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73
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 53 (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 37 (14 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.
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 27 (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.
Asymptotic Estimates for Bound States in Quantum Waveguides Coupled Laterally Through a Narrow Window
, 1995
"... . Consider the Laplacian in a straight planar strip of width d , with the Neumann boundary condition at a segment of length 2a of one of the boundaries, and Dirichlet otherwise. For small enough a this operator has a single eigenvalue ffl(a) ; we show that there are positive c 1 ; c 2 such that \G ..."
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Cited by 27 (6 self)
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. Consider the Laplacian in a straight planar strip of width d , with the Neumann boundary condition at a segment of length 2a of one of the boundaries, and Dirichlet otherwise. For small enough a this operator has a single eigenvalue ffl(a) ; we show that there are positive c 1 ; c 2 such that \Gammac 1 a 4 ffl(a) \Gamma (ß=d) 2 \Gammac 2 a 4 . An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length 2a in the common boundary. 1 Introduction Recent progress in "mesoscopic" physics brought not only new physical effects but also some interesting spectral problems. One of them concerns the existence of bound states which appear if a Dirichlet tube of a constant cross section is locally deformed, e.g., bent or protruded, or coupled to another tube  see [BGRS, ES, DE, SRW] and references therein. In this paper we are concerned with another system of this type, which consists of a pair of parallel Dirichlet strips cou...
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 18 (2 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.
Optimal Eigenvalues For Some Laplacians And Schrödinger Operators Depending On Curvature
 Proceedings of QMath7 (Prague
, 1998
"... We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fun ..."
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Cited by 17 (3 self)
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We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round. We also comment on the use of coordinate transformations for these operators and mention some open problems.
Boundstate asymptotic estimate for windowcoupled Dirichlet strips and layers
, 1997
"... We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel straight strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lo ..."
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Cited by 15 (4 self)
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We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel straight strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lower asymptotic bounds on the gap between the eigenvalue and the essential spectrum in the planar case, as well as for N = 1 in three dimensions. Based on these results, we formulate a conjecture on the weak–coupling asymptotic behaviour of such bound states. 1
INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS
, 2003
"... We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators ..."
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Cited by 12 (2 self)
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We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schrödinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.
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 12 (3 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.
Exponential bounds on curvatureinduced resonances in a twodimensional Dirichlet tube
 Helv. Phys. Acta
, 1998
"... Abstract. We consider curvature–induced resonances in a planar two–dimensional Dirichlet tube of a width d. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as d → 0+, provided the curvature of the strip axis satisfies certain analyticity ..."
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Cited by 12 (1 self)
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Abstract. We consider curvature–induced resonances in a planar two–dimensional Dirichlet tube of a width d. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as d → 0+, provided the curvature of the strip axis satisfies certain analyticity and decay requirements. 1