Existence of complex eigenvalues near the isolated essential spectrum

Diomba Sambou (1)

1 IMB - Institut de Mathématiques de Bordeaux

We prove complex eigenvalue upper bounds and asymptotics for non-self-adjoint relatively compact perturbations of certain operators of mathematical physics. In particular, these asymptotics describe the distribution (the rate convergence) and prove the existence of the complex eigenvalues in a neighborhood of the essential spectrum of the operators. For instance, we apply our results to quantum Landau Hamiltonians $(-i\nabla - A)^{2} - b$ with constant magnetic field of strength $b>0$. We obtain the main asymptotic term of the complex eigenvalues counting function for an annulus centered at a Landau level $2bq$, $q \in \bn$. On this way, we prove that they are localized in certain sectors adjoining the Landau levels and that they accumulate to these thresholds asymptotically.

Mots clés: Non-self-adjoint relatively compact perturbations; self-adjoint operators; complex eigenvalues; Landau Hamiltonians; upper bounds; asymptotic expansions

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Diomba Sambou. Existence of complex eigenvalues near the isolated essential spectrum. 2014. ⟨hal-01053771v1⟩