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Wilhelm Schlag

Publications and source records attributed to Wilhelm Schlag.

87 records · Page 5Linked to original sources
arXiv2006

Dispersive estimates for Schroedinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any specific properties of NLS. Rather, it is axiomatic on the linear level, and our results are obtained from four assumptions (which are of course motivated by NLS). This work is in three dimensions.

math.AP↗
arXiv2006

Agmon-Kato-Kuroda theorems for a large class of perturbations

We extend the classical Agmon theorem on asymptotic completeness of two body Schroedinger operators to cover a larger class of perturbations. This is accomplished by means of a suitable limiting absorption principle. The proof of the latter relies on methods from harmonic analysis centered around the Stein-Tomas and Bochner-Riesz theorems.

math.AP↗
arXiv2005

Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues

We develop some non-perturbative methods for studying the IDS in almost Mathieu and related models. Assuming positive Lyapunov exponents, and assuming that the potential function is a trigonometric polynomial of degree k, we show that the Holder exponent of the IDS is almost 1/2k. We also show that this is stable under small perturbations of the potential (e.g., potentials which are close to that of almost Mathieu again give rise to almost 1/2 Holder continuous IDS). Moreover, off a set of Hausdorff dimension zero the IDS is Lipschitz. We further deduce from these properties that the IDS is absolutely continuous for almost every shift angle. The proof is based on large deviation theorems for the entries of the transfer matrices in combination with the avalanche principle for non-unimodular matrices. These two are combined to yield a multiscale approach to counting zeros of determinants which fall into small disks. In addition, we develop a new mechanism of eliminating resonant phases, and use it to obtain lower bounds on the gap between Dirichlet eigenvalues. The latter also uses the exponential localization of the eigenfunctions.

math-ph↗
arXiv2005

On the focusing critical semi-linear wave equation

The focusing critical wave equation in three dimensions exhibits a special class of static solutions which are linearly unstable. These solutions decay like an inverse first power. We construct small codimension one stable manifolds in the class of radial perturbations for these static solutions. We show that initial data on this manifold lead to global solutions. Moreover, the long-time behavior of these solutions can be described explicitly. In particular, the family of static solutions acts as an attractor for these solutions. On a technical level, we remark that the linearized operator exhibits a zero energy resonance, and we develop the tools needed for that case.

math.AP↗
arXiv2005

Non-generic blow-up solutions for the critical focusing NLS in 1-d

This paper addresses the existence of codimension one stable manifolds for the pseudo-conformal blow-up solution for critical one-dimensional NLS. By the work of Perelman and Merle, Raphael, the blow-up rate of these solutions is far from the generic one. However, Bourgain,Wang and Perelman formulated the possibility of codimension one stable manifolds for the non-generic blow-up solutions. This paper established the existence of such manifolds, albeit only in the measurable category.

math.AP↗
arXiv2005

A remark on Littlewood-Paley theory for the distorted Fourier transform

We show that the usual Mikhlin multiplier theorem relative to the distorted Fourier transform holds in the range (3/2,3) at least for radial functions and potentials. The restricted range is due to the fact that the Schroedinger operator which gives rise to the distorted Fourier transform may have a zero energy resonance. Consequently, the Littlewood-Paley theorem is shown also only for the range (3/2,3).

math.CA↗
arXiv2005

Numerical verification of a gap condition for linearized NLS

We make a detailed numerical study of the spectrum of two Schroedinger operators L_- and L_+ arising in the linearization of the supercritical nonlinear Schroedinger equation (NLS) about the standing wave, in three dimensions. This study was motivated by a recent result of the second author on conditional asymptotic stability of solitary waves in the case of a cubic nonlinearity. Underlying the validity of this result is a spectral condition on the operators L_- and L_+, namely that they have no eigenvalues nor resonances in the gap (a region of the positive real axis between zero and the continuous spectrum,) which we call the gap property. The present numerical study verifies this spectral condition, and further shows that the gap property holds for NLS exponents of the form 2*beta + 1, as long as beta_* < beta <= 1, where beta_* = 0.913958905 +- 1e-8. Our strategy consists of rewriting the original eigenvalue problem via the Birman-Schwinger method. From a numerical analysis viewpoint, our main contribution is an efficient quadrature rule for the kernel 1/|x-y| in R^3, i.e., provably spectrally accurate. As a result, we are able to give similar accuracy estimates for all our eigenvalue computations. We also propose an improvement of the Petviashvili's iteration for the computation of standing wave profiles which automatically chooses the radial solution.

math.AP↗
arXiv2004

Stable manifolds for an orbitally unstable NLS

We construct a local Lipschitz graph around a soliton of the cubic focusing NLS in three dimensions on which global solutions exist, and asymptotic stability as well as scattering holds.

math.AP↗
arXiv2004

Dispersive estimates for Schroedinger operators in dimension two

We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular point for the perturbed resolvent.

math.AP↗