Browsing by Author "Lukić, Milivoje"
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Item Orthogonal rational functions with real poles, root asymptotics, and GMP matrices(American Mathematical Society, 2023) Eichinger, Benjamin; Lukić, Milivoje; Young, GiorgioThere is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for . We extend aspects of this theory in the setting of rational functions with poles on, obtaining a formulation which allows multiple poles and proving an invariance with respect to preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.Item Spectral analysis of Schrödinger operators with decaying distributional potentials(2024-04-05) Wang, Xingya; Lukić, MilivojeThe 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.Item Stahl-Totik regularity and exotic spectra of Dirac operators(2023-04-04) Gwaltney, Ethan; Lukić, MilivojeThis thesis motivates and presents three novel results in the spectral theory of one-dimensional Dirac operators, each of which concerns various forms of exotic or distinguished spectral characteristics. First, we consider the possibility of embedded eigenvalues in the absolutely continuous spectrum of a Dirac operator with operator data of Wigner-von Neumann type. Second, we demonstrate the genericity of Cantor spectrum when the operator data is chosen to be limit-periodic. Third, we provide for the Dirac operator setting an analogue of Stahl-Totik regularity, which, among other things, provides a lower bound on the thickness of the spectrum in terms of the operator data when the data is taken to be uniformly locally square integrable.