Transport in the one-dimensional Schrödinger equation

We prove a dispersive estimate for the Schrödinger equation on the real line, mapping between weighted spaces with stronger time-decay ( versus ) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that be integrable and not have a resonance at zero energy. If a resonance is present (for example, in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance.

     

Unique continuation for the Schrödinger equation with gradient vector potentials

We obtain unique continuation results for Schrödinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear Schrödinger equations.

     

On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .

     

Imbedded singular continuous spectrum for Schrödinger operators

We construct examples of potentials satisfying where the function is growing arbitrarily slowly, such that the corresponding Schrödinger operator has an imbedded singular continuous spectrum. This solves one of the fifteen ``twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We also prove that if the singular continuous spectrum is empty. Therefore our result is sharp.

     

Strichartz estimates for the Schrödinger equation with radial data

We prove an endpoint Strichartz estimate for radial solutions of the two-dimensional Schrödinger equation:

     

Finite speed of propagation and local boundary conditions for wave equations with point interactions

We show that the boundary conditions entering in the definition of the self-adjoint operator describing the Laplacian plus a finite number of point interactions are local if and only if the corresponding wave equation has finite speed of propagation.

     

On the singular spectrum of Schrödinger operators with decaying potential

The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the so-called Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal.

     

Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation

We establish that the initial value problem for the quadratic non-linear Schrödinger equation

where , is locally well-posed in when . The critical exponent for this problem is , and previous work by Colliander, Delort, Kenig and Staffilani, 2001, established local well-posedness for .

     

Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity

This paper is concerned with the inhomogeneous nonlinear Shrödinger equation (INLS-equation)

In the critical and supercritical cases with it is shown here that standing-wave solutions of (INLS-equation) on perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small 0.$">

     

Solutions to a class of Schrödinger equations

We establish existence and multiplicity of solutions to a class of nonlinear Schrödinger equations with, e.g., ``atomic' Hamiltonians, via critical point theory.

     

Asymptotical behavior of ground state solutions for critical quasilinear Schrödinger equation

This paper is concerned with the existence and asymptotical behavior of positive ground state solutions for a class of critical quasilinear Schrodinger equation.By using a change of variables and variational argument,we prove the existence of positive ground state solution and discuss their asymptotical behavior。     

Remark on well-posedness for the fourth order nonlinear Schrödinger type equation

We consider the initial value problem for the fourth order nonlinear Schrödinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper.

     

A bound for ratios of eigenvalues of Schrödinger operators with single-well potentials

For Schrödinger operators with nonnegative single-well potentials ratios of eigenvalues are extremal only in the case of zero potential. To prove this, we investigate some monotonicity properties of Prüfer-type variables.

     

The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Let be a noncompact complete Riemannian manifold. We consider the Schrödinger operator acting on , where is a nonnegative, locally integrable function on . We obtain some simple conditions which imply that , the bottom of the spectrum of , is strictly positive. We also establish upper and lower bounds for the counting function .

     

Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

Foundations of Computational Mathematics - We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and...     

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with being a critical frequency in the sense that We show that if the zero set of has isolated connected components such that the interior of is not empty and is smooth, has isolated zero points, , , and has critical points such that , then for small, there exists a standing wave solution which is trapped in a neighborhood of Moreover the amplitudes of the standing wave around , and are of a different order of . This type of multi-scale solution has never before been obtained.

     

Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities

We study a time-independent nonlinear Schrödinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.

     

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.     

Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

We establish an asymptotic estimate of the lowest eigenvalue of the Schrödinger operator with a magnetic field in a bounded -dimensional domain, where curl vanishes non-degenerately, and is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity.