Asymptotics of the Weyl function for Schrödinger operators with measure-valued potentials - Luger, Annemarie; Teschl, Gerald; Wöhrer, Tobias (Monatshefte für Mathematik, 2016)
We derive an asymptotic expansion for the Weyl function of a one-dimensional Schrödinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.
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Keywords: Asymptotics; Distributional coefficients; Schrödinger operators; Weyl function

Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation - Arnold, Anton; Einav, Amit; Signorello, Beatrice; Wöhrer, Tobias (Journal of Statistical Physics, 2021)
The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher’s equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to those found in [8].
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Keywords: BGK equation; Exponential decay; Hypocoercivity; Large time behaviour; Lyapunov functional

On the rates of decay to equilibrium in degenerate and defective Fokker–Planck equations - Arnold, Anton; Einav, Amit; Wöhrer, Tobias (Journal of Differential Equations, 2018)
We establish sharp long time asymptotic behaviour for a family of entropies to defective Fokker–Planck equations and show that, much like defective finite dimensional ODEs, their decay rate is an exponential multiplied by a polynomial in time. The novelty of our study lies in the amalgamation of spectral theory and a quantitative non-symmetric hypercontractivity result, as opposed to the usual approach of the entropy method.
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Keywords: Fokker–Planck equations; Long time behaviour; Non-symmetric hypercontractivity; Spectral theory

Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations - Arnold, Anton; Jin, Shi; Wöhrer, Tobias (Journal of Differential Equations, 2020)
We review the Lyapunov functional method for linear ODEs and give an explicit construction of such functionals that yields sharp decay estimates, including an extension to defective ODE systems. As an application, we consider three evolution equations, namely the linear convection-diffusion equation, the two velocity BGK model and the Fokker–Planck equation. Adding an uncertainty parameter to the equations and analyzing its linear sensitivity leads to defective ODE systems. By applying the Lyapunov functional construction, we prove sharp long time behavior of order (1+tM)e−μt, where M is the defect and μ is the spectral gap of the system. The appearance of the uncertainty parameter in the three applications makes it important to have decay estimates that are uniform in the non-defective limit.
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Keywords: Defective ODEs; Kinetic equations; Long time behavior; Lyapunov functionals; Sensitivity analysis; Uncertainty quantification

Sharpening of decay rates in Fourier based hypocoercivity methods - Wöhrer, Tobias; Dolbeault, Jean; Arnold, Anton; Schmeiser, Christian (Recent Advances in Kinetic Equations and Applications, 2021)