Existence of bound states in quantum waveguides under weak conditions

We apply results derived in other contexts for the spectrum of the Laplace operator in curved geometries to the study of an ideal polymer chain confined to a spherical annulus in arbitrary space dimension D and conclude that the free energy compared to its value for an uncurved box of the same thickness and volume, is lower when D < 3, stays the same when D = 3, and is higher when D > 3. Thus confining an ideal polymer chain to a cylindrical shell, lowers the effective bending elasticity of the walls, and might induce spontaneous symmetry breaking, i.e. bending. (Actually, the above mentioned results show that any shell in D = 3 induces this effect, except for a spherical shell). We compute the contribution of this effect to the bending rigidities in the Helfrich free energy expression.

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“…(11). Noticing that − ∂h(k) ∂k = O(k), and h(0) = π 2 12 , one can expand h using the expressions derived above:…”

Section: Effect On κ and κmentioning

confidence: 99%

Polymers in Curved Boxes

Yaman¹,

Pincus²,

Solis³

et al. 1997

Macromolecules

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“…The study was motivated by mesoscopic physics where a reasonable model for the dynamics of a particle in quantum waveguides is given by the Laplacian in hard-wall tubular neighbourhoods of infinite curves in R d , d = 2, 3 (quantum strips, tubes), or surfaces in R 3 (quantum layers); see [3,4] for the physical background and references. Nowadays, it is well known that any non-trivial curvature of the reference curve, that is asymptotically straight, produces bound states below the essential spectrum in the strips and tubes, [3,5,6].…”

Section: Introductionmentioning

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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“…Although this sufficient condition is very natural and in perfect accordance with the intuition, it is for the first time in this paper when the essential spectrum is localized without imposing any additional conditions (e.g., about the decay of the derivatives of curvature at infinity, cf [18,72,17]). The progress has become possible due to a general characterization of essential spectrum adopted from a paper by Y. Dermenjian et al [15] (cf our Lemma 5.1), which is for our purposes more suitable than the classical Weyl criterion.…”

Section: Scope Of the Papermentioning

confidence: 70%

“…Although the case of Dirichlet strips, i.e. ι = D, has already been considered in many works, our Theorem 4.1 represents a new result in this situation as well, since it is for the first time when the whole essential spectrum has been localized under a condition which does not contain derivatives of k. Some decay assumptions about the derivatives of the curvature were even required in order to localize the threshold inf σ ess (H D ) itself in the previous works, cf [18,72]. (An exception is the paper [58] where, however, only a lower bound on the threshold is given.)…”

Section: Theorem 41 (Essential Spectrum) Suppose H If the Strip Smentioning

confidence: 87%

On the Spectrum of Curved Planar Waveguides

Krejčiřı́k¹,

Křı́ž²

2005

Publ. Res. Inst. Math. Sci.

102

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The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound to the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.

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Existence of bound states in quantum waveguides under weak conditions

Cited by 53 publications

References 7 publications

Polymers in Curved Boxes

Polymers in Curved Boxes

Topologically nontrivial quantum layers

On the Spectrum of Curved Planar Waveguides

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