Spectral analysis of the Schrödinger operator with a PT-symmetric periodic optical potential

What is it about?

For the first time I constructed perturbation theory of the multidimensional Schrodinger operator with a periodic potential and obtained asymptotic formulas for the Bloch eigenvalues and Bloch functions. Then using this perturbation theory I proved the validity of the famous Bethe-Sommerfeld conjecture for arbitrary dimension and arbitrary lattice. This conjecture was formulated in 1928. My method is a first and unique (for the present) by which the validity of this conjecture for arbitrary lattice and dimension is proved. Moreover, using this perturbation theory, I constructively determined new spectral invariants and found the potential of the there- dimensional Schrödinger operator from the given Bloch eigenvalues. Besides, I constructed the spectral expansion of the ordinary nonself-adjoint differential operators with complex-valued periodic coefficients in all real line which appears to have been open since 1950 years. For this I suggested a new method for the spectral expansion and introduced new notions as essential spectral singularities and singular quasimomenta. Then I used these consideration for the investigations of the Schrodinger operator with PT-simmetric periodic potential

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Why is it important?

These investigations can be applied in Physics. See Monographs 1. Oktay Veliev, Multidimensional Periodic Schrödinger Operator (Perturbation Theory and Applications), Springer Cham Heidelberg New York Dordrecht London 2015. http://link.springer.com/book/10.1007/978-3-319-16643-8 2. Oktay Veliev, Multidimensional Periodic Schrödinger Operator (Perturbation Theory and Applications), Springer Cham, second edition 2019. https://www.springer.com/gp/book/9783030245771