Multi-dimensional Schrödinger operators with some negative spectrum

We obtain conditions on the negative spectra of Schrödinger operators with potentials V and −V which guarantee that the positive real line is covered by the absolutely continuous spectrum.     

Estimates for multiparameter maximal operators of Schrödinger type

Multiparameter maximal estimates are considered for operators of Schrödinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which naturally appears with a TT^?$T{T}^{?}$ argument and discuss the behavior at the endpoints. We treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear Schrödinger equation and make some comments on an open problem.     

Schrödinger operators and unique continuation. Towards an optimal result

In this article we prove the property of unique continuation (also known for C^∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|?|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?     

Absolutely continuous spectrum of Schrödinger operators with potentials slowly decaying inside a cone

For a large class of multi-dimensional Schrödinger operators it is shown that the absolutely continuous spectrum is essentially supported by [0,∞). We require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non-negative bounded potential outside the cone. In particular, we do not require the existence of wave operators. The result and method of proof extends previous work by Laptev, Naboko and Safronov.     

Classes of weights related to Schrödinger operators

In this work we obtain boundedness on weighted Lebesgue spaces on Rd of the semi-group maximal function, Riesz transforms, fractional integrals and g-function associated to the Schrödinger operator −Δ+V, where V satisfies a reverse Hölder inequality with exponent greater than d/2. We consider new classes of weights that locally behave as Muckenhoupt's weights and actually include them. The notion of locality is defined by means of the critical radius function of the potential V given in Shen (1995) [8].     

A variational approach to dislocation problems for periodic Schrödinger operators

As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential V=V(x,y) on R² with period lattice Z² by setting Wt(x,y)=V(x+t,y) for x<0 and Wt(x,y)=V(x,y) for x?0, for t∈[0,1]. For Lipschitz-continuous V it is shown that the Schrödinger operators Ht=−Δ+Wt have spectrum (surface states) in the spectral gaps of H₀, for suitable t∈(0,1). We also discuss the density of these surface states as compared to the density of the bulk. Our approach is variational and it is first applied to the well-known dislocation problem (Korotyaev (2000, 2005) [15] and [16]) on the real line. We then proceed to the dislocation problem for an infinite strip and for the plane. In Appendix A, we discuss regularity properties of the eigenvalue branches in the one-dimensional dislocation problem for suitable classes of potentials.     

On a class of Schrödinger systems with discontinuous nonlinearities

Using a non-smooth critical point theory for locally Lipschitz functionals, we investigate a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The existence of nontrivial solution is obtained.     

Spectral properties of a class of reflectionless Schrödinger operators

We prove that one-dimensional reflectionless Schrödinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schrödinger operators. In addition, we construct examples of reflectionless Schrödinger operators with more general types of spectra, given by the complement of a Denjoy-Widom-type domain in C, which exhibit a singular component.     

Schrödinger operators with nonlocal point interactions

Schrödinger operators with nonlocal point interactions are considered as new solvable models with point interactions. Examples in one and three dimensions are discussed. Corresponding direct and inverse scattering problems in one dimension are also discussed.