On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $

Abstract

In this paper we investigate the non-self-adjoint operator H generated in L2(−∞, ∞) by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.

Spectral Analysis of the Schrodinger Operator with an Optical Potential

“…In paper [16] we found the explicit conditions on the potential q such that L(q) is an asymptotically spectral operator. In [24] we find a criterion for asymptotic spectrality of the operator L(q) with the potential (7) stated in term of a and b. Moreover, in [24], we obtained the following result Summary 3 If ab ∈ R, then the operator L(q) with the potential ( 7) is a spectral operator if and only if it is self adjoint.…”

“…In this section we introduce some notations, give the precise definitions of the spectral singularity and ESS and formulate some results of the papers [7,11,12,17,[19][20][21][22][23][24] as summaries which are necessary for this paper. Moreover, here we give some results for the operator L(V ) which immediately and readily follows from the results of those papers.…”

“…Thus to consider the spectral expansion it is necessary to investigate the ESS. In this paper we use the following results of [21,24] about ESS formulated here as summary.…”

“…First, in Section 2 we define the spectral singularities and ESS and give some results of the papers [7,11,12,17,[19][20][21][22][23][24] that are used in the next sections. Using it in Section 3 we consider the Bloch eigenvalues and spectrum of L(V ).…”

“…Then in Section 4 using the results of Section 3, we study the spectral singularity, ESS and investigate the spectral expansion of L(V ). Note that in the papers [21,23,24] we have proved that only the special type of the spectral singularities called as the essential spectral singularities (ESS) determines the form of the spectral expansion. Therefore in Section 4 we consider the spectral expansion by investigating the ESS of L(V ).…”