Multidimensional Periodic Schrödinger Operator

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Focusing on the multidimensional Schrödinger operator with a periodic potential, this book delves into both direct and inverse problems central to perturbation theory and spectral analysis. It meticulously derives asymptotic formulas for Bloch eigenvalues and functions across dimensions, while also estimating iso-energetic surfaces in high-energy scenarios. A novel method validates the Bethe-Sommerfeld conjecture and introduces an algorithm for uniquely determining the potential from Bloch eigenvalues, with some spectral invariants expressed through Fourier coefficients of the potential.