On the existence of complex eigenvalues near the isolated essential spectrum
Résumé
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 $2bq$, $q \in \mathbb{N}$. On this way, we prove
that they are localized in certain sectors adjoining the Landau levels and that
they accumulate to these thresholds asymptotically.
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