arXiv subjects
Publications and source records attributed to T. Kappeler.
We consider the torsion function for the Dirichlet Laplacian $-Δ$, and for the Schrödinger operator $- Δ+ V$ on an open set $Ω\subset \R^m$ of finite Lebesgue measure $0<|Ω|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.
We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H^{s}_{0}(\mathbb{T},\mathbb{R})$, $s > -1/2$, to the scale of weighted $\ell^2-$sequence spaces, $\mathfrak{h}^{s +1/2}_{r,0}(\mathbb{N},\mathbb{C})$, $s >-1/2$. As an application we show that for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation $\mathcal{S}_0^t : H^{s}_{0}(\mathbb{T},\mathbb{R})\to H^{s}_{0}(\mathbb{T},\mathbb{R})$ is {\em nowhere locally uniformly continuous} in $H^{s}_{0}(\mathbb{T},\mathbb{R})$.
The Thinking Wave is an ongoing development of visualization concepts showing the real-time effort and confidence of semi-autonomous vehicle (AV) systems. Offering drivers access to this information can inform their decision making, and enable them to handle the situation accordingly and takeover when necessary. Two different visualizations have been designed, Concept one, Tidal, demonstrates the AV systems effort through intensified activity of a simple graphic which fluctuates in speed and frequency. Concept two, Tandem, displays the effort of the AV system as well as the handling dynamic and shared responsibility between the driver and the vehicle system. Working collaboratively with mobility research teams at the University of Tokyo, we are prototyping and refining the Thinking Wave and its embodiments as we work towards building a testable version integrated into a driving simulator. The development of the thinking wave aims to calibrate trust by increasing the drivers knowledge and understanding of vehicle handling capacity. By enabling transparent communication of the AV systems capacity, we hope to empower AV-skeptic drivers and keep over-trusting drivers on alert in the case of an emergency takeover situation, in order to create a safer autonomous driving experience.
In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in $H^{s}_{0}(\T, \R)$ for any $s>-1/2$ where $H^{s}_{0}(\T, \R)$ denotes the subspace of the Sobolev space $H^{s}(\T, \R)$ of elements with mean $0$. As an application we show that for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation $\mathcal{S}_0^t : H^{s}_{0}(\T, \R)\to H^{s}_{0}(\T, \R)$ is nowhere locally uniformly continuous in a neighborhood of zero in $H^{s}_{0}(\T, \R)$.
We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(\mathbb{T},\mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(\mathbb{T},\mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results.
We prove that there exist potentials so that near them the focusing non-linear Schrödinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.
In this paper we prove approximation properties of the solutions of the defoucsing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces $H^s$ with $s\geq 1$ remain bounded for all times.
In this paper we prove that the non-linear Fourier transform of the defocusing NLS equation on the circle is linear up to terms which are one order smoother.
In this paper we provide new asymptotic estimates of various spectral quantities of Zakharov-Shabat operators on the circle. These estimates are uniform on bounded subsets of potentials in Sobolev spaces.
We develop a new approach for constructing normalized differentials on hyperelliptic curves of infinite genus and obtain uniform asymptotic estimates for the distribution of their zeros.
Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c > 0$. For any $0<θ<1$, denote by $X_θ= [X_0, X_1]_θ$ and $Y_θ= [Y_0, Y_1]_θ$ the complex interpolation spaces and by $B(r, X_θ)$, $0 \le θ\le 1,$ the open ball of radius $r>0$ in $X_θ$, centered at zero. Then for any analytic map $Φ: B(r, X_0) \to Y_0+ Y_1$ such that $Φ: B(r, X_0)\to Y_0$ and $Φ: B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $Φ$ to $B(c^{-θ}r, X_θ)$, $0 < θ< 1,$ is shown to be a map with values in $Y_θ$ which is analytic and bounded by $M_0^{1-θ} M_1^θ$.
We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schrödinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.
In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree.
In this paper we develop tools to study families of non-selfadjoint operators $L(φ), φ\in P$, characterized by the property that the spectrum of $L(φ)$ is (partially) simple. As a case study we consider the Zakharov-Shabat operators $L(φ)$ appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials $φ$ of Sobolev class $H^N, N \geq 0$, so that all small eigenvalues of $L(φ)$ are simple, is path connected and dense.
In this paper we provide new asymptotic estimates of the Floquet exponents of Schrödinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.
We construct normalized differentials on families of curves of infinite genus. Such curves are used to investigate integrable PDE's such as the focusing Nonlinear Schr{ö}dinger equation.
We prove that the Birkhoff map $\Om$ for KdV constructed on $H^{-1}_0(\T)$ can be interpolated between $H^{-1}_0(\T)$ and $L^2_0(\T)$. In particular, the symplectic phase space $H^{1/2}_0(\T)$ can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential $q\in H^{-1}(\T)$ in terms of the decay of the gap lengths of the periodic spectrum of Hill's operator on the interval $[0,2]$.
In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax L-A pair. Using it we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and may even include functions which tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schr{ö}dinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.