Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator: The method of positive commutators

We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for 1-D Pseudo-differential self-adjoint Hamiltonians within the algebraic and microlocal framework of Helffer and Sj\"ostrand; BS holds precisely when the Gram matrix consisting of scalar products of WKB solutions with respect to the "flux norm" is not invertible

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian with PT symmetry: generalized Bohr-Sommerfeld quantization rules

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions, 2) constructing local sections of the fibre bundle of microlocal solutions, 3) normalizing these solutions for the "flux norm" associated to the microlocal Wronskians, 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm, 5) from this we deduce Bohr-Sommerfeld (BS) quantization rules that hold precisely when the fibre bundle of microlocal solutions (depending on the energy parameter E) has trivial holonomy. Such a semi-classical treatement reveals interesting continuous symetries related to monodromy.Comment: IOP Conference series GROUP3

Bohr-Sommerfeld Quantization Rules Revisited: The Method of Positive Commutators

International audienceWe 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. 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

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian in the Theory of Supraconductivity. A more rigorous approach taking also into account tunneling corrections, will be discussed elsewhere

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian in the Theory of Supraconductivity. A more rigorous approach taking also into account tunneling corrections, will be discussed elsewhere

Hamiltonien de Bogoliubov-de Gennes en 1-D avec symétrie PT: règles de quantification généralisées de Bohr-Sommerfeld.

International audienceWe present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions; 2) constructing local sections of the fibre bundle of microlocal solutions; 3) normalizing these solutions for the "flux norm" associated to the microlocal Wronskians; 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm; 5) from this we deduce Bohr-Sommerfeld (BS) quantization rules that hold precisely when the fibre bundle of microlocal solutions (depending on the energy parameter E) has trivial holonomy. Such a semi-classical treatement reveals interesting continuous symetries related to monodromy. Details will appear elsewhere

Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator. The method of positive commutators

International audienceWe revisit in this Note 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.Dans le cadre algébrique et microlocal élaboré par Helffer et Sjöstrand, on propose une ré-écriture de la règle de quantification de Bohr-Sommerfeld pour un opérateur auto-adjoint h-Pseudo-différentiel 1-D; elle s'exprime par la non-inversibilité de la matrice de Gram d'un couple de solutions WKB dans une base convenable, pour le produit scalaire associé à la " norme de flux "

Bohr-Sommerfeld Quantization Rules Revisited: The Method of Positive Commutators

International audienceWe 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. 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

Bohr-Sommerfeld Quantization Rules Revisited: The Method of Positive Commutators

International audienceWe 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. 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