One-dimensional Stark Research Articles

On the Inverse Spectral Problem for the One-dimensional Stark Operator on the Semiaxis

On the Inverse Spectral Problem for the One-dimensional Stark Operator on the Semiaxis

Trace formulas for the one-dimensional Stark operator and integrals of motion for the cylindrical Korteweg–de Vries equation

Trace formulas for the one-dimensional Stark operator and integrals of motion for the cylindrical Korteweg–de Vries equation

On Spectral Properties of the One-Dimensional Stark Operator on the Semiaxis

We consider a one-dimensional Stark operator on the semiaxis with Dirichlet boundary condition at the origin. The asymptotic behavior of eigenvalues at infinity is analyzed.

Stark–Wannier ladders and cubic exponential sums

Given a one-dimensional Stark–Wannier operator, we study the reflection coefficient and its poles in the lower half of the complex plane far from the real axis. In particular, the reflection coefficient is described asymptotically in terms of regularized infinite cubic exponential sums.

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)−1/4−e; in the smoothness direction, a sufficient condition in Holder classes isq∈C 1/2+e(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)−1/4 and belong toC 1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.

On the Absolutely Continuous Spectrum of Stark Operators

The stability of the absolutely continuous spectrum of the one-dimensional Stark operator $$$$ under perturbations of the potential is discussed. The focus is on proving this stability under minimal assumptions on smoothness of the perturbation. A general criterion is presented together with some applications. These include the case of periodic perturbations where we show that any perturbation vL 1 (𝕋)∩H −1/2 (𝕋) preserves the a.c. spectrum.

Absolutely continuous spectrum of perturbed Stark operators

Absolutely continuous spectrum of perturbed Stark operators

Imaginary parts of Stark–Wannier resonances

We consider a one-dimensional Stark–Wannier Hamiltonian, H=−d2/dx2+p(x)−εx, x∈R, where p is a smooth periodic, finite-gap potential, and ε>0 is small enough. We compute rigorously the imaginary parts of the spectral resonances. For this purpose we develop some related elements of the adiabatic approach to the equations of the form −ψ″+p(x)ψ+q(εx)ψ=Eψ, ε→0.

Some properties of spectral expansions related to the one-dimensional Stark effect Hamiltonian

We consider the one-dimensional Stark effect Hamiltonian defined over the whole line R by the differential expression Hu = − u″ + Exu + q( x) u, E ≠ 0. This paper analyses the equiconvergence property of the related spectral expansion with the Fourier integral expansion in the uniform over any compact set or over the whole line R metric.

Scattering theory for Stark Hamiltonians

We give an introduction to the spectral and scattering theory for Schrodinger operators. An abstract short range scattering theory is developed. It is applied to perturbations of the Laplacian. Particular attention is paid to the study of Stark Hamiltonians. The main result is an explanation of the discrepancy between the classical and the quantum scattering theory for one-dimensional Stark Hamiltonians.

Resonances in one-dimensional Stark effect and continued fractions

The Stieltjes type continued fraction (i.e., any diagonal Padé approximants sequence) of the perturbation series for the resonances of the so-called one-dimensional Stark effect converges to the resonances.