Uniform resolvent estimates and absence of eigenvalues for Lamé operators with complex potentials

Abstract

We consider the 0-order perturbed Lamé operator −∆ * +V (x). It is well known that if one considers the free case, namely V = 0, the spectrum of −∆ * is purely continuous and coincides with the non-negative semi-axis. The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lamé operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about V .

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.…”

“…Let A ∈ L 2 loc (R 3 ; R 3 ) be such that B ∈ L 2 loc (R 3 ; R 3 ). Suppose that V ∈ L 1 loc (R 3 ; C 2×2 ) admits the decomposition V = V (1) (2) with components V (1) ∈ L 1 loc (R 3 ) and V (2) = V (2) I C 2 , where V (2) ∈ L 1 loc (R 3 ) is such that [∂ r (r Re V (2) )] + ∈ L 1 loc (R 3 ) and r V (1) , r (Re V (2) ) − , r Im V (2) ∈ L 2 loc (R 3 ). such that, for all two-vector u with components in C ∞ 0 (R 3 ), the inequalities…”

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

“…Concerning the Schrödinger equation we should quote [14,16,20,21] and for the Dirac equation [9,10]. Moreover, we mention [15] for an adaptation of this method to an elasticity setting.…”

“…Moreover, in [20] the two-dimensional electromagnetic Schrödinger operators has been covered by the same authors. We also mention [15] where an adaptation of this method to an elasticity setting was performed by the first author.…”

“…The origin of the present work was to continue with the research of [15,20,21] by investigating the robustness of the method of multipliers in the setting of Schrödinger operators acting in subdomains of the Euclidean space. It turns out that the generalization is not obvious due to the presence of boundaries, at least in the case of complex potentials.…”