Lukić, Milivoje2024-05-212024-05-212024-052024-04-05May 2024Wang, Xingya. Spectral analysis of Schr�dinger operators with decaying distributional potentials. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/116141https://hdl.handle.net/1911/116141The primary theme of this thesis is to extend various classical techniques and spectral results regarding 1-dimensional Schrödinger operators with locally integrable potentials to the more general setting of distributional potentials which are locally in the Sobolev space H^{-1}. We will start by reviewing the classical spectral theoretical framework along with relevant results obtained therein. Next, we proceed to establish the corresponding framework in the distributional setting, and recover Last–Simon-type description of the absolutely continuous spectrum and sufficient conditions for different spectral types. In the last chapter, we focus on potentials which are decaying in a locally H^{−1} sense. In particular, we prove a spectral transition between short-range and long-range in the class of sparse distributional potentials, and we establish WKB-type asymptotic behavior of eigenfunctions and spectral properties for locally H^{−1} potentials whose decay rate is between L^1 and L^2.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Schrödinger operatorsdistributional potentialsspectral typesdecaying potentialsThesis2024-05-21