PT-Symmetric Schrödinger Operators with Unbounded Potentials [electronic resource] /
Nesemann, Jan.author.
Following the pioneering work of Carl M. Bender et al. (1998), there has been an increasing interest in theoretical physics in so-called PT-symmetric Schrödinger operators. In the physical literature, the existence of Schrödinger operators with PT-symmetric complex potentials having real spectrum was considered a surprise and many examples of such potentials were studied in the sequel. From a mathematical point of view, however, this is no surprise at all – provided one is familiar with the theory of self-adjoint operators in Krein spaces. Jan Nesemann studies relatively bounded perturbations of self-adjoint operators in Krein spaces with real spectrum. The main results provide conditions which guarantee the spectrum of the perturbed operator to remain real. Similar results are established for relatively form-bounded perturbations and for pseudo-Friedrichs extensions. The author pays particular attention to the case when the unperturbed self-adjoint operator has infinitely many spectral gaps, either between eigenvalues or, more generally, between separated parts of the spectrum.
Description based upon print version of record.
Acknowledgment; Table of Contents; Introduction; Chapter 1 Relatively Bounded Perturbations inKrein Spaces; 1.1 Linear Operators in Krein Spaces; 1.2 Stability Theorems; 1.2.1 Relatively Bounded and Relatively CompactOperators; 1.2.2 The Case of Relative Bound 0; 1.2.3 Stability of Self-Adjointness in Krein Spaces; 1.3 Continuity of Separated Parts of theSpectrum; 1.3.1 Continuity of Resolvents; 1.3.2 Perturbation of Isolated Parts of the Spectrum; 1.3.3 Perturbation of Spectra of Self-Adjoint Operatorsin Hilbert Spaces; 1.3.4 Perturbation of Spectra of Self-Adjoint Operatorsin Krein Spaces
Chapter 2 Relatively Form-BoundedPerturbations in Krein Spaces2.1 Stability Theorems; 2.1.1 Accretive and Sectorial Operators; 2.1.2 Quadratic Forms and Associated Operators; 2.1.3 Relatively Form-Bounded andRelatively Form-Compact Operators; 2.2 Continuity of Separated Parts of theSpectrum; 2.2.1 Perturbation of Spectra of Self-Adjoint Operatorsin Hilbert Spaces; 2.2.2 Perturbation of Spectra of Self-Adjoint Operatorsin Krein Spaces; 2.3 Pseudo-Friedrichs Extensions; 2.3.1 Perturbation of Spectra of Self-Adjoint Operatorsin Krein Spaces; Chapter 3Examples; 3.1 Example 1; 3.2 Example 2
3.3 A Class of Schrödinger Operators withRelatively Bounded Complex Potentialsand Real SpectrumBibliography
Dissertation Universität Bern, 2010.
Includes bibliographical references.
Following the pioneering work of Carl M. Bender et al. (1998), there has been an increasing interest in theoretical physics in so-called PT-symmetric Schrödinger operators. In the physical literature, the existence of Schrödinger operators with PT-symmetric complex potentials having real spectrum was considered a surprise and many examples of such potentials were studied in the sequel. From a mathematical point of view, however, this is no surprise at all – provided one is familiar with the theory of self-adjoint operators in Krein spaces. Jan Nesemann studies relatively bounded perturbations of self-adjoint operators in Krein spaces with real spectrum. The main results provide conditions which guarantee the spectrum of the perturbed operator to remain real. Similar results are established for relatively form-bounded perturbations and for pseudo-Friedrichs extensions. The author pays particular attention to the case when the unperturbed self-adjoint operator has infinitely many spectral gaps, either between eigenvalues or, more generally, between separated parts of the spectrum.
Operator theory.
Mathematics.
Operator Theory.
Mathematics, general.