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 .

In this paper we study uniqueness properties of solutions to the Zakharov-Kuznetsov equation of plasma physic.Given two sufficiently regular solutions u 1 , u 2 , we prove that, if u 1´u2 decays fast enough at two distinct times, then u 1 " u 2 .The equation was introduced in the context of plasma physic by Zakharov and Kuznetsov in [38], where they formally deduced that the propagation of nonlinear ion-acoustic waves in magnetized plasma is governed by this mathematical model. A rigorous derivation of equation (1) was given by Lannes, Linares and Saut in [29].The problem of local and global well-posedness for the Cauchy problem associated to (1) has extensively been studied. Up to date the best local well-posedness result available in the literature was obtained independently by Molinet and Pilod [33] and Grünrock and Herr [16] for initial data in H s pR 2 q, s ą 1 2 . Then the global theory follows by standard arguments based on L 2 and H 1 conservation laws. We refer to [14,[30][31][32] and references therein for other results of this type and several additional remarks concerning with properties of this equation.Our main goal is to prove uniqueness properties from two distinct times for equation (1). More precisely we want to deduce sufficient conditions on the behavior of the difference u 1´u2 of two solutions u 1 , u 2 of (1) at two different times, t 0 " 0 and t 1 " 1, which guarantee that u 1 " u 2 . This kind of results is inspired to the program performed in [7][8][9][10][11][12] for Schrödinger and KdV (see also [37] and Remark 1.1 below for further details).The main motivation for our study is a recent work by Bustamante, Isaza and Mejía [4] where an upper bound for the possible decay at two different times of a non-trivial difference of two solutions of (1) was given. More precisely they prove the following: 1 ( [4] ). Suppose that for some small ε ą 0 u 1 , u 2 P C`r0, 1s; H 4 pR 2 q X L 2 pp1`x 2`y2 q 4 3`ε dxdyq˘X C 1`r 0, 1s; L 2 pR 2 q˘, are solutions of (1). Then there exists a universal constant a 0 ą 0, such that if for some a ą a 0 u 1 p0q´u 2 p0q, u 1 p1q´u 2 p1q P L 2 pe apx 2`y2 q 3{4 dxdyq, then u 1 " u 2 .

By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators.

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.