Bohr-Sommerfeld quantization rules revisited: the method of positive commutators

2017/02/22 by Ifa, Abdelwaheb, Louati, Hanen, Rouleux, Michel · 2 citations
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1702.06758

Abstract

We revisit the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the "flux norm" is not invertible. The interest of this procedure lies in its possible generalization to matrix-valued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian. It is simplified in the scalar case by using action-angle variables.

Cited by

• Bohr-Sommerfeld quantization conditions for Schrodinger operator: the Method of Microlocal Wronskian and Gram Matrix
• Bohr-Sommerfeld Quantization Rules for 1-D Semiclassical Pseudo-Differential Operator: the Method of Microlocal Wronskian and Gram Matrix

Related

• Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator: The method of positive commutators
• Bohr-Sommerfeld Quantization Rules for 1-D Semiclassical Pseudo-Differential Operator: the Method of Microlocal Wronskian and Gram Matrix
• Bohr-Sommerfeld quantization conditions for Schrodinger operator: the Method of Microlocal Wronskian and Gram Matrix
• A remark on Bohr-Sommerfeld Quantization Rules for a self-adjoint 1-D h-Pseudo-differential operator
• Bohr--Sommerfeld rules for systems