PT-Symmetric Schrödinger Operators with Unbounded Potentials

✍ Scribed by Jan Nesemann

• Publisher: Vieweg+Teubner
• Year: 2011
• Tongue: English
• Leaves: 92
• Edition: 2011
• Category: Library

✦ Synopsis

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 Schr?¶dinger operators. In the physical literature, the existence of Schr?¶dinger 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 - providing 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.

✦ Table of Contents

Cover......Page 1
PT-Symmetric
Schrödinger Operators
with Unbounded Potentials......Page 4
ISBN 9783834817624......Page 5
Acknowledgment......Page 6
Table of Contents......Page 8
Introduction......Page 10
1.1 Linear Operators in Krein Spaces......Page 18
1.2.1 Relatively Bounded and Relatively CompactOperators......Page 23
1.2.2 The Case of Relative Bound 0......Page 26
1.2.3 Stability of Self-Adjointness in Krein Spaces......Page 28
1.3.1 Continuity of Resolvents......Page 29
1.3.2 Perturbation of Isolated Parts of the Spectrum......Page 32
1.3.3 Perturbation of Spectra of Self-Adjoint Operatorsin Hilbert Spaces......Page 35
1.3.4 Perturbation of Spectra of Self-Adjoint Operatorsin Krein Spaces......Page 39
2.1.1 Accretive and Sectorial Operators......Page 44
2.1.2 Quadratic Forms and Associated Operators......Page 45
2.1.3 Relatively Form-Bounded andRelatively Form-Compact Operators......Page 51
2.2 Continuity of Separated Parts of theSpectrum......Page 55
2.2.1 Perturbation of Spectra of Self-Adjoint Operatorsin Hilbert Spaces......Page 56
2.2.2 Perturbation of Spectra of Self-Adjoint Operatorsin Krein Spaces......Page 62
2.3.1 Perturbation of Spectra of Self-Adjoint Operatorsin Krein Spaces......Page 68
3.1 Example 1......Page 74
3.2 Example 2......Page 78
3.3 A Class of Schrödinger Operators withRelatively Bounded Complex Potentialsand Real Spectrum......Page 81
Bibliography......Page 88