About

Wilhelm Schlag is the Phillips Professor of Mathematics at Yale University. His research interests encompass a broad range of topics in mathematical analysis and differential equations. Specifically, he focuses on linear and nonlinear wave equations, investigating the stability of solitons and topological solitons. His work also includes spectral theory of differential operators and the distorted Fourier transform, as well as the spectral theory of lattice Schrödinger operators with quasiperiodic potentials. Additionally, Schlag engages in computer-assisted proofs in differential equations and spectral problems, and his expertise extends to classical and harmonic analysis. At Yale, he contributes to the academic community through teaching various courses in mathematics.

Research topics

• Quantum mechanics
• Combinatorics
• Pure mathematics
• Mathematics
• Mathematical analysis

Selected publications

• An introduction to the distorted Fourier transform

  Advanced Nonlinear Studies · 2025-01-16

  articleOpen accessSenior authorCorresponding

  Abstract This article is intended as an introduction to the distorted Fourier transform associated with a Schrödinger operator on the line or the half-line. This versatile tool has seen numerous applications in nonlinear PDE in recent years. It typically arises in the asymptotic stability analysis of topological solitons in classical field theory, such as kinks in the sine-Gordon or ϕ 4 models. The distorted Fourier transform is also a natural technique in the analysis of dispersive equations on manifolds with symmetries. Such models appear in general relativity, for example in the study of waves on a black-hole background. While microlocal methods have proven to be powerful in such applications, the more classical Weyl–Titchmarsh spectral theory and with it, the distorted Fourier transform, continue to play an essential role in the analysis of evolution PDEs. This article explain how it can be derived from Stone’s formula, which also establishes the Plancherel and inversion theorems.

  PublisherOA PDFDOI

• Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices

  ArXiv.org · 2025-11-10

  preprintOpen accessSenior author

  In this paper, we prove Hölder continuity of the integrated density of states for discrete quasiperiodic Jacobi $d\times d$ block matrices with Diophantine frequencies. The Hölder exponent is shown to be any $β$ such that $0<β<1/(2κ^d)$, where $κ^d$ is the acceleration, i.e., the slope of the sum of the top $d$ Lyapunov exponents in the imaginary direction of the phase. This generalizes the Hölder continuity results in the Schrödinger operator setting in \cites{GS2,HS1}, and also strengthens them in that setting by covering more Diophantine frequencies. The proof is built on a new scheme for obtaining a local zero count for finite-volume characteristic polynomials from a global one.

  PublisherOA PDFDOI

• On Continuous Time Bubbling for the Harmonic Map Heat Flow in Two Dimensions

  Abel symposia · 2025-01-01

  book-chapterSenior author
  PublisherDOI

• Continuous in time bubble decomposition for the harmonic map heat flow

  Forum of Mathematics Pi · 2025-01-01 · 1 citations

  articleOpen accessSenior author

  Abstract We consider the harmonic map heat flow for maps $\mathbb {R}^{2} \to \mathbb {S}^2$ . It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.

  PublisherOA PDFDOI

• Avila's acceleration via zeros of determinants and applications to Schrödinger cocycles

  Communications on Pure and Applied Mathematics · 2025-10-22 · 1 citations

  articleSenior author

  Abstract In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove ‐Hölder continuity of the integrated density of states for supercritical quasi‐periodic Schrödinger operators restricted to the th stratum, for any and . We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.

  PublisherDOI

• The cubic NLS on the line with an inverse square potential

  ArXiv.org · 2025-08-03

  preprintOpen access

  We establish modified scattering for solutions of the cubic NLS on the line with a repulsive inverse square potential and small localized data. The method is based on a comparison between the free and distorted Galilei vector fields and a wave packet transform.

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• Correction to: On Localization and the Spectrum of Multi-frequency Quasi-periodic Operators

  Vietnam Journal of Mathematics · 2025-03-27

  articleOpen access
  PublisherOA PDFDOI

• Uniqueness of excited states to −Δu + u−u3= 0in three dimensions

  Analysis & PDE · 2024-07-19 · 2 citations

  articleOpen accessSenior author

  We prove the uniqueness of several excited states to the ODE (t) + (2/t) (t) + f (y(t)) = 0, y(0) = b, and (0) = 0, for the model nonlinearity f (y) = y 3 -y.The n-th excited state is a solution with exactly n zeros and which tends to 0 as t .These represent all smooth radial nonzero solutions to the PDE u + f (u) = 0 in H 1 .We interpret the ODE as a damped oscillator governed by a double-well potential, and the result is proved via rigorous numerical analysis of the energy and variation of the solutions.More specifically, the problem of uniqueness can be formulated entirely in terms of inequalities on the solutions and their variation, and these inequalities can be verified numerically. (y b (t), b (t)) (1, 0), (y b (t), b (t)) (-1, 0), (y b (t), b (t)) (0, 0);

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• On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

  Communications of the American Mathematical Society · 2024-02-28 · 9 citations

  articleOpen accessSenior author

  We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]). The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172].

  PublisherDOI

• Asymptotic stability of the sine-Gordon kink under odd perturbations

  Duke Mathematical Journal · 2023-10-01 · 13 citations

  articleSenior author

  We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sine-Gordon kink, a remarkable nonresonance property exhibited by the quadratic nonlinearity in the Klein–Gordon equation for the perturbation, and a variable coefficient quadratic normal form. We emphasize that the restriction to odd perturbations does not bypass the effects of the odd threshold resonance of the linearized operator. Our techniques have applications to soliton stability questions for several well-known nonintegrable models, for instance, to the asymptotic stability problem for the kink of the ϕ4 model as well as to the conditional asymptotic stability problem for the solitons of the focusing quadratic and cubic Klein–Gordon equations in one space dimension.

  PublisherDOI

Recent grants

• Harmonic Analysis with Applications to Mathematical Physics

  NSF · $85k · 2005–2007

• Long-Term Dynamics of Nonlinear Evolution Partial Differential Equations

  NSF · $375k · 2015–2018

• Harmonic Analysis, Mathematical Physics, and Nonlinear PDE

  NSF · $288k · 2007–2013

• Long-Term Dynamics of Nonlinear Evolution Partial Differential Equations

  NSF · $24k · 2018–2019

• Global dynamics for nonlinear dispersive equations

  NSF · $333k · 2012–2016

Frequent coauthors

• Joachim Krieger

  École Polytechnique Fédérale de Lausanne

  52 shared

• Carlos E. Kenig

  University of Chicago

  38 shared

• Andrew Lawrie

  30 shared

• Camil Muscalu

  Romanian Academy

  27 shared

• Baoping Liu

  State Grid Corporation of China (China)

  25 shared

• Kenji Nakanishi

  24 shared

• Roland Donninger

  University of Vienna

  20 shared

• Avy Soffer

  19 shared