Solvable Models in Quantum Mechanics

Albeverio, Sergio; Gesztesy, Fritz; Høegh-Krohn, Raphaël; Holden, Helge

doi:10.1007/978-3-642-88201-2

Cited by 1,416 publications

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“…[28]), the self-adjoint extension of operator h is defined on the domain of functions of the form f nc,s (r) g nc,s (r) = ξ 0 (r) + c ξ + (r) − e −iΘ ξ − (r) , (B.5)…”

Section: Discussionmentioning

confidence: 99%

Electronic properties of graphene with a topological defect

Sitenko

Власій

2007

Nuclear Physics B

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Various types of topological defects in graphene are considered in the framework of the continuum model for long-wavelength electronic excitations, which is based on the Dirac-Weyl equation. The condition for the electronic wave function is specified, and we show that a topological defect can be presented as a pseudomagnetic vortex at the apex of a graphitic nanocone; the flux of the vortex is related to the deficit angle of the cone. The cases of all possible types of pentagonal defects, as well as several types of heptagonal defects (with the numbers of heptagons up to three, and six), are analyzed. The density of states and the ground state charge are determined.

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“…[28]), the self-adjoint extension of operator h is defined on the domain of functions of the form f nc,s (r) g nc,s (r) = ξ 0 (r) + c ξ + (r) − e −iΘ ξ − (r) , (B.5)…”

Section: Discussionmentioning

confidence: 99%

Electronic properties of graphene with a topological defect

Sitenko

Власій

2007

Nuclear Physics B

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“…In Appendix 2 we prove that the above boundary condition is admissible. It is also possible to demonstrate that the above boundary condition reduces to the one given by [20], page 53, equation 2.1.9. (It may be necessary to use the technique described in [5], Appendix B).…”

Section: Example 13) the Operator Momentummentioning

confidence: 84%

“…The physical significance of this boundary condition is that, in addition to the Coulomb interaction, we have an additional point interaction [20], with zero range, at the origin. At this point it is necessary to note that the modulus square of the wave function is a probability density, and so it can assume infinite values, at isolated points, if its integral, the probability, is finite.…”

Section: Example 13) the Operator Momentummentioning

confidence: 99%

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The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

2008

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We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems -the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone´s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the timedependent Schrödinger equation.

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“…In particular, the quantum mechanical solvable models describing a particle moving in a local singular potential concentrated at one or a discrete number of points have been extensively discussed, see e.g. [6,7,8] and references therein. One dimensional problems with point interactions at, say, the origin (x = 0) can be characterized by the boundary conditions imposed on the wave function at x = 0, which is equivalent to two particles with contact interactions.…”

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confidence: 99%

Integrability and PT-Symmetry of N-Body Systems with Spin-Coupling -Interactions

Fei

2003

Czechoslovak Journal of Physics

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We study the PT-symmetric boundary conditions for "spin"-related δ-interactions and the corresponding integrability for both bosonic and fermionic many-body systems. The spectra and bound states are discussed in detail for spin-1 2 particle systems. Integrable models play significant roles in statistical and condensed matter physics. Many of these models can be exactly solved in terms of an algebraic or coordinate "Bethe Ansatz method"[1], see e.g., [2,3] for spin chain and ladder models and [4,5] for (continuous variable) quantum mechanical models. In particular, the quantum mechanical solvable models describing a particle moving in a local singular potential concentrated at one or a discrete number of points have been extensively discussed, see e.g. [6,7,8] and references therein. One dimensional problems with point interactions at, say, the origin (x = 0) can be characterized by the boundary conditions imposed on the wave function at x = 0, which is equivalent to two particles with contact interactions. The integrability of one dimensional quantum mechanical many-body problems with general contact interactions between two particles, has been studied in [9] and the bound states and scattering matrices are calculated for both bosonic and fermionic statistics. The results are generalized to the case of quantum mechanical systems with "spin"-related contact interactions, namely, the boundary conditions describing the contact interactions are dependent on the spin states of the particles [10].Recently, the complex generalization of conventional quantum mechanics has been investigated [11]. In stead of the standard formulation of quantum mechanics in terms of HermitianHamiltonians, quantum mechanical models with space-time reflection symmetry (PT symmetry) have been constructed and studied for continuous interaction potentials [12]. For point interaction potentials a systematic description of the boundary conditions and the spectra properties for self-adjoint, PT-symmetric systems and systems with real spectra have been presented, and the corresponding integrability of one dimensional many body systems with these kinds of point 1

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Solvable Models in Quantum Mechanics

Abstract: Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.

Cited by 1,416 publications

References 0 publications

Electronic properties of graphene with a topological defect

Electronic properties of graphene with a topological defect

The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

Integrability and PT-Symmetry of N-Body Systems with Spin-Coupling -Interactions

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