arXiv subjects

Peter Schlosser

Publications and source records attributed to Peter Schlosser.

At least 19 recordsLinked to original sources

arXiv2026

Nonlocal Fourier Laws for Heat Propagation via Fractional powers of Vector Operators

The present work is devoted to the study of fractional powers of vector operators, with particular emphasis on the gradient operator with non-constant coefficients. Within the setting of Clifford algebra $\mathbb{R}_n$, this operator turns out to have bisectorial properties. By applying the spectral theory on the $S$-spectrum, we address a fundamental mathematical challenge: unlike sectorial operators, bisectorial operators involve fractional powers that are not analytic on the negative real line. To circumvent this, we introduce a novel definition of the fractional power function in this setting. Building upon previous works on bisectorial vector operators and weak solutions, we extend the definition of fractional powers to abstract vector operators. The core contribution of this work is the application of the functional calculus for vector operators to the gradient operator, showing that these fractional powers provide a rigorous mathematical foundation for nonlocal Fourier laws in heat propagation.

math.FA↗

arXiv2025

The discrete Laplace asymptotic method and its application to the 3XOR satisfiability problem

A standard way to calculate the asymptotic behavior of integrals of the form \int_Wg(x)e^{-nh(x)}dx is the (continuous) Laplace asymptotic method. However, also discrete sums like \sum_{x\in W\capΛ_n}g_n(x)e^{-nh_n(x)} have similar behavior, when Λ_n is a discrete grid which becomes infinitely fine, and the functions g_n and h_n converge to g and h respectively. We go even further, and also derive the asymptotic formula for sums of the form \sum_{x\in W\capΛ_n}S_n(x), where the summand S_n asymptotically behaves as g_ne^{-nh_n}. The motivation, and also an immediate application, will be filling in all details in the classical breakthrough paper of Dubois and Mandler from 2002, which gives the solvability (phase transition) threshold of the 3XOR-SAT problem using the second moment method. Various analytical arguments there were lightly described, but the appendix of this paper combines recent results to fill all of them in. We would expect our theorems on asymptotics to apply to other (especially combinatorial) problems as well. For example, they seem effective on 3XOR-GAME problems.

math.CO↗

arXiv2025

Quadratic estimates for the $H^\infty$-functional calculus of bisectorial Clifford operators

The $H^\infty$-functional calculus is a two-step procedure, introduced by A. McIntosh, that allows the definition of functions of sectorial operators in Banach spaces. It plays a crucial role in the spectral theory of differential operators, as well as in their applications to evolution equations and various other fields of science. An extension of the $H^\infty$-functional calculus also exists in the hypercomplex setting, where it is based on the notion of $S$-spectrum. Originally this was done for sectorial quaternionic operators, but then also generalized all the way to bisectorial fully Clifford operators. In the latter setting and in Hilbert spaces, this paper now characterizes the boundedness of the $H^\infty$-functional calculus through certain quadratic estimates. Due to substantial differences in the definitions of the $S$-spectrum and the $S$-resolvent operators, the proofs of quadratic estimates in this setting face additional challenges compared to the classical theory of complex operators.

math.SP↗

arXiv2025

The $H^\infty$-functional calculus for right slice hyperholomorphic functions and right linear Clifford operators

In 2016, the spectral theory on the $S$-spectrum was used to establish the $H^\infty$-functional calculus for quaternionic or Clifford operators. This calculus applies for example to sectorial or bisectorial right linear operators $T$ and left slice hyperholomorphic functions $f$ that can grow as polynomials. It relies on the product of the two operators $e(T)^{-1}$ and $(ef)(T)$, both defined via some underlying $S$-functional calculus (also called $ω$-functional calculus). For left slice holomorphic functions $f$ this definition does not depend on the choice of the regularizer function $e$. However, due to the non-commutative multiplication of Clifford numbers, it was unclear how to extend this definition to right slice hyperholomorphic functions. This paper addresses this significant unresolved issue and shows how right linear operators can possess the $H^\infty$-functional calculus also for right slice hyperholomorphic functions.

