Schrödinger Operators and Applications to Evolution Equations

Abstract

This thesis explores the spectral theory of Schr¨odinger operators and their applications to evolution equations. The primary focus is on the spectral decomposition ofunbounded self-adjoint operators on Hilbert spaces, with particular emphasis on theLaplacian and Schr¨odinger operators of the form −∆ + V , where V is a potentialfunction. The work begins with a detailed exposition of the spectral theorem forunbounded self-adjoint operators, utilizing tools from functional analysis, includingprojection-valued measures, the Cayley transform, and the theory of Banach C∗-algebras. The spectral theorem is then applied to Schr¨odinger operators, where thepotential V is assumed to satisfy various growth conditions, such as V (x) → ∞ as|x| → ∞ or V (x) → 0 as |x| → ∞.The thesis also investigates the essential self-adjointness of these operators and theirspectral properties, including the discrete and essential spectra. Specific examples,such as the quantum harmonic oscillator and the Coulomb potential, are analyzed indetail, with explicit calculations of eigenvalues and eigenfunctions provided. Thework further extends to the study of evolution equations, where the semigrouptheory is employed to solve initial value problems associated with PDEs involvingSchr¨odinger operators.In the final chapters, the thesis delves into the theory of spectral multipliers, whichare operators of the form m(T), where m is a bounded measurable function and Tis a self-adjoint operator. The boundedness of these multipliers on Lpspaces is established using techniques from Fourier analysis, including the Mikhlin-H¨ormandermultiplier theorem and the Calder´on-Zygmund decomposition. The results are applied to Schr¨odinger operators, providing sufficient conditions for the Lp-boundednessof spectral multipliers associated with these operators. Overall, this thesis bridges the gap between abstract spectral theory and concreteapplications in PDEs, offering a comprehensive treatment of the spectral propertiesof Schr¨odinger operators and their implications for solving evolution equations. Theresults presented have potential applications in quantum mechanics, mathematicalphysics, and the analysis of PDEs.