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.     

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.     

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.     

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?     

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

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

     

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].     

Schrödinger operators on periodic discrete graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrödinger operators, constructed in the paper.     

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).     

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.     

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.     

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.     

Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator Hh, depending on the semiclassical parameter h>0, on a noncompact Riemannian manifold M such that H¹(M,R)=0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ₀(Hh) of the spectrum of the operator Hh in L²(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator Hh in the region close to λ₀(Hh), as h→0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.     

Semiclassical spectral estimates for Schrödinger operators at a critical energy level. Case of a degenerate minimum of the potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on Rn. We assume here that the potential has a totally degenerate critical point associated to a local minimum. The main result, which computes the contribution of this equilibrium, is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point.     

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.     

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.     

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.     

Schrödinger operators on regular metric trees with long range potentials: Weak coupling behavior

Consider a regular d-dimensional metric tree Γ with root o. Define the Schrödinger operator −Δ−V, where V is a non-negative, symmetric potential, on Γ, with Neumann boundary conditions at o. Provided that V decays like |x|^−γ at infinity, where 1<γ?d?2, γ≠2, we will determine the weak coupling behavior of the bottom of the spectrum of −Δ−V. In other words, we will describe the asymptotic behavior of infσ(−Δ−αV) as α→0+.