The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave-packet approach

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α>0. The high-frequency (or: semi-classical) parameter is ?>0. We let ? and α go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution u^? radiates in the outgoing direction, uniformly in ?. In particular, the function u^?, when conveniently rescaled at the scale ? close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in ?) of the limiting absorption principle.Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) for very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above-mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in ?.     

An improved Hardy-Sobolev inequality in W 1,p and its application to Schrödinger operators

In this paper we prove new Hardy-like inequalities with optimal potential singularities for functions in W^1,p(Ω), where Ω is either bounded or the whole space and also some extensions to arbitrary Riemannian manifolds. We also study the spectrum of perturbed Schr?dinger operators for perturbations which are just below the optimality threshold for the corresponding Hardy inequality.     

Quantum Hamiltonians with quasi-ballistic dynamics and point spectrum

Consider the family of Schrödinger operators (and also its Dirac version) on ?²(Z) or ?²(N)

     

On matrix representations of deformed Lie algebras for quantized Hamiltonians

Consider the Lie algebras L:[K₁,K₂]=F(K₃)+G(K₄),[K₃,K₁]=uK₁,[K₃,K₂]=-uK₂,[K₄,K₁]=vK₁,[K₄,K₂]=-vK₂,[K₃,K₄]=0, subject to the physical conditions, K₃ and K₄ are real diagonal operators and († is for hermitian conjugation). Matrix representations are discussed and faithful representations of least degree for L satisfying the physical requirements are given for appropriate values of u,v∈R and certain conditions for the polynomials F(K₃) and G(K₄). Representations satisfying K₁+K₂ to be real are separately considered.     

SchröDinger operators - geometric estimates in terms of the occupation time

The difference of Schrödinger and Dirichlet semigroups is expressed in terms of the Laplace transform of the Brownian motion occupation time. This implies quantitative upper and lower bounds for the operator norms of the corresponding resolvent differences. One spectral theoretical consequence is an estimate for the eigenfunction for a Schrödinger operator in a ball where the potential is given as a cone indicator function.     

Mourre's inequality and embedded bound states

A self-adjoint operator H with an eigenvalue λ embedded in the continuum spectrum is considered. Boundedness of all operators of the form AnP is proved, where P is the eigenprojection associated with λ and A is any self-adjoint operator satisfying Mourre's inequality in a neighborhood of λ and such that the higher commutators of H with A up to order n+2 are relatively bounded with respect to H.     

Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space

We study a semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. Key results are semiboundedness theorem of the Schrödinger operator, Laplace-type asymptotic formula and IMS localization formula. We also make a remark on the semiclassical problem of a Schrödinger operator on a path space over a Riemannian manifold.     

Semiclassical measure for the solution of the dissipative Helmholtz equation

We study the semiclassical measure for the solution of the high-frequency Helmholtz equation in Rn with non-constant absorption index and a source term concentrated on a bounded submanifold of Rn. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption index is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagrangian distributions.     

On the transition to the normal phase for superconductors surrounded by normal conductors

For a cylindrical superconductor surrounded by a normal material, we discuss transition to the normal phase of stable, locally stable and critical configurations. Associated with those phase transitions, we define critical magnetic fields and we provide a sufficient condition for which those critical fields coincide. In particular, when the conductivity ratio of the superconducting and the normal material is large, we show that the aforementioned critical magnetic fields coincide, thereby proving that the transition to the normal phase is sharp. One key-ingredient in the paper is the analysis of an elliptic boundary value problem involving ‘transmission’ boundary conditions. Another key-ingredient involves a monotonicity result (with respect to the magnetic field strength) of the first eigenvalue of a magnetic Schrödinger operator with discontinuous coefficients.     

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.     

Shell interactions for Dirac operators

The self-adjointness of H+V$H+V$ is studied, where H=−iα⋅∇+mβ$H=-i\alpha \cdot \mathrm{\nabla }+m\beta$ is the free Dirac operator in R³${\mathbb{R}}^3$ and V is a measure-valued potential. The potentials V under consideration are given by singular measures with respect to the Lebesgue measure, with special attention to surface measures of bounded regular domains. The existence of non-trivial eigenfunctions with zero eigenvalue naturally appears in our approach, which is based on well known estimates for the trace operator defined on classical Sobolev spaces and some algebraic identities of the Cauchy operator associated to H.     

Global subsonic Euler flows in an infinitely long axisymmetric nozzle

In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blow-up argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.     

Existence of an extremal ground state energy of a nanostructured quantum dot

This paper is concerned with two rearrangement optimization problems. These problems are motivated by two eigenvalue problems which depend nonlinearly on the eigenvalues. We consider a rational and a quadratic eigenvalue problem with Dirichlet’s boundary condition and investigate two related optimization problems where the goal function is the corresponding first eigenvalue. The first eigenvalue in the rational eigenvalue problem represents the ground state energy of a nanostructured quantum dot. In both the problems, the admissible set is a rearrangement class of a given function.     

Quasilinear scalar field equations with competing potentials

We are concerned with the existence and non-existence of nontrivial weak solutions for a class of quasilinear scalar field equations in RN driven by competing nonlinearities with general potentials which can be unbounded or decaying to zero as |x|→+∞. Furthermore, the existence of ground states and/or bound states is considered.     

Quasi-periodic solutions in a nonlinear Schrödinger equation

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

iut−uxx+mu+⁴|u|u=0     

Preparing materials with a desired refraction coefficient

A recipe is given for creating material with a desired refraction coefficient by embedding many small particles in a given material. To implement this recipe practically, some technological problems have to be solved. These problems are formulated.     

Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations

The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.Supported by Contract MM-31 with Bulgarian Ministry of Culture, Science and Education and Alexander Von Humboldt Foundation.Partially supported by NSF grant DMS-9114456.     

Stationary solutions of non-autonomous Maxwell–Dirac systems

The Maxwell–Dirac system describes the interaction of a particle with its self-generated electromagnetic field. In this paper, we study the existence of least energy stationary solutions for non-autonomous Maxwell–Dirac system with a nonlinear term in (3+1)$(3+1)$-Minkowski space–time via variational methods. This problem is strongly indefinite and presents a lack of compactness. To overcome these difficulties, we will use the linking and concentration compactness arguments.     

Spectral statistics of random Schrödinger operator with growing potential

In this work we investigate the spectral statistics of random Schrödinger operators acting on where are i.i.d random variables distributed uniformly on [0,1].