Searcharxiv⌕ Search

arXiv subjects

A. Melin

Publications and source records attributed to A. Melin.

2 recordsLinked to original sources
arXiv2001

Determinants of pseudodifferential operators and complex deformations of phase space

Consider an h-pseudodifferential operator P, whose symbol extends holomorphically to a tubular neighborhood of the real phase space and converges sufficiently fast to 1, so that the determinant of P is well-defined. We show that the modulus of this determinant is asymptotically bounded by an exponential of the integral of the logarithm of the modulus of the symbol along a certain complex deformation of the real phase space. Since there are many possible such deformations, we get a variational problem. The paper is devoted to the corresponding variational calculus.

math.SP↗
arXiv2001

Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2

For a class of non-selfadjoint h-pseudodifferential operators in dimension 2, we determine all eigenvalues in an h-independent domain in the complex plane and show that they are given by a Bohr-Sommerfeld quantization condition. No complete integrability is assumed, and as a geometrical step in our proof, we get a KAM-type theorem (without small divisors) in the complex domain.

math.SP↗