arXiv subjects

Frank Rösler

Publications and source records attributed to Frank Rösler.

12 recordsLinked to original sources

arXiv2024

Computing scattering resonances of rough obstacles

This paper is concerned with the numerical computation of scattering resonances of the Laplacian for Dirichlet obstacles with rough boundary. We prove that under mild geometric assumptions on the obstacle there exists an algorithm whose output is guaranteed to converge to the set of resonances of the problem. The result is formulated using the framework of Solvability Complexity Indices. The proof is constructive and provides an efficient numerical method. The algorithm is based on a combination of a Glazman decomposition, a polygonal approximation of the obstacle and a finite element method. Our result applies in particular to obstacles with fractal boundary, such as the Koch Snowflake and certain filled Julia sets. Finally, we provide numerical experiments in MATLAB for a range of interesting obstacle domains.

math.NA↗

arXiv2023

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counter example showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

math.AP↗

arXiv2023

On the complexity of the inverse Sturm-Liouville problem

This paper explores the complexity associated with solving the inverse Sturm-Liouville problem with Robin boundary conditions: given a sequence of eigenvalues and a sequence of norming constants, how many limits does a universal algorithm require to return the potential and boundary conditions? It is shown that if all but finitely many of the eigenvalues and norming constants coincide with those for the zero potential then the number of limits is zero, i.e. it is possible to retrieve the potential and boundary conditions precisely in finitely many steps. Otherwise, it is shown that this problem requires a single limit; moreover, if one has a priori control over how much the eigenvalues and norming constants differ from those of the zero-potential problem, and one knows that the average of the potential is zero, then the computation can be performed with complete error control. This is done in the spirit of the Solvability Complexity Index. All algorithms are provided explicitly along with numerical examples.

math.NA↗

arXiv2022

Computing Klein-Gordon Spectra

We study the computational complexity of the eigenvalue problem for the Klein-Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein-Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein-Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.

math.SP↗

arXiv2021

A Strange Vertex Condition Coming From Nowhere

We prove norm-resolvent and spectral convergence in $L^2$ of solutions to the Neumann Poisson problem $-Δu_\varepsilon = f$ on a domain $Ω_\varepsilon$ perforated by Dirichlet-holes and shrinking to a 1-dimensional interval. The limit $u$ satisfies an equation of the type $-u''+μu = f$ on the interval $(0,1)$, where $μ$ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighbourhood and the vertex neighbourhood is chosen correctly, the constant $μ$ will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation.

math.AP↗

arXiv2021

Universal algorithms for computing spectra of periodic operators

Schrödinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a "one-size-fits-all" algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schrödinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schrödinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits.

math.SP↗

arXiv2021

Computing the Sound of the Sea in a Seashell

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $\mathcal C^2$. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.

math.SP↗

arXiv2020

Computing Scattering Resonances

The question of whether it is possible to compute scattering resonances of Schrödinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to be known a priori is the size of the support of the potential. The potential itself is merely required to be $\mathcal{C}^1$. The proof is constructive, providing a universal algorithm which only needs to access the values of the potential at any requested point.

math.SP↗

arXiv2019

On The Solvability Complexity Index for Unbounded Selfadjoint Operators and Schrödinger Operators

We study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential spectrum of a selfadjoint operator is convex, then the SCI for computing its spectrum is equal to 1. This result is then extended to relatively compact perturbations of such operators and applied to Schrödinger operators with compactly supported (complex valued) potentials to obtain SCI=1 in this case, as well.

math.SP↗

arXiv2018

A Note on Spectral Convergence in Varying Hilbert Spaces

We prove sufficient conditions for Hausdorff convergence of the spectra of sequences of closed operators defined on varying Hilbert spaces and converging in norm-resolvent sense, i.e. $\|J_\varepsilon(1+A_\varepsilon)^{-1} - (1+A)^{-1}J_\varepsilon\|\to 0$ as $\varepsilon\to 0$, where $J_\varepsilon$ is a suitable identification operator between the domains of the operators. This is an extension of results of [Mugnolo-Nittka-Post(2013)], who proved absence of spectral pollution for sectorial operators.

math.SP↗

arXiv2018

Norm-Resolvent Convergence in Perforated Domains

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-Δ$ in the perforated domain $Ω\setminus \bigcup_{ i\in 2\varepsilon\mathbb Z^d }B_{a_\varepsilon}(i),$ $a_\varepsilon\ll\varepsilon,$ to the limit operator $-Δ+μ_ι$ on $L^2(Ω)$, where $μ_ι\in\mathbb C$ is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Pro c. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.

math.AP↗

arXiv2017

A Bound on the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential

We are concerned with the non-normal Schrödinger operator $$ H=-Δ+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this operator is discrete and contained in the positive half plane. In general, the $\varepsilon$-pseudospectrum of $H$ will have an unbounded component for any $\varepsilon>0$ and thus will not approximate the spectrum in a global sense. By exploiting the fact that the semigroup $e^{-tH}$ is immediately compact, we show a complementary result, namely that for every $δ>0$, $R>0$ there exists an $\varepsilon>0$ such that the $\varepsilon$-pseudospectrum $$ σ_\varepsilon(H)\subset \{z:\operatorname{Re}(z) \geq R\}\cup\bigcup_{λ\inσ(H)}\{z:|z-λ|<δ\}. $$ In particular, the unbounded part of the pseudospectrum escapes towards $+\infty$ as $\varepsilon$ decreases. Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.

math-ph↗