Quantum Waveguides

Exner, Pavel; Kovařík, Hynek

doi:10.1007/978-3-319-18576-7

Cited by 146 publications

(140 citation statements)

References 276 publications

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“…Schrödinger operators with δ-interactions are frequently used in mathematical physics to model interactions of quantum particles; we refer to the monographs [1] and [23], to the review article [18] and to [2, 3, 8, 16, 17, 19-22, 24, 25, 32, 33, 36, 37, 42] for a small selection of related papers on spectral analysis of such operators. Let A free be the usual self-adjoint realization of −∆ in L 2 (R n ) and let A δ,α be the self-adjoint operator with δ-interaction on C in (1).…”

Section: Introductionmentioning

confidence: 99%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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The self-adjoint Schrödinger operator A δ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L 2 (R n ). The aim of this note is to construct a boundary triple for S * and a self-adjoint parameter Θ δ,α in the boundary space L 2 (C) such that A δ,α corresponds to the boundary condition induced by Θ δ,α . As a consequence the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of A δ,α in terms of the Weyl function and Θ δ,α .

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Section: Introductionmentioning

confidence: 99%

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Behrndt¹,

Langer²,

Lotoreichik³

2016

Nanosystems: Phys. Chem. Math.

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“…Our present choice of the elementary illustrative example with compact and rectangular R changes the physics and becomes more intimately related to the problems of the so called quantum waveguides with impenetrable walls [30]. Most of the mathematical problems solved in the latter context are very close to the present ones.…”

Section: The Context Of Generalized Quantum Waveguidesmentioning

confidence: 83%

Markov constant and quantum instabilities

Pelantová

Starosta

Znojil

2016

J. Phys. A: Math. Theor.

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For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight in the phenomenologically relevant occurrence of anomalies in the spectra. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of certain hiddenly unitary quantum evolution models up to the clarification of the mechanisms of occurrence of ghosts in quantum cosmology.

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“…Further, the similarity between the wave-propagation Equation (2) in optics and the Schrödinger equation in quantum mechanics suggests a similarity between the guided transmission of waves in the guiding channel and propagation of real quantum particles in holding channel potentials [21]. The consideration of the transport of quantum particles in such channels gives rise to many intriguing peculiarities, such as the consideration of curved guiding channels.…”

Section: Linear Waveguidesmentioning

confidence: 99%

Editorial: Guided-Wave Optics

Malomed

2017

Applied Sciences

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Guided waves represent a vast class of phenomena in which the propagation of collective excitations in various media is steered in required directions by fixed (or, sometimes, reconfigurable) conduits. Arguably, the most well-known and practically important waveguides are single-mode and multi-mode optical fibers [1,2], including their more sophisticated version in the form of photonic crystal fibers [3] and hollow metallic structures transmitting microwave radiation [4]. Light pipes, in the form of hollow tubes with reflecting inner surfaces, are used in illumination techniques. On the other hand, medical stethoscopes offer a commonly known example of a practically important acoustic waveguide. New directions of studies in photonics are focused on waveguides for plasmonic waves on metallic surfaces [5][6][7] (which provide the possibility of using wavelengths much smaller than those corresponding to the traditional optical range, and thus offer opportunities to build much more compact photonic devices) and on the other hand, on the guided transmission of terahertz waves, which also have a great potential for applications [8].Outside of the realm of photonics (optics and plasmonics) and acoustics, wave propagation plays a profoundly important role in many other areas; accordingly, waveguiding settings have drawn a great deal of interest in those areas as well. In particular, as concerns hydrodynamics, natural waveguides-which may be very long-exist for internal waves propagating in stratified liquids (e.g., in the ocean) [9]. Various settings in the form of waveguides for matter waves are well known in studies of Bose-Einstein condensates in ultracold bosonic gases [10,11]. In solid-state physics, guided propagation regimes for magnon waves in ferromagnetic media are a subject of theoretical and experimental studies [12]. In superconductivity, long Josephson junctions are waveguides for plasma waves [8,13]. The significance of waveguiding in plasma physics is also well-known; e.g., Ref [14][15][16].Below, a very brief overview of basic theoretical models and experimental realizations of various physical implementations of the waveguiding phenomenology is given. The text is structured according to the character of the guided wave propagation: linear or nonlinear and conservative or dissipative, as well as according to the materials used in the underlying settings, natural or artificial.This presentation definitely does not aim to include an exhaustive bibliography on this vast research area. References are given chiefly to review articles and books summarizing the known results, rather than to original papers where the results were first published. However, in some cases original papers are also cited if it is necessary in the context of the presentation. Linear WaveguidesThe basic waveguiding structure is a single-mode conduit, designed with a sufficiently small transverse size and boundary conditions at the boundary between the guiding core and surrounding cladding, which admits the propagation of a single tr...

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Quantum Waveguides

Cited by 146 publications

References 276 publications

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Markov constant and quantum instabilities

Editorial: Guided-Wave Optics

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