Absence of eigenvalues of non‐self‐adjoint Robin Laplacians on the half‐space

Abstract

By developing the method of multipliers, we establish sufficient conditions which guarantee the total absence of eigenvalues of the Laplacian in the half‐space, subject to variable complex Robin boundary conditions. As a further application of this technique, uniform resolvent estimates are derived under the same assumptions on the potential. Some of the results are new even in the self‐adjoint setting, where we obtain quantum‐mechanically natural conditions.

Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers

“…The two-dimensional situation was covered later in [15]. The robustness of the method of multipliers has been demonstrated in its successful application to the half-space instead of the whole Euclidean space in [4] and to Lamé instead of Schrödinger operators in [3]. In the present paper, we push the analysis forward by investigating how the unconventional method provides meaningful and interesting results in the same direction also in the less explored setting of the spinorial Hamiltonians.…”

“…The purpose of this subsection is to provide, in a unified and rigorous way, the proof of the common crucial starting point of the series of works [3,4,15,16] for proving the absence of the point spectrum of the electromagnetic Hamiltonians H A,V in various settings.…”

“…Indeed, though on the one hand the unboundedness of the radial function φ does not pose any problems because the support of u is assumed to be compact at this step, on the other hand ∇ A u does not necessarily belong to D A . Following the strategy developed in [4], we replace (4.47) by its regularised version with N > 0 is the usual truncation function. After the second equality of (4.48) and in the sequel, we use the Einstein summation convention.…”

Localization of eigenvalues for non-self-adjoint Dirac and Klein-Gordon operators

“…Before moving to the main results of the present note, let us observe that despite the robustness of the Birman-Schwinger principle, it is not the only tool for obtaining spectral enclosures for non-self-adjoint operators. Indeed, another powerful technique which has been recently employed in various related problems is the method of multipliers (see, e.g., [18,19,6,7,5]).…”

Keller-type bounds for Dirac operators perturbed by rigid potentials

“…However, despite the robustness of the Birman-Schwinger principle, it is not the only tool one could use to obtain spectral enclosures for non-selfadjoint operators: another powerful technique is the method of multipliers, see e.g. [25,24,6,7,8].…”