Spectral monodromy of non-self-adjoint operators

Abstract

In the present paper, we build a combinatorial invariant, called the “spectral monodromy” from the spectrum of a single (non-self-adjoint) h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self-adjoint h-pseudodifferential operators, given by S. Vu Ngoc [“Quantum monodromy in integrable systems,” Commun. Math. Phys. 203(2), 465–479 (1999)]. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex case we propose is a small perturbation of a self-adjoint operator with a classical integrability property. We show that the discrete spectrum (in a small band around the real axis) also has a combinatorial monodromy. The main difficulty in this case is that we do not know the description of the spectrum everywhere, but only in a Cantor type set. In addition, we also show that the corresponding monodromy can be identified with the classical monodromy, defined by J. Duistermaat [“On global action-angle coordinates,” Commun. Pure Appl. Math. 33(6), 687–706 (1980)].