Category Universite Paris 7

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On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

Aug 26, ’11 1:12 PM

D. Krejcirik, P. Siegl, J. Zelezny

We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

http://arxiv.org/abs/1108.4946
Spectral Theory (math.SP); Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)

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Perfect transmission scattering as a PT-symmetric spectral problem

Nov 19, ’10 8:14 AM

H. Hernandez-Coronado, D. Krejcirik, P. Siegl

Transmissions |T|^2 as a function of energy k2 for the step-like potential <i>v</i> with a = Pi/4, epsilon_1 = 0.2, epsilon_3 = 0.5, beta_3 = -100, beta_2 = 0, beta_1 = -120 (continuous red line), and beta_1 = -200 (dashed blue line). See [5] for animated plots of |T|^2 as a function of potential.We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the other hand, the scattering data can serve to describe the spectrum of a large class of Schroedinger operators with complex Robin boundary conditions.

http://arxiv.org/abs/1011.4281
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

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