Bohr-Sommerfeld quantization rules revisited: The method of positive commutators
Vol. 25 (2018), No. 2, Page 91-127.
Ifa, Abdelwaheb; Louati, Hanen; Rouleux, Michel
Bohr-Sommerfeld quantization rules revisited: The method of positive commutators
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Abstract:
We revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D $h$-Pseudo-differential operator 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. It is simplified by using action-angle variables. The interest of this procedure lies in its possible generalization to matrix-valued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian.
Keywords: Bohr Sommerfeld; Weyl quantization; positive commutators; flux norm; microlocal Wronskian
Mathematics Subject Classification (2010): 81S10, 81S30
Mathematical Reviews Number: MR3792787
Received: 2017-04-04
