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

Ifa, Abdelwaheb

Rouleux, Michel

English

arXiv

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 " 

Rouleux, Michel, L.

HAL AMU

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

Rouleux, Michel,

HAL Descartes

Hal-Diderot

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

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

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HAL AMU

HAL Descartes

Hal-Diderot