Given a self-adjoint 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 self-adjoint operators such that any Aτ Θ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreĭn-like 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−2-construction” 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 pseudo-differential operators, thus unifying and extending previously known results. 1.

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 self-adjointness of the Schrödinger Hamiltonian.

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

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We review a variety of recently obtained trace formulas for one- and multidimensional Schrödinger operators. Some of the results are extended to Sturm-Liouville and matrix-valued Schrodinger operators. Furthermore, we recall a set of trace formulas in one, two, and three dimensions related to point interactions as well as a new uniqueness result for three-dimensional Schrödinger operators with spherically symmetric potentials.

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 self--adjoint extensions of H can have inside the gap J and provide methods how to construct self--adjoint 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) self--adjoint operator M 0 in the Hilbert space H and every open subset J 0 of the gap J of H there exists a self--adjoint extension ~ H of H such that inside J the self--adjoint 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 ...

. We study the point limit of the linearized Maxwell--Lorentz 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 Maxwell--Lorentz 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 self--adjoint 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 three--dimensional. We obtain that this three--dimensional subspace is nothing but the velocity particle space, the particle dynamics being therefore completely determined -- in an explicit way -- by the behaviour of ...

We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et. al. [1], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finitedimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz. To fit into Arxiv’s file size requirements, low-resolution versions of certain large figures were used. A version of this paper with the full figures is available at

We study the diffusion in R 3 of a particle interacting with N fixed points through point interactions whose strength varies in time. Under mild assumptions on the time dependence of the strengths we prove existence for all times and uniqueness of the solution, for which we provide a rather explicit expression. We also prove that, under a suitable rescaling of the interaction strengths, the solution converges, when N ! 1, to the solution of a diffusion equation with a regular killing term (potential). We use properties of the local self-adjoint extensions of the laplacian, and results from the theory of fractional integrals and derivatives. 1. Introduction and results In recent years singular perturbations of selfadjoint differential operators have attracted increasing attention in Applied Physics as well as in Pure Mathematics. Models of "small" obstacles for scattering of waves or particles, faults in rock fracture dynamics, etc. are examples of situations where one is naturally...

In this note we present a proof of the Wegner estimate for random Schrödinger operators of Anderson type that do not have necessarily overlapping single site potentials. We use the spectral averaging procedure via trace formula and spectral shift function.