Resolvent Estimates and Bounds on Eigenvalues for Schrödinger and Dirac Operators
Alexandra Enblom · 2018
This thesis concerns the spectral theory of Schrödinger and Dirac operators.The main results relate to the problems of estimating perturbed eigenvalues.The thesis is based on four papers.The first paper focuses on the problem of localization of perturbed eigenvalues for multidimensional Schrödinger operators.Bounds for eigenvalues, lying outside the essential spectrum, are obtained in terms of the Lebesgue's classes.The methods used make it possible to consider the general case of non-self-adjoint operators, and involve the weak Lebesgue's potentials.The results are extended to the case of the polyharmonic operators.In the second paper, the problem of location of the discrete spectrum is solved for the class of Schrödinger operators considered on the half-line.The general case of complex-valued potentials, imposing various boundary conditions, typically Dirichlet and Neumann conditions, is considered.General mixed boundary conditions are also treated.The third paper is devoted to Dirac operators.The case of spherically symmetric potentials is considered.Estimates for the eigenvalues are derived from the asymptotic behaviour of the resolvent of the free Dirac operator.For the massless Dirac operators, whose essential spectrum is the whole real line, optimal bounds for the imaginary part of the eigenvalues are established.In the fourth paper, new Hardy-Carleman type inequalities for Dirac operators are proven.Concrete Carleman type inequalities, useful in applications, Agmon and also Treve type inequalities are derived from the general results by involving special weight functions.The results are extended to the case of the Dirac operator describing the relativistic particle in a potential magnetic field.i I would like to express my gratitude to my supervisor Ari Laptev.Thank you for introducing me to this interesting scientific topic, for all the useful discussions and for giving me such an opportunity, as to work in Linköping University.It was also an adventure to be a part of your journey and meet you in various places.I would also want to thank Vladimir Kozlov for always being near and helping me with everything I had problems with.Department of Mathematics at LIU is a very nice place to work at and therefore I would like to thank all the colleagues, the administration and all the PhD students for making it more enjoyable.Finally, I would like to thank my parents, my dad for teaching me to joke, my mom for being my best friend and my sister for always being there for me.