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102
Weakly coupled bound states in quantum waveguides
 PROC. AM. MATH. SOC
, 1997
"... We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain Ω obtained by adding an arbitrarily small “bump ” to the tube Ω0 = R × ..."
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Cited by 56 (0 self)
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We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain Ω obtained by adding an arbitrarily small “bump ” to the tube Ω0 = R × (0, 1) (i.e., Ω � Ω0, Ω ⊂ R2 open and connected, Ω =Ω0 outside a bounded region) produces at least one positive eigenvalue below the essential spectrum [π 2, ∞) of the Dirichlet Laplacian − ∆ D Ω.ForΩ\Ω0  sufficiently small ( .  abbreviating Lebesgue measure), we prove uniqueness of the ground state EΩ of − ∆ D Ω and derive the “weak coupling” result EΩ = π 2 − π 4 Ω\Ω0  2 + O(Ω\Ω0  3). As a corollary of these results we obtain the following surprising fact: Starting from the tube Ω0 with Dirichlet boundary conditions at ∂Ω0, replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment (a, b) ×{1}, a<bof ∂Ω0. If H(a, b) denotes the resulting Laplace operator in L²(Ω0), then H(a, b) has a discrete eigenvalue in [π 2 /4,π 2) no matter how small b − a > 0 is.
Convergence of spectra of graphlike thin manifolds
 J. Geom. Phys
"... Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at th ..."
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Cited by 50 (13 self)
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Abstract. We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the LaplaceBeltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices. 1.
Geometrically Induced Discrete Spectrum in Curved Tubes
 Differential Geom. Appl
, 2005
"... The Dirichlet Laplacian in curved tubes of arbitrary crosssection rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a ..."
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Cited by 38 (3 self)
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The Dirichlet Laplacian in curved tubes of arbitrary crosssection rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincides with the spectrum of the straight tube of the same crosssection and that the discrete spectrum is not empty. MSC2000: 81Q10; 58J50; 53A04.
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
 Journal of Physics A: Mathematical and General
"... Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) ..."
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Cited by 33 (7 self)
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Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature. 1.
A Vugalter: Asymptotic Estimates for Bound States in Quantum Waveguide Coupled laterally through a boundary window
, 1996
"... ..."
Bound States in Curved Quantum Layers
 Comm. Math. Phys
"... We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a noncompact surface embedded in R 3 . We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hardwall boundary. Under the assumption that the s ..."
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Cited by 29 (8 self)
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We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a noncompact surface embedded in R 3 . We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hardwall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of geometrically induced bound states. KeyWords: waveguides, layers, constrained systems, Dirichlet Laplacian, bound states, surface geometry, curvature, integral curvatures, geodesic polar coordinates 1 Introduction Relations between the geometry of a region\Omega in R n , boundary conditions at @ and spectral properties of the corresponding Laplacian are one of the vintage problems of mathematical physics. Recent years brought new motivations and focused attention to aspects of the problem which attracted little attention earlier. A strong impetus comes from mesoscopic physics, where new e...
A Hardy inequality in twisted waveguides
, 2008
"... We show that twisting of an infinite straight threedimensional tube with noncircular crosssection gives rise to a Hardytype inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. ..."
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Cited by 26 (0 self)
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We show that twisting of an infinite straight threedimensional tube with noncircular crosssection gives rise to a Hardytype inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary crosssections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a nonempty discrete spectrum. 1 1
Coupling in the singular limit of thin quantum waveguides
 J. Math. Phys
"... Abstract. We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by ..."
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Cited by 22 (5 self)
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Abstract. We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e. a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a non trivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling. 1.
Stability of the magnetic Schrödinger operator in a waveguide
 Comm. Partial Differential Equations
"... The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum [BGRS]. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum [DE]. ..."
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Cited by 20 (1 self)
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The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum [BGRS]. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum [DE]. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardytype inequality in the waveguide, which is interesting in its own. 1