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In this chapter, we give complete spectral analysis of the Schrödinger operatorSchrödinger operator L(q) with an arbitrary locally square integrable periodic complex-valued potentialComplex-valued potential q by introducing new concepts and approaches. First, in the introduction section, we introduce the required notations and discuss the results of this chapter. Next, in Sect. 2.2, we study the Floquet solutions of the equation \(L(q)y={\lambda } y\) and consider the general property of the spectrum of L(q). In Sect. 2.3, we obtain asymptotic formulasAsymptotic formulas of arbitrary order for all Bloch eigenvaluesBloch eigenvalue and Bloch functionsBloch function. In Sect. 2.4, we consider the projections of the operator L(q) and investigate the spectral singularitiesSpectral singularities and essential spectral singularitiesEssential spectral singularity (ESS). Then, in Sect. 2.5, using the results of the Sects. 2.3 and 2.4, we construct the spectral expansion for L(q). In Sect. 2.6, we find necessary and sufficient condition for the elegant form of the spectral expansion. In Sect. 2.7, we find the conditions on the potential q for the asymptotic spectrality of L(q). Some calculations and estimations of this chapter are given in Sect. 2.8 (Appendices).

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Spectral Theory for the Schrödinger Operator with a Complex-Valued Periodic Potential

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  • 发表日期：

    2025年

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  • Oktay Veliev

摘要

In this chapter, we give complete spectral analysis of the Schrödinger operatorSchrödinger operator L(q) with an arbitrary locally square integrable periodic complex-valued potentialComplex-valued potential q by introducing new concepts and approaches. First, in the introduction section, we introduce the required notations and discuss the results of this chapter. Next, in Sect. 2.2, we study the Floquet solutions of the equation \(L(q)y={\lambda } y\) and consider the general property of the spectrum of L(q). In Sect. 2.3, we obtain asymptotic formulasAsymptotic formulas of arbitrary order for all Bloch eigenvaluesBloch eigenvalue and Bloch functionsBloch function. In Sect. 2.4, we consider the projections of the operator L(q) and investigate the spectral singularitiesSpectral singularities and essential spectral singularitiesEssential spectral singularity (ESS). Then, in Sect. 2.5, using the results of the Sects. 2.3 and 2.4, we construct the spectral expansion for L(q). In Sect. 2.6, we find necessary and sufficient condition for the elegant form of the spectral expansion. In Sect. 2.7, we find the conditions on the potential q for the asymptotic spectrality of L(q). Some calculations and estimations of this chapter are given in Sect. 2.8 (Appendices).