraic and microlocal framework of B. Helffer and J. Sjostrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the "flux norm" is not invertible. This condition is obtained using the microlocal Wronskian and does not rely on traditional matching techniques. It is simplified by using action-angle variables. The interest of this procedure lies in its possible generalization to matrixvalued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian.

Bohr-Sommerfeld quantization conditions for Schrodinger operator: the Method of Microlocal Wronskian and Gram Matrix

Abstract

In this work, we present a new formulation of the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a Schrodinger operator within the algebraic and microlocal framework of B. Helffer and J. Sjostrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the "flux norm" is not invertible. This condition is obtained using the microlocal Wronskian and does not rely on traditional matching techniques. It is simplified by using action-angle variables. The interest of this procedure lies in its possible generalization to matrixvalued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian.

Paper Structure

This paper contains 10 sections, 3 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. The microlocal Wronskian
  3. Semiclassical Quantization Rules, Quasi-Modes, and the Flux Norm
  4. Normalized Quasi-Modes
  5. Asymptotic solutions in the Fourier representation
  6. Normalisation
  7. The homology class of the generalized action: Fourier representation
  8. Asymptotic solutions in the spatial representation
  9. Quantization Rules and Fredholm Property
  10. Bohr-Sommerfeld and Action-Angle Variables

Key Result

Proposition 2.1

Let $u^{a}, v^{a}\in K_{h}(E)$, and denote by $\widehat{u}$ the $h$-Fourier (unitary) transform of $u$. Then mod $\mathcal{O}(h^{\infty})$: Moreover, $\mathcal{W}^{a}_{\rho}(u^{a}, \overline{v^{a}})$ does not depend on the choice of $\chi^{a}$ above.

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.1
  • Remark 4.1