1-D Schrödinger operators with δ and δ′ interactions on non-discrete sets and complex problems (2024)
This dissertation explores the spectral properties of one-dimensional Schrödinger operators with δ and δ′ interactions on non-discrete sets. We extend classical Sturm-Liouville theory to these singular perturbations and analyze the self-adjoint realizations of differential expressions involving Dirac delta distributions and their derivatives. Using Weyl's classification, we establish criteria for the limit-point and limit-circle cases in complex Sturm-Liouville problems, answering fundamental questions about spectral properties in non-Hermitian quantum mechanics. We prove the absence of embedded eigenvalues in certain cases and extend Levinson’s theorem for Borel measures. The findings have implications for quantum mechanics and mathematical physics, particularly in understanding spectral phenomena in PT-symmetric systems.