Reviews in Mathematical Physics

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schrödinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof rely on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [10].

Starting from the complete Mellin representation of Feynman amplitudes for noncommutative vulcanized scalar quantum field theory, introduced in a previous publication, we generalize to this theory the study of asymptotic behaviors under scaling of arbitrary subsets of external invariants of any Feynman amplitude. This is accomplished in both convergent and renormalized amplitudes.

We develop a mathematical theory of time operators of a Hamiltonian with purely discrete spectrum. The main results include boundedness, unboundedness and spectral properties of them. In addition, possible connections of a time operator of H with regular perturbation theory are discussed.

We study the spectral geometry of the quantum projective plane , a deformation of the complex projective plane ℂP², the simplest example of spin^c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0⁺-summable triple, equivariant under U_q(su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum.

We establish the existence of stationary states for the following nonlinear Dirac equation