arXiv subjects

Kasper S. Sørensen

Publications and source records attributed to Kasper S. Sørensen.

6 recordsLinked to original sources

arXiv2025

Sharp iteration asymptotics for transfer operators induced by greedy $β$-expansions

We consider base-$β$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<β<a_0+1$ is the positive solution to $β^2 = a_0β+ a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto βx-\lfloor βx\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\mathscr{P}u=u$ and $\mathscr{P}v=β^{-1} v$ such that for every sufficiently smooth $F$ which is supported in $[0,1]$ and satisfies $\int_0^1 F \; \mathrm{d} x=1$, we have $\mathscr{P}^kF= u +β^{-k}\big ( F(1)-F(0)\big )v +o(β^{-k})$ in $L^\infty$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.

math.DS↗

arXiv2023

Bulk-edge correspondence for unbounded Dirac-Landau operators

We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying functions of the bulk and edge operators. One of these corollaries leads to the so-called Středa formula: if the bulk operator has an isolated compact spectral island, then the integrated density of states of the corresponding bulk spectral projection varies linearly with the magnetic field as long as the gaps between the spectral island and the rest of the spectrum are not closed, and the slope of this variation is given by the Chern character of the projection. The same bulk Chern character is related to the number of edge states which appear in the gaps of the bulk operator.

math-ph↗

arXiv2022

Singular distribution functions for random variables with stationary digits

Let $F$ be the cumulative distribution function (CDF) of the base-$q$ expansion $\sum_{n=1}^\infty X_n q^{-n}$, where $q\ge2$ is an integer and $\{X_n\}_{n\geq 1}$ is a stationary stochastic process with state space $\{0,\ldots,q-1\}$. In a previous paper we characterized the absolutely continuous and the discrete components of $F$. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: $F$ is then either a uniform or a singular CDF on $[0,1]$. Moreover, we study mixtures of such models. In most cases expressions and plots of $F$ are given.

math.PR↗

arXiv2022

Characterization of random variables with stationary digits

Let $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$, and $F$ the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$. We show that stationarity of $\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by $F$ and use this to characterize the characteristic function of $X$ and the structure of $F$ in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of $F$ can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that $\mathrm{d} F$ is a Rajchman measure if and only if $F $ is the uniform CDF on $[0,1]$.

math.PR↗

arXiv2022

Optimal Profile Design for Acoustic Black Holes using Timoshenko beam Theory

We revisit the problem of constructing one-dimensional acoustic black holes. Instead of considering the Euler-Bernoulli beam theory, we use Timoshenko's approach instead, which is known to be more realistic at higher frequencies. Our goal is to minimize the reflection coefficient under a constraint imposed on the normalized wave number variation. We use the calculus of variations in order to derive the corresponding Euler-Lagrange equation analytically and then use numerical methods to solve this equation in order to find the optimal height profile for different frequencies. We then compare these profiles to the corresponding ones previously found using the Euler-Bernoulli beam theory and see that in the lower range of the dimensionless frequency $Ω$ (defined using the largest height of the plate), the optimal profiles almost coincide, as expected. For higher such frequencies, even for values where Euler-Bernoulli theory should still be marginally valid, the profiles predicted using Euler-Bernoulli differ substantially from the correct ones predicted by Timoshenko theory. One explanation for this phenomenon is that unlike in the constant height case, in our setting the wave numbers also depend on the ratio between the smallest and the largest heights.

physics.class-ph↗

arXiv2018

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

First, we reconsider the magnetic pseudodifferential calculus and show that for a large class of non-decaying symbols, their corresponding magnetic pseudodifferential operators can be represented, up to a global gauge transform, as generalized Hofstadter-like, bounded matrices. As a by-product, we prove a Calderón-Vaillancourt type result. Second, we make use of this matrix representation and prove sharp results on the spectrum location when the magnetic field strength $b$ varies. Namely, when the operators are self-adjoint, we show that their spectrum (as a set) is at least $1/2$-Hölder continuous with respect to $b$ in the Hausdorff distance. Third, when the magnetic perturbation comes from a constant magnetic field we show that their spectral edges are Lipschitz continuous in $b$. The same Lipschitz continuity holds true for spectral gap edges as long as the gaps do not close.

math.AP↗