Optimization of the lowest eigenvalue of a soft quantum ring

Exner, Pavel; Lotoreichik, Vladimir

doi:10.1007/s11005-021-01369-2

Cited by 8 publications

(6 citation statements)

References 31 publications

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“…While the two situations are of primary interest to us, they are particular cases of a more general channel profile. In the spirit of [EL21] one can consider the operator (e) H Γ,µ associated with the quadratic form…”

Section: Problem Settingmentioning

confidence: 99%

“…Recently an alternative, more realistic model of 'soft' waveguides attracted attention, see [Ex20] and further developments in [KKK21,EL21,EKL22,EV23], where the interaction term is a regular potential 'ditch' the axis of which is a fixed curve. In such a description the tunneling between different parts of the guide is not suppressed while its transverse width need not be zero.…”

Section: Introductionmentioning

confidence: 99%

“…The above two questions remained open again, and goal of this paper is to address them in the less restrictive geometrical setting of [EV23] where the 'spine' of the book may have an arbitrary shape provided it is finite and sufficiently regular. As in [EL21] we characterize the ditch profile by a measure type potential which allows us to treat on the same footing regular and singular interactions; recall that for the latter the said two questions are also open.…”

Section: Introductionmentioning

confidence: 99%

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Tunneling in soft waveguides: closing a book

Exner,

Spitzkopf

2024

J. Phys. A: Math. Theor.

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We investigate the spectrum of a soft quantum waveguide in two dimensions of the generalized `bookcover' shape, that is, Schr"odinger operator with the potential in the form of a ditch consisting of a finite curved part and straight asymptotes which are parallel or almost parallel pointing in the same direction. We show how the eigenvalues accumulate when the angle between the asymptotes tends to zero. In case of parallel asymptotes the existence of a discrete spectrum depends on the ditch profile. We prove that it is absent in the weak-coupling case, on the other hand, it exists provided the transverse potential is strong enough. We also present a numerical example in which the critical strength can be assessed.

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Section: Problem Settingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tunneling in soft waveguides: closing a book

Exner,

Spitzkopf

2024

J. Phys. A: Math. Theor.

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“…In the mentioned paper it was shown that two-dimensional soft waveguides share an important property with their more idealized counterparts, namely that a bend of the potential channel can give rise to bound states of such a Hamiltonian, although the existence condition derived in [Ex20] lacks the universality known in the said two models. Other recent results concern the existence of a discrete spectrum in the example of a 'bookcover' guide [KKK21] and a spectral optimization for ring-shaped soft guides [EL21]; related results can be also found in [EKP20,WT14].…”

Section: Introductionmentioning

confidence: 96%

Soft quantum waveguides in three dimensions

Exner

2021

Preprint

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We discuss a three-dimensional soft quantum waveguide, in other words, Schrödinger operator in R 3 with an attractive potential supported by an infinite tube and keeping its transverse profile fixed. We show that if the tube is asymptotically straight, the distance between its ends is unbounded, and its twist satisfies the so-called Tang condition, the esential spectrum is not affected by smooth bends. Furthermore, we derive a sufficient condition, expressed in terms of the tube geometry, for the discrete spectrum of such an operator to be nonempty.

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“…The results obtained there concern mainly the existence of the discrete spectrum points below the threshold of the essential spectrum. For example, it has been proved in [12,14,23,25] that the local bending acts as an attractive potential inducing a new spectrum below the essential threshold. In [10], it has been shown that for a weak deformation the existence of the discrete spectrum points depends on the total area of deformation which is assumed to be positive if the deformation is localized on Σ \ Ω and negative if the deformation is situated on Ω \ Σ.…”

Section: Introductionmentioning

confidence: 99%

Quantum soft waveguides with resonances induced by broken symmetry

Kondej

2024

J. Phys. A: Math. Theor.

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We consider two-dimensional, non-relativistic quantum system with asymptotically straight soft waveguide. We show that the local deformation of the symmetric waveguide can lead to the emerging of the embedded eigenvalues living in the continuous spectrum. The main problem of this paper is devoted to the analysis of weak perturbation of the symmetric system. We show that the original embedded eigenvalues turn to the second sheet of the resolvent analytic continuation and constitute resonances. We describe the asymptotics of the real and imaginary components of the complex resonant pole depending on deformation. Finally, we generalize the problem to three dimensional system equipped with a soft layer.

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Optimization of the lowest eigenvalue of a soft quantum ring

Cited by 8 publications

References 31 publications

Tunneling in soft waveguides: closing a book

Tunneling in soft waveguides: closing a book

Soft quantum waveguides in three dimensions

Quantum soft waveguides with resonances induced by broken symmetry

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