Quantum bound states in open geometries

Avishai, Y.; Bessis, D.; Giraud, B. G.; Mantica, Giorgio

doi:10.1103/physrevb.44.8028

Cited by 118 publications

(107 citation statements)

References 11 publications

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“…Consequently, we know that −∆ Ω ′ D possesses at least one discrete eigenvalue below κ 2 1 due to Theorem 2. This is a non-trivial result for flat enough conical layers only, since using a trick analogous to that of [11] one can check that the cardinality of σ disc (−∆ …”

Section: Theoremmentioning

confidence: 99%

Topologically nontrivial quantum layers

Carron

Exner

Krejčiřı́k³

2004

Journal of Mathematical Physics

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Given a complete non-compact surface Σ embedded in R 3 , we consider the Dirichlet Laplacian in the layer Ω that is defined as a tubular neighbourhood of constant width about Σ. Using an intrinsic approach to the geometry of Ω, we generalise the spectral results of the original paper [1] by Duclos et al. to the situation when Σ does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain Ω.

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Section: Theoremmentioning

confidence: 99%

Topologically nontrivial quantum layers

Carron

Exner

Krejčiřı́k³

2004

Journal of Mathematical Physics

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“…In §2 we check that if H < H * (H * ∈ (1, 2] is some threshold), then the operator of the problem in the domain Ξ H has a unique eigenvalue μ H below the continuous spectrum [π 2 min{1, H −2 }, ∞); also, we study the properties of the function H → μ H . We mention the papers [10,11,12,13,14,15], where the spectra of broken, bent, and branching waveguides were investigated. In §3 we list all solutions of the homogeneous problem in Ξ 1 ξ that have at most exponential growth O(exp |ξ| π 2 − μ 1 at infinity.…”

Section: §1 Problem Settingmentioning

confidence: 99%

On the spectrum of the Laplace operator on the infinite Dirichlet ladder

Назаров

2012

St. Petersburg Math. J.

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Abstract. The spectrum of the Dirichlet problem is studied in the case of a periodic infinite planar domain having the form of an accommodation ladder: two parallel strips-uprights of thickness h > 0 are linked by treads of the same thickness. It is shown that, for h small, a gap is always opened between the second and third segments of the essential spectrum of the problem operator. The gap between the first and second segments is also discussed: its presence and characteristics depend on the distance between the uprights. It is shown that variation of the thickness of finitely many treads leads to the arising of any prescribed number of discrete spectrum points below the essential spectrum as well as inside the open gap. §1. Problem setting Let Γ be the union of the lines {x = (x 1 , x 2 ) ∈ R 2 : x 1 = ±l/2, x 2 ∈ R} and the infinite collection of segments γ p = {x : |x 1

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“…These waveguides are often called Y-junctions. Our intention is to investigate the existence of the non-trivial solutions of the Helmholtz equation: (1) −∆u(x) = λu(x), x ∈ Y, u(x) = 0, x ∈ ∂Y.…”

Section: Physical Backgroundmentioning

confidence: 99%

“…The same reasoning as in the earlier section ensures the non-emptiness of the discrete spectrum σ d (Y α R ). For example, in [1] and [14] the two-dimensional broken waveguides were studied with the varying angle. Altogether it was noticed that the multiplicity of the discrete spectrum increases infinitely as the angle approaches to zero.…”

Section: The Discrete Spectrum When the Angle Is Varyingmentioning

confidence: 99%

“…The existence of the eigenvalues in the discrete spectrum is studied in several types of waveguides. In the paper [1] it was shown with the help of the comparison principle that for any planar and broken waveguide there exists at least one eigenvalue in the discrete spectrum (see also [7]). In the paper [6], the non-emptiness of the discrete spectrum was verified for two and three dimensional bend tubes with equidistant width.…”

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confidence: 99%

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The bound states of 3D Y-junction waveguides

Uusitalo¹

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. The spectrum of the Dirichlet problem for Laplace operator is studied in three terminal tubes. The cross-section of the tubes is either a circle or a square. We show that these Y-shaped waveguides always have at least one eigenvalue in the discrete spectrum. In the regular case, that is, the angle between the cylinders is 2 3 π, there exists exactly one eigenvalue in the discrete spectrum. While the angle 2α between the arms is varying, we show that the number of the bound states remains to be one for α ∈ arctan 3 4 , π 2 when the cross-section of the tubes is square. However, when the angle becomes sharp enough, the number of eigenvalues in the discrete spectrum increases. Moreover, it is shown that the eigenvalues are monotonously increasing when the angle 2α is in the interval 0, 2 3 π and are monotonously decreasing when 2α ∈ 2 3 π, π . Physical backgroundThe development of nanotechnology has lead to smaller and smaller devices in electrical engineering. Over the last few decades there have been several both theoretical and applied studies to understand the physical behavior of such small nanometer sized waveguides.The nanometer scale devices have attracted due to their excellent attributes. They have light weight, but they are still strong because of their elasticity. Moreover, they have exceptional electrical properties. These waveguides have either metallic or semi-conducting behavior depending on their geometrical structure.These devices have a high purity and crystalline structure. Thus, the electron mean free path is greater than the diameter of the system. Therefore the scattering is unsubstantial and derives us to model the electron motion as a free particle in the infinite waveguide where the motion is limited inside the waveguide by posing the Dirichlet condition on the boundaries.In this study we concentrate on the three-dimensional and three terminal nanotubes with either rectangular or circular cross-section. These waveguides are often called Y-junctions. Our intention is to investigate the existence of the non-trivial solutions of the Helmholtz equation:In particular, those solutions which have the finite energy and the eigenvalue corresponding to such an eigenfunction is located below the continuous spectrum. The continuous spectrum is where the electron propagation can occur as the contrary to the discrete spectrum where the eigenfunction is localized.

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Quantum bound states in open geometries

Cited by 118 publications

References 11 publications

Topologically nontrivial quantum layers

Topologically nontrivial quantum layers

On the spectrum of the Laplace operator on the infinite Dirichlet ladder

The bound states of 3D Y-junction waveguides

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