Abstract
<abstract><p>Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity $ -\Delta^\ast + V $ in terms of $ L^p $-norms of the potential. Original results within the self-adjoint framework are provided too.</p></abstract>
Highlights
This paper is devoted to providing bounds on the location of discrete eigenvalues of operators of the form−∆∗ + V acting on [L2(Rd)]d, the Hilbert space of vector fields with components in L2(Rd)
We address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity −∆∗ + V in terms of Lp-norms of the potential
−∆∗ denotes the Lamé operator of elasticity, that is a linear, symmetric, differential operator of second order that acts on smooth L2-vector fields u = (u1, u2, . . . , ud) on Rd, say [C0∞(Rd)]d, in this way:
Summary
−∆∗ + V acting on [L2(Rd)]d, the Hilbert space of vector fields with components in L2(Rd). The main aim of our paper is to investigate on spectral properties in the elasticity setting, providing bounds on the distribution of discrete (possibly complex) eigenvalues of Lamé operators (1) in terms of Lp-norm of the potential. To the much more investigated Schrödinger operator, up to our knowledge, eigenvalue bounds of the form (3) for the perturbed Lamé operator are unknown even in the self-adjoint situation In this case the proof of (3) follows almost verbatim the instead-well-known one for Schrödinger, we decided to dedicate Section 3 to prove it anyhow. The notation T E→F will be used to denote the operator norm of T
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