35 research outputs found

    The spectral density of the scattering matrix for high energies

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    We determine the density of eigenvalues of the scattering matrix of the Schrodinger operator with a short range potential in the high energy asymptotic regime. We give an explicit formula for this density in terms of the X-ray transform of the potential.Comment: 11 pages, Latex 2

    On localization of pseudo-relativistic energy

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    We present a Kato-type inequality for bounded domain Omega \subset R^n, n>1.Comment: 17 page

    The spectral shift function and spectral flow

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    This paper extends Krein's spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow.Comment: 47 page

    On the negative spectrum of two-dimensional Schr\"odinger operators with radial potentials

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    For a two-dimensional Schr\"odinger operator HαV=ΔαVH_{\alpha V}=-\Delta-\alpha V with the radial potential V(x)=F(x),F(r)0V(x)=F(|x|), F(r)\ge 0, we study the behavior of the number N(HαV)N_-(H_{\alpha V}) of its negative eigenvalues, as the coupling parameter α\alpha tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N(HαV)=O(α)N_-(H_{\alpha V})=O(\alpha) and for the validity of the Weyl asymptotic law.Comment: 13 page

    On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2

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    Let Ei(H)E_i(H) denote the negative eigenvalues of the one-dimensional Schr\"odinger operator Hu:=uVu, V0,Hu:=-u^{\prime\prime}-Vu,\ V\geq 0, on L2(R)L_2({\Bbb R}). We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R} V^{\gamma+1/2}(x)dx, (1) for the "limit" case γ=1/2.\gamma=1/2. This will imply improved estimates for the best constants Lγ,1L_{\gamma,1} in (1), as $1/2<\gamma<3/2.Comment: AMS-LATEX, 15 page

    Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States

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    We prove general comparison theorems for eigenvalues of perturbed Schrodinger operators that allow proof of Lieb--Thirring bounds for suitable non-free Schrodinger operators and Jacobi matrices.Comment: 11 page

    Global Bounds for the Lyapunov Exponent and the Integrated Density of States of Random Schr\"odinger Operators in One Dimension

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    In this article we prove an upper bound for the Lyapunov exponent γ(E)\gamma(E) and a two-sided bound for the integrated density of states N(E)N(E) at an arbitrary energy E>0E>0 of random Schr\"odinger operators in one dimension. These Schr\"odinger operators are given by potentials of identical shape centered at every lattice site but with non-overlapping supports and with randomly varying coupling constants. Both types of bounds only involve scattering data for the single-site potential. They show in particular that both γ(E)\gamma(E) and N(E)E/πN(E)-\sqrt{E}/\pi decay at infinity at least like 1/E1/\sqrt{E}. As an example we consider the random Kronig-Penney model.Comment: 9 page

    Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

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    Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

    Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

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    We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.Comment: 26 page

    Cantor and band spectra for periodic quantum graphs with magnetic fields

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    We provide an exhaustive spectral analysis of the two-dimensional 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 Kronig-Penney 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 Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte
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