Singular Bohr-Sommerfeld conditions for 1D Toeplitz operators: elliptic case

Yohann Le Floch (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this article, we state the Bohr-Sommerfeld conditions around a global minimum of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface, using an argument of normal form which is obtained thanks to Fourier integral operators. These conditions give an asymptotic expansion of the eigenvalues of the operator in a neighbourhood of fixed size of the singularity. We also recover the usual Bohr-Sommerfeld conditions away from the critical point. We end by investigating an example on the two-dimensional torus.

Mots clés: semiclassical analysis; spectral theory; toepliz operators

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Yohann Le Floch. Singular Bohr-Sommerfeld conditions for 1D Toeplitz operators: elliptic case. Communications in Partial Differential Equations, 2014, 39 (2), pp.213-243. ⟨10.1080/03605302.2013.845045⟩. ⟨hal-00736232⟩