A Szeg? condition for a multidimensional Schrödinger operator

We consider spectral properties of a Schrödinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szeg?-type condition.     

A local trace formula for resonances of perturbed periodic Schrödinger operators

We give a local trace formula for the pair (P₁(h)=P₀+W(hy),P₀), where P₀ is a periodic Schrödinger operator, W is a decreasing perturbation and h is a small positive parameter. We apply this result to establish the existence of ∼h⁻ⁿ resonances near some energy λ of σ(P₀).     

Energy critical fourth-order Schrödinger equations with subcritical perturbations

In this paper, we study the global well-posedness and scattering theory of an 8-D cubic nonlinear fourth-order Schrödinger equation, which is perturbed by a subcritical nonlinearity. We utilize the strategies in Tao et al. (2007) [16] and Zhang (2006) [17] to obtain when the cubic term is defocusing, the solution is always global no matter what the sign of the subcritical perturbation term is. Moreover, scattering will occur either when the pertubation is defocusing and 1<p<2 or when the mass of the solution is small enough and 1≤p<2.     

Quasi-periodic solutions in a nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation

iut−uxx+mu+⁴|u|u=0     

On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions

General elliptic boundary value problems with the spectral parameter appearing linearly both in the elliptic equation and in boundary conditions are considered. It is proved that the corresponding matrix operator from the Boutet de Monvel algebra is similar to an almost diagonal operator. This result is applied to prove the completeness and the summability (in the sense of Abel) of the root vectors of this operator.The support of the Rashi Foundation is gratefully acknowledged.The support of the Israel Ministry of Science and Technology is gratefully acknowledged.     

A ground state alternative for singular Schrödinger operators

Let a be a quadratic form associated with a Schrödinger operator L=-∇·(A∇)+V on a domain Ω⊂R^d. If a is nonnegative on , then either there is W>0 such that for all , or there is a sequence and a function ?>0 satisfying L?=0 such that a[?_k]→0, ?_k→? locally uniformly in Ω?{x₀}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W>0 such that for every satisfying there exists a constant C>0 such that for all .     

Weighted estimates for commutators of some singular integrals related to Schrödinger operators

Let L=−Δ+V$L=-\mathrm{\Delta }+V$ be a Schrödinger operator with a non-negative potential V satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to L such as the Riesz transforms and fractional integrals with the new BMO functions introduced in Bongioanni et al. (2011) [1] on the weighted spaces L^p(w)$L^p(w)$ where w belongs to the new classes of weights introduced by Bongioanni et al. (2011) [2].     

Bound states of the Schrödinger-Newton model in low dimensions

We prove the existence of quasi-stationary symmetric solutions with exactly n≥0 zeros and uniqueness for n=0 for the Schrödinger-Newton model in one dimension and in two dimensions along with an angular momentum m≥0. Our result is based on an analysis of the corresponding system of second-order differential equations.     

Biharmonic nonlinear Schrödinger equation and the profile decomposition

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L²-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H²(R^N), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given.     

Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential

This paper is devoted to the study of essential self-adjointness of a relativistic Schrödinger operator with a singular homogeneous potential. From an explicit condition on the coefficient of the singular term, we provide a sufficient and necessary condition for essential self-adjointness.     

Macroscopic current induced boundary conditions for Schrödinger-type operators

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.     

Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order , d?5. We prove that if a maximal-lifespan radial solution obeys supt∈I‖Δu(t)_‖2<‖ΔW_‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters.     

Coupled nonlinear Schrödinger systems with potentials

Coupled nonlinear Schrödinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of concentrating solutions of a singularly perturbed coupled nonlinear Schrödinger system, in presence of potentials. We show how the location of the concentration points depends strictly on the potentials.     

Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees

We construct conjugate operators for the real part of a completely non-unitary isometry and we give applications to the spectral and scattering theory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrödinger operators on trees. We consider C^*-algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with respect to the ideal of compact operators, and give formulas for the essential spectrum of these Hamiltonians.     

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)$(e^{2i\omega t}\phi ,0,0)$, (0,e2iωtφ,0)$(0,e^{2i\omega t}\phi ,0)$, (0,0,e2iωtφ)$(0,0,e^{2i\omega t}\phi )$, where φ is a ground state of the scalar nonlinear Schrödinger equation.     

Convex hypersurfaces and L estimates for Schrödinger equations

This paper is concerned with Schrödinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the L^p-L^q estimate of the solution operator in the free case. This estimate, combined with the results of fractionally integrated groups, allows us to further obtain the L^p estimate of solutions for the initial data belonging to a dense subset of L^p in the case of integrable potentials.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.