math.SP↗

arXiv2025

The $H^\infty$-functional calculus for bisectorial Clifford operators

The aim of this article is to introduce the H-infinity functional calculus for unbounded bisectorial operators in a Clifford module over the algebra R_n. While recent studies have focused on bounded operators or unbounded paravector operators, we now investigate unbounded fully Clifford operators and define polynomially growing functions of them. We first generate the omega-functional calculus for functions that exhibit an appropriate decay at zero and at infinity. We then extend to functions with a finite value at zero and at infinity. Finally, using a subsequent regularization procedure, we can define the H-infinity functional calculus for the class of regularizable functions, which in particular include functions with polynomial growth at infinity and, if T is injective, also functions with polynomial growth at zero.

math.FA↗

arXiv2024

An introduction to the fine structures on the $S$-spectrum

Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum.

math.SP↗

arXiv2024

Spectral properties of the gradient operator with nonconstant coefficients

In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier's law for heat propagation and Fick's first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider $n\geq 3$ orthogonal unit vectors $e_1,\dots,e_n\in\mathbb{R}^n$, and let $Ω\subseteq\mathbb{R}^n$ be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator $T=\sum_{i=1}^ne_ia_i(x)\frac{\partial}{\partial x_i}$ with nonconstant positive coefficients $a_i:\overlineΩ\to(0,\infty)$. Under certain regularity and growth conditions on the $a_i$, we identify bisectorial or strip-type regions that belong to the $S$-resolvent set of $T$. Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the $S$-spectrum, designed to study the operators acting in Clifford modules $V$ over the Clifford algebra $\mathbb{R}_n$, with vector operators being a specific crucial subclass. The spectral properties related to the $S$-spectrum of $T$ are linked to the inversion of the operator $Q_s(T):=T^2-2s_0T+|s|^2$, where $s\in\mathbb{R}^{n+1}$ is a paravector, i.e., it is of the form $s=s_0+s_1e_1+\dots+s_ne_n$. This spectral problem is substantially different from the complex one, since it allows to associate general boundary conditions to $Q_s(T)$, i.e., to the squared operator $T^2$.

math.FA↗

arXiv2024

On a class of oscillatory integrals and their application to the time dependent Schrödinger equation

In this paper a class of oscillatory integrals is interpreted as a limit of Lebesgue integrals with Gaussian regularizers. The convergence of the regularized integrals is shown with an improved version of iterative integration by parts that generates additional decaying factors and hence leads to better integrability properties. The general abstract results are then applied to the Cauchy problem for the one dimensional time dependent Schrödinger equation, where the solution is expressed for C^n-regular initial conditions with polynomial growth at infinity via the Green's function as an oscillatory integral.

math.FA↗

arXiv2024

The $H^\infty$-functional calculi for the quaternionic fine structures of Dirac type

In this paper, we utilize various integral representations derived from the Fueter-Sce extension theorem, to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, due to the different factorizations of the Laplace opertor with respect to the Cauchy-Fueter operator and its conjugate, we identify four distinct classes of functions: Slice hyperholomorphic functions (leading to the $S$-functional calculus), axially harmonic functions (leading to the $Q$-functional calculus), axially polyanalytic functions of order $2$ (leading to the $P_2$-functional calculus), and axially monogenic functions (leading to the $F$-functional calculus). By applying the respective product rule, we establish the four different $H^\infty$-versions of these functional calculi.

math.FA↗

arXiv2024

Interpolation between domains of powers of operators in quaternionic Banach spaces

In contrast to the classical complex spectral theory, where the spectrum is related to the invertibility of $λ-A:D(A)\subseteq X_\mathbb{C}\rightarrow X_\mathbb{C}$, in the noncommutative quaternionic $S$-spectral theory one uses the invertibility of the second order polynomial $Q_s(T):=T^2-2\text{Re}(s)T+|s|^2:D(T^2)\subseteq X\rightarrow X$ to define the $S$-spectrum, where $X$ is a quaternionic Banach space. In this paper we will consider quaternionic operators $T$, for which at least one ray $\{te^{iω}\;|\;t>0\}$, $ω\in[0,π]$, $i\in\mathbb{S}$ is contained in the $S$-resolvent set, and the inverse operator $Q_s^{-1}(T)$ admits certain decay properties on this ray. Utilizing the $K$-interpolation method, we then demonstrate that the domain $D(T^k)$ of the $k$-th power of $T$ is an intermediate space between $D(T^n)$ and $D(T^m)$, whenever $n<k<m\in\mathbb{N}_0$. Moreover, also a characterization of the interpolation space $(X,D(T^n))_{θ,p}$, $θ\in(0,1)$, $p\in[1,\infty]$, in is given in terms of integrability conditions on the pseudo $S$-resolvent $Q_s^{-1}(T)$.

