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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 39 (11 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 36 (12 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.
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
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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 30 (16 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 22 (12 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 20 (4 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.
Heat Trace Asymptotics with Transmittal Boundary Conditions and Quantum Brane–world Scenario
, 2001
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SINGULAR PERTURBATIONS OF SELFADJOINT OPERATORS
, 2002
"... Abstract. Singular ¯nite rank perturbations of an unbounded selfadjoint operator A0 in a Hilbert space H0 are de¯ned formally as A(®) = A0 + G®G ¤,whereG is an injective linear mapping from H = C d to the scale space H¡k(A0), k 2 N, of generalized elements associated with the selfadjoint operator A ..."
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Cited by 12 (2 self)
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Abstract. Singular ¯nite rank perturbations of an unbounded selfadjoint operator A0 in a Hilbert space H0 are de¯ned formally as A(®) = A0 + G®G ¤,whereG is an injective linear mapping from H = C d to the scale space H¡k(A0), k 2 N, of generalized elements associated with the selfadjoint operator A0,andwhere ® is a selfadjoint operator in H. Thecasesk =1 and k = 2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k =2n>1 has been considered recently by various authors from a mathematical point of view. In this paper singular ¯nite rank perturbations A(®) in the general setting ran G H¡k(A0), k 2 N, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application singular perturbations of the Dirac operator are considered. 1.
Cantor and band spectra for periodic quantum graphs with magnetic fields
 Comm. Math. Phys
"... ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lya ..."
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Cited by 11 (3 self)
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ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable KronigPenney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the BetheSommerfeld conjecture fails in this case.