Transformation Operators for Sturm-Liouville Operators with Singular Potentials

We construct transformation operators for Sturm-Liouville operators with singular potentials from the space W ⁻¹ ₂(0,1) and show that these transformation operators naturally appear during factorisation of Fredholm operators of a special form. Some applications to the spectral analysis of Sturm-Liouville operators with singular potentials under consideration are also given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Singular Spectrum Near a Singular Point of Friedrichs Model Operators of Absolute Type

In we consider a family of selfadjoint operators of the Friedrichs model: . Here is the operator of multiplication by the corresponding function of the independent variable , and (perturbation) is a trace-class integral operator with a continuous Hermitian kernel satisfying some smoothness condition. These absolute type operators have one singular point of order . Conditions on the kernel are found guaranteeing the absence of the point spectrum and the singular continuous one of such operators near the origin. These conditions are actually necessary and sufficient. They depend on the finiteness of the rank of a perturbation operator and on the order of singularity . The sharpness of these conditions is confirmed by counterexamples.     

D-bar Operators on Quantum Domains

We study the index problem for the d-bar operators subject to Atiyah- Patodi-Singer boundary conditions on noncommutative disk and annulus.     

Jacobi Operators with Singular Continuous Spectrum

We consider a semilinear elliptic system which includes the model system of the W-strings in the cosmology as a special case. We prove existence of multi-string solutions and obtain precise asymptotic decay estimates near infinity for the solutions. As a special case of this result we solve an open problem posed by Yang [14]. AMS Subject Classifications (2000): 35J45, 35J60, 37K40, 70S15     

Root System of Singular Perturbations of the Harmonic Oscillator Type Operators

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space \({\mathcal{H}}\), i.e. of the self-adjoint operator with simple positive eigenvalues μ_k satisfying μ_k+1 ? μ_k ≥ Δ > 0. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system contains a Riesz basis.     

Spectral Analysis of One-Dimensional Dirac Operators with Slowly Decreasing Potentials

We prove that the absolutely continuous spectrum of Dirac operators on the half-line with square integrable potentials fills the whole real axis. We also establish an estimate on the number of eigenvalues for Coulomb-like potentials.     

Wegner Estimates for Some Random Operators with Anderson-type Surface Potentials

For Schrödinger operator with random potentials concentrated near a surface, Wegner-type estimates are proven by using the spectral averaging method of Combes, Hislop and Klopp. These estimates allow us to show the local regularity of the integrated density of surface states at the gap of the background periodic operator. Acoustic operator with random surface potentials is treated similarly.     

Bound States for Hypercentral Singular and Exponential Potentials

With a view to obtaining an exact closed form solution to the Schrodinger equation for a variety of hypercentral potentials, we investigate further application of an ansatz. This method is good enough for many kinds of potentials, but in this article it applies to a type of the hypercentral singular potentials V(x) = ax2 bx-4 cx-6 and exponential hypercentral Morse potential U(x) = U0 ( e-2ax -2 e-ax) for three interacting particles. The Morse potential is used for diatomic molecule while this method will be successfully used for many atomic molecules. The three-body potentials are more easily introduced and treated within the hyperspherical harmonic formalism. The internal particle motion is usually described by means of Jacobi relative coordinates ρ, λ, and R, in terms of three particle positions r1,r2, and r3. We discuss some results obtained by using harmonic and anharmonic oscillators, however the hypercentral potential can be easily generalized in order to allow a systematic analysis, which admits an exact solution of the wave equation. This method is also applied to some other types of three-body, four-body, ..., interacting potentials.     

The Spectral Flow for Dirac Operators on Compact Planar Domains with Local Boundary Conditions

Let D_t, ${0\;\leqslant\;t\;\leqslant\;1}$ be a 1-parameter family of Dirac type operators on a two-dimensional disk with m ? 1 holes. Suppose that all operators D_t have the same symbol, and that D₁ is conjugate to D₀ by a scalar gauge transformation. Suppose that all operators D_t are considered with the same elliptic local boundary condition. Our main result is a computation of the spectral flow for such a family of operators. The answer is obtained up to multiplication by an integer constant depending only on the number of holes in the disk. This constant is calculated explicitly for the case of the annulus (m = 2).     

Some Schrödinger Operators with Power-Decaying Potentials and Pure Point Spectrum

We construct (deterministic) potentials such that the Schr?dinger equation on has dense pure point spectrum in for almost all boundary conditions at . As a by-product, we also obtain power-decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions. Received: 8 November 1996 / Accepted: 8 January 1997     

A Probabilistic Approach to One-Dimensional Schrödinger Operators with Sparse Potentials

We consider the one-dimensional Schr?dinger equation with sparse potential V (i.e.\ mainly V= 0). It is shown that the asymptotics of the solutions corresponding to positive energies E can be approximately described by an infinite sum of independent random variables. Using results from probability theory, we can then determine the spectral properties of the operators under consideration. We prove absolute continuity for a general class of potentials, and we also have examples with singular continuous spectrum. Received: 19 June 1996 / Accepted: 11 September 1996     

Lieb–Thirring Inequalities for Schrödinger Operators with Complex-valued Potentials

Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential.     

Number of Bound States of Schrödinger Operators with Matrix-Valued Potentials

We give a Cwikel–Lieb–Rozenblum type bound on the number of bound states of Schrödinger operators with matrix-valued potentials using the functional integral method of Lieb. This significantly improves the constant in this inequality obtained earlier by Hundertmark.     

On Eigenvalues of Discrete Schrödinger Operators with Potentials of Coulomb-Type Decay

We study the distribution of the eigenvalues inside of the essential spectrum for discrete one-dimensional Schrödinger operators with potentials of Coulomb type decay.     

The Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Decaying Potentials

We investigate one-dimensional Schr?dinger operators with asymptotically small potentials. It will follow from our results that if with , then is an essential support of the absolutely continuous part of the spectral measure. We also prove that if , then the spectrum is purely absolutely continuous on . These results are optimal. Received: 25 June 1997 / Accepted: 29 July 1997     

Trigonometric Phase-Difference Operators of Quantum Electromagnetic Fields

Journal of Experimental and Theoretical Physics - Quantum-mechanical operators of phase difference between two electromagnetic fields are proposed and their properties are analyzed. The Hermitian...     

Anderson Localization for Schrödinger Operators on ℤ with Potentials Given by the Skew–Shift

In this paper we study one-dimensional Schr?dinger operators on the lattice with a potential given by the skew shift. We show that Anderson localization takes place for most phases and frequencies and sufficiently large disorders. Received: 19 September 2000 / Accepted: 15 February 2001     

Relativistic Two-Fermion Problem with the Most General Electric and Magnetic Potentials

The energy equation of two spin-1/2 particles interacting with their charges and anomalous magnetic moments is examined. Starting with the most general Hamiltonian already obtained by other authors, the relevant wave equation has been written in terms of the generators of the de Sitter group. The radial equations for four different two-fermion systems are derived and the positronium case is studied in detail. Their bound state solutions are discussed and the similarity to the sixteen radial equations arrived at by other authors in completely different manner is pointed out.     

Power Law Subordinacy and Singular Spectra.¶II. Line Operators

We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schr?dinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19]. Received: 21 October 1999 / Accepted: 21 December 1999