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153
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 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.
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 32 (13 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
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
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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 20 (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 (5 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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The Wave Equation with One Point Interaction and the (Linearized) Classical Electrodynamics of a Point Particle
 Ann. Inst. Henri Poincar'e
"... . We study the point limit of the linearized MaxwellLorentz equations describing the interaction, in the dipole approximation, of an extended charged particle with the electromagnetic field. We find that this problem perfectly fits into the framework of singular perturbations of the Laplacian; ind ..."
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Cited by 11 (7 self)
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. We study the point limit of the linearized MaxwellLorentz equations describing the interaction, in the dipole approximation, of an extended charged particle with the electromagnetic field. We find that this problem perfectly fits into the framework of singular perturbations of the Laplacian; indeed we prove that the solutions of the MaxwellLorentz equations converge  after an infinite mass renormalization which is necessary in order to obtain a non trivial limit dynamics  to the solutions of the abstract wave equation defined by the selfadjoint operator describing the Laplacian with a singular perturbation at one point. The elements in the corresponding form domain have a natural decomposition into a regular part and a singular one, the singular subspace being threedimensional. We obtain that this threedimensional subspace is nothing but the velocity particle space, the particle dynamics being therefore completely determined  in an explicit way  by the behaviour of ...
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.
On Inverse Spectral Theory for SelfAdjoint Extensions: Mixed Types of Spectra
 J. Funct. Anal
, 1996
"... Let H be a symmetric operator in a separable Hilbert space H. Suppose that H has some gap J . We shall investigate the question about what spectral properties the selfadjoint extensions of H can have inside the gap J and provide methods how to construct selfadjoint extensions of H with prescrib ..."
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Cited by 10 (3 self)
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Let H be a symmetric operator in a separable Hilbert space H. Suppose that H has some gap J . We shall investigate the question about what spectral properties the selfadjoint extensions of H can have inside the gap J and provide methods how to construct selfadjoint extensions of H with prescribed spectral properties inside J . Under some weak assumptions about the operator H which are satisfied, e. g., provided the deficiency indices of H are infinite and the operator (H \Gamma ) \Gamma1 is compact for one regular point of H, we shall show that for every (auxiliary) selfadjoint operator M 0 in the Hilbert space H and every open subset J 0 of the gap J of H there exists a selfadjoint extension ~ H of H such that inside J the selfadjoint extension ~ H of H has the same absolutely continuous and the same point spectrum as the given operator M 0 and the singular continuous spectrum of ~ H in J equals the closure of J 0 in J . Moreover we shall present a method how to ...