Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs(R), s?1, and global well-posedness in H¹(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation

     

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.     

Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

This paper studies the scattering matrix S(E;?) of the problem

−?²ψ^″(x)+V(x)ψ(x)=Eψ(x)     

Two realizations of Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression P=ΔM+V on a complete Riemannian manifold (M,g) with metric g, where ΔM is the scalar Laplacian on M and V is a real-valued locally integrable function on M. We study two self-adjoint realizations of P in L²(M) and show their equality. This is an extension of a result of S. Agmon.     

Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential

Dissipative Schrödinger operators with a matrix potential are studied in L₂((0,∞);E)(dimE=n<∞) which are extension of a minimal symmetric operator L₀ with defect index (n,n). A selfadjoint dilation of a dissipative operator is constructed, using the Lax-Phillips scattering theory, the spectral analysis of a dilation is carried out, and the scattering matrix of a dilation is founded. A functional model of the dissipative operator is constructed and its characteristic function's analytic properties are determined, theorems on the completeness of eigenvectors and associated vectors of a dissipative Schrödinger operator are proved.     

Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents

We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov exponent is a continuous strictly positive function of the energy. It is possible to include a coupling constant in the model and these results then hold for every non-zero value of the coupling constant.     

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.     

Almost periodic Schrödinger operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of irreducible interval exchange transformations in a dense set, have pure singular continuous spectrum for Lebesgue almost all points of the interval. Such potentials are natural generalizations of the Sturmian case.     

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.     

“Localized” self-adjointness of Schrödinger type operators on Riemannian manifolds

We prove self-adjointness of the Schrödinger type operator , where ∇ is a Hermitian connection on a Hermitian vector bundle E over a complete Riemannian manifold M with positive smooth measure dμ which is fixed independently of the metric, and V∈L_loc¹(EndE) is a Hermitian bundle endomorphism. Self-adjointness of H_V is deduced from the self-adjointness of the corresponding “localized” operator. This is an extension of a result by Cycon. The proof uses the scheme of Cycon, but requires a refined integration by parts technique as well as the use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.     

Internal Lifshits tails for random magnetic Schrödinger operators

This paper is devoted to the study of Lifshits tails for weak random magnetic perturbations of periodic Schrödinger operators acting on L²(Rd) of the form Hλ,w=(−i∇−λ_∑γ∈ZdwγA²(⋅−γ))+V, where V is a Zd-periodic potential, λ is positive coupling constants, (wγ)_γ∈Zd are i.i.d and bounded random variables and is the single site vector magnetic potential. We prove that, for λ small, at an open band edge, a true Lifshits tail for the random magnetic Schrödinger operator occurs if a certain set of conditions on H₀=−Δ+V and on A holds.     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.     

Trace asymptotics for fractional Schrödinger operators

This paper proves an analogue of a result of Bañuelos and Sá Barreto [5] on the asymptotic expansion for the trace of Schrödinger operators on R^d${\mathbb{R}}^d$ when the Laplacian −Δ, which is the generator of the Brownian motion, is replaced by the non-local integral operator (−Δ)^α/2${(-\mathrm{\Delta })}^{\alpha /2}$, 0<α<2$0<\alpha <2$, which is the generator of the symmetric stable process of order α. These results also extend recent results of Bañuelos and Yildirim [6] where the first two coefficients for (−Δ)^α/2${(-\mathrm{\Delta })}^{\alpha /2}$ are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed-stable processes are obtained.     

Inverse scattering for the magnetic Schrödinger operator

We show that fixed energy scattering measurements for the magnetic Schrödinger operator uniquely determine the magnetic field and electric potential in dimensions n?3. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston (1995) [5] which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schrödinger equation via a pseudodifferential conjugation argument.     

Non-trapping magnetic fields and Morrey-Campanato estimates for Schrödinger operators

We prove some uniform in ? a priori estimates for solutions of the equation

     

Threshold properties of matrix-valued Schrödinger operators, II. Resonances

We present some results on the perturbation of eigenvalues embedded at a threshold for a matrix-valued Hamiltonian with three-dimensional dilation analytic Schrödinger operators as entries and with a small off-diagonal perturbation. The main result describes how a threshold eigenvalue generates resonances (that is, poles of the meromorphic continuation of the perturbed Hamiltonian).