We consider the $0$-order perturbed Lamé operator $-\Delta^\ast + V(x)$. It is well known that if one considers the free case, namely $V=0,$ the spectrum of $-\Delta^\ast$ 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 $V$.

Uniform resolvent estimates and absence of eigenvalues for Lamé operators with complex potentials / Cossetti, Lucrezia. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - STAMPA. - 455:I(2017), pp. 336-360. [10.1016/j.jmaa.2017.05.063]

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

Lucrezia Cossetti
2017

Abstract

We consider the $0$-order perturbed Lamé operator $-\Delta^\ast + V(x)$. It is well known that if one considers the free case, namely $V=0,$ the spectrum of $-\Delta^\ast$ 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 $V$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1016363
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