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226
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
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A Kreinlike formula for singular perturbations of selfadjoint operators and applications
"... Given a selfadjoint operator A: D(A) ⊆ H → H and a continuous linear operator τ: D(A) → X with Range τ ′ ∩ H ′ = {0}, X a Banach space, we explicitly construct a family A τ Θ of selfadjoint operators such that any Aτ Θ coincides with the original A on the kernel of τ. Such a family is obtained ..."
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Cited by 52 (16 self)
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Given a selfadjoint operator A: D(A) ⊆ H → H and a continuous linear operator τ: D(A) → X with Range τ ′ ∩ H ′ = {0}, X a Banach space, we explicitly construct a family A τ Θ of selfadjoint operators such that any Aτ Θ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreĭnlike formula where the role of the deficiency spaces is played by the dual pair (X, X ′); the parameter Θ belongs to the space of symmetric operators from X ′ to X. When X = C one recovers the “ H−2construction” of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which H = L 2 (R n) and τ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudodifferential operators, thus unifying and extending previously known results. 1.
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 51 (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 Spectrum in Curved Leaky Wires
 J. Phys. A34
, 2001
"... Introduction The aim of the present paper is to elucidate some geometrically induced spectral properties for the Laplacian in L 2 (R 2 ) perturbed by a negative multiple of the Dirac measure of an infinite curve \Gamma in the plane. This problem has at least two motivations. On the physics side ..."
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Cited by 43 (17 self)
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Introduction The aim of the present paper is to elucidate some geometrically induced spectral properties for the Laplacian in L 2 (R 2 ) perturbed by a negative multiple of the Dirac measure of an infinite curve \Gamma in the plane. This problem has at least two motivations. On the physics side we note that quantum mechanics of electrons confined to narrow tubelike regions has attracted a considerable interest, because such systems represent a natural model for semiconductor "quantum wires". In some examples the region in question is a strip or tube with hard walls  see, e.g., [DE] and references therein  while other treatments assume even stronger localization to a curve 1 or a graph  a rich bibliography to such models can be found in [KS]. Various interesting spectral effects were found in such a setting related to the geometry and topology of the underlying restricted configuration space. One of th
On the global existence of Bohmian mechanics
 Comm. Math. Phys
, 1995
"... Abstract. We show that the particle motion in Bohmian mechanics, given by the solution of an ordinary differential equation, exists globally: For a large class of potentials the singularities of the velocity field and infinity will not be reached in finite time for typical initial values. A substant ..."
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Cited by 30 (14 self)
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Abstract. We show that the particle motion in Bohmian mechanics, given by the solution of an ordinary differential equation, exists globally: For a large class of potentials the singularities of the velocity field and infinity will not be reached in finite time for typical initial values. A substantial part of the analysis is based on the probabilistic significance of the quantum flux. We elucidate the connection between the conditions necessary for global existence and the selfadjointness of the Schrödinger Hamiltonian.
Spectra of Schrödinger operators on equilateral quantum graphs
, 2006
"... We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum g ..."
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Cited by 28 (7 self)
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We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum graph is the preimage of the combinatorial spectrum under a certain entire function. Using this correspondence we show that that the number of gaps in the spectrum of the Schrödinger operators admits an estimate from below in terms of the Hill operator independently of the graph structure.
Resolvents of selfadjoint extensions with mixed boundary conditions
, 2006
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Sergii: JSelfAdjoint Operators with CSymmetries: Extension Theory Approach
"... gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Oktober 2008 Jselfadjoint operators with Csymmetries: extension theory approach ..."
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Cited by 17 (5 self)
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gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Oktober 2008 Jselfadjoint operators with Csymmetries: extension theory approach
Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential
, 2008
"... We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overco ..."
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Cited by 17 (5 self)
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We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standingwave solution is stable in H 1 rad (R) and unstable in H 1 (R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finitewidth instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finitewidth instability or a blowup instability.