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.

     

On the location of the essential spectrum of Schrödinger operators

We give estimates on the bottom of the essential spectrum of Schrödinger operators in .

     

A version of Gordon's theorem for multi-dimensional Schrödinger operators

We consider discrete Schrödinger operators in with , and study the eigenvalue problem for these operators. It is shown that the point spectrum is empty if the potential is sufficiently well approximated by periodic potentials. This criterion is applied to quasiperiodic and to so-called Fibonacci-type superlattices.

     

Reflection symmetries and absence of eigenvalues for one-dimensional Schrödinger operators

We prove a criterion for absence of decaying solutions for one-dimensional Schrödinger operators. As necessary input, we require infinitely many centers of local reflection symmetry and upper and lower bounds for the traces of the associated transfer matrices.

     

Bound states of discrete Schrödinger operators with super-critical inverse square potentials

We consider discrete one-dimensional Schrödinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of the number of eigenvalues below a given energy as this energy tends to the bottom of the essential spectrum.

     

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

     

Embedded singular continuous spectrum for one-dimensional Schrödinger operators

We investigate one-dimensional Schrödinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.

     

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.

     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

On decay of solutions to nonlinear Schrödinger equations

We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. Under certain natural assumptions we show that any such solution is continuous and vanishes at infinity. This allows us to interpret the solution as a finite multiplicity eigenfunction of a certain linear Schrödinger operator and, hence, apply well-known results on the decay of eigenfunctions.

     

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 propagation speed and kernel estimates for Schrödinger operators

We point out finite propagation speed phenomena for discrete and continuous Schrödinger operators and discuss various types of kernel estimates from this point of view.

     

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 .

     

Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations

Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in , . This result implies that best result concerning local well-posedness for the IVP is in . It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.

     

On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an -regular solution, a first-order error bound in the norm is shown and used to derive a second-order error bound in the norm. For the cubic Schrödinger equation with an -regular solution, first-order convergence in the norm is used to obtain second-order convergence in the norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and -conditional stability for error propagation, where for the Schrödinger-Poisson system and for the cubic Schrödinger equation.

     

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.

     

Asymptotic Dirichlet problem for the Schrödinger operator via rough isometry

We pose and solve the asymptotic Dirichlet problem for the Schrödinger operator via rough isometries on a certain class of Riemannian manifolds. With suitable potentials, we give the solvability of the problem for a naturally defined class of data functions.

     

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators. We prove sub-exponential decay for functions in Gevrey classes and exponential decay for real analytic functions.

     

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

Standing wave solutions of the one-dimensional nonlinear Schrödinger equation

with are well known to be unstable. In this paper we show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult -critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors' companion paper Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005.