It is well known that the formal Aharonov-Bohm Hamiltonian operator, describing the interaction of a charged particle with a magnetic vortex, has a four-parameter family of self-adjoint extensions, which reduces to a two-parameter family if one requires that the Hamiltonian commutes with the angular momentum operator. The question we study here is which of these self-adjoint extensions can be c...

Let X = Γ\G/K be a compact locally symmetric space. In this paper we establish a version of the Selberg trace formula for non-unitary representations of the lattice Γ. On the spectral side appears the spectrum of the “flat Laplacian” ∆, acting in the space of sections of the associated flat bundle. In general, this is a non-self-adjoint operator.

What is known today as the Lee-Friedrichs model is characterized by a self-adjoint operator H on a Hilbert space H, which is the sum of two self-adjoint operators H0 and V , such that H,H0 and V have common domain; H0 has absolutely continuous spectrum (of uniform multiplicity) except for the end-point of the semi-bounded from below spectrum, and one or more eigenvalues which may or may not be ...

We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each ch...

LetA be a ν-vector of self-adjoint, pairwise commuting operators andB a bounded operator of classCn0 (A). We prove a Taylor-like expansion of the commutator [B, f(A)] for a large class of functions f : Rν → R, generalising the one-dimensional result where A is just a self-adjoint operator. This is done using almost analytic extensions and the higher-dimensional Helffer-Sjöstrand formula.

We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each ch...

In this paper we consider generalized eigenvalue problems for a family of operators with a polynomial dependence on a complex parameter. This problem is equivalent to a genuine non self-adjoint operator. We discuss here existence of non trivial eigenstates for models coming from analytic theory of smoothness for P.D.E. We shall review some old results and present recent improvements on this sub...

A fairly general class of nonlinear evolution equations with a self-adjoint or non self-adjoint linear operator is considered, and a family of approximate inertial manifolds (AIMs) is constructed. Two cases are considered: when the spectral gap condition (SGC) is not satisfied and an exact inertial manifold is not known to exist the construction is such that the AIMs have exponential order, whi...

A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle, where the velocity profile is assumed to be monotonic and analytic. It is shown that unstable eigenvalues of Rayleigh's equation (which is a non-self-adjoint eigenvalue problem) can be associated with positive eigenvalues of a certain self-adjoint ...