29 research outputs found
The spectral density of the scattering matrix for high energies
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
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
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
For a two-dimensional Schr\"odinger operator
with the radial potential , we study the behavior of
the number of its negative eigenvalues, as the coupling
parameter tends to infinity. We obtain the necessary and sufficient
conditions for the semi-classical growth 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
Let denote the negative eigenvalues of the one-dimensional
Schr\"odinger operator on . 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 This will imply
improved estimates for the best constants 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
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
Schrödinger operators with δ and δ′-potentials supported on hypersurfaces
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
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
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