math.FA↗

arXiv2023

Characterization of continuous homomorphisms on entire slice monogenic functions

This paper is inspired by a class of infinite order differential operators arising in the time evolution of superoscillations. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e., functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra. We introduce the concept of proximate order and establish some fundamental properties of entire hyperholomorphic functions that are crucial for this characterization.

math.FA↗

arXiv2023

The harmonic $H^\infty$-functional calculus based on the S-spectrum

The aim of this paper is to introduce the $H^\infty$-functional calculus for harmonic functions over the quaternions. More precisely, we give meaning to Df(T) for unbounded sectorial operators T and polynomially growing functions of the form Df, where f is a slice hyperholomorphic function and $D=\partial_{q_0}+e_1\partial_{q_1}+e_2\partial_{q_2}+e_3\partial_{q_3}$ is the Cauchy-Fueter operator. The harmonic functional calculus can be viewed as a modification of the well known S-functional calculus f(T), with a different resolvent operator. The harmonic $H^\infty$-functional calculus is defined in two steps: First, for functions with a certain decay property, one can make sense of the bounded operator Df(T) directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators Df(T). The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the S-spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter-Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hypercomplex analysis.

math.FA↗

arXiv2023

Infinite order differential operators associated with superoscillations in the half-plane barrier

Superoscillations are a phenomenon in physics, where linear combinations of low-frequency plane waves interfere almost destructively in such a way that the resulting wave has a higher frequency than any of the individual waves. The evolution of superoscillatory initial datum under the time dependent Schrödinger equation is stable in free space, but in general it is unclear whether it can be preserved in the presence of an external potential. In this paper, we consider the two-dimensional problem of superoscillations interacting with a half-plane barrier, where homogenous Dirichlet or Neumann boundary conditions are imposed on the negative $x_2$-semiaxis. We use the Fresnel integral technique to write the wave function as an absolute convergent Green's function integral. Moreover, we introduce the propagator of the Schrödinger equation in form of an infinite order differential operator, acting continuously on the function space of exponentially bounded entire functions. In particular, this operator allows to prove that the property of superoscillations is preserved in the form of a similar phenomenon called supershift, which is stable over time.

math-ph↗

arXiv2023

Integral representation of superoscillations via complex Borel measures and their convergence

In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space $\mathcal{A}_1(\mathbb{C})$ of exponentially bounded entire functions.

math-ph↗

arXiv2022

Time evolution of superoscillations for the Schrödinger equation in $\mathbb{R}\setminus\{0\}$

In the context of quantum mechanics superoscillations, or the more general supershifts, appear as initial conditions of the time dependent Schrödinger equation. Already in \cite{ABCS21_2} a unified approach was developed, which yields time persistence of the supershift property under certain holomorphicity and growth assumptions on the corresponding Green's function. While that theory considers the Schrödinger equation on the whole real line $\mathbb{R}$, this paper takes the natural next step and considers $\mathbb{R}\setminus\{0\}$ instead, and allow boundary conditions at $x=0^\pm$ in addition. In particular the singular $\frac{1}{x^2}$-potential as well as the very important $δ$ and $δ'$ distributional potentials are covered.

math-ph↗

arXiv2022

Schrödinger operators with $δ$-potentials supported on unbounded Lipschitz hypersurfaces

In this note we consider the self-adjoint Schrödinger operator $\mathsf{A}_α$ in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $δ$-potential supported on a Lipschitz hypersurface $Σ\subseteq\mathbb{R}^d$ of strength $α\in L^p(Σ)+L^\infty(Σ)$. We show the uniqueness of the ground state and, under some additional conditions on the coefficient $α$ and the hypersurface $Σ$, we determine the essential spectrum of $\mathsf{A}_α$. In the special case that $Σ$ is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schrödinger operator as Birman-Schwinger operator. As an application we prove an optimization result for the bottom of the spectrum of $\mathsf{A}_α$.

math.SP↗

arXiv2021

A unified approach to Schrödinger evolution of superoscillations and supershifts

Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper we give a unified approach to determine the supershift property for the solution of the time dependent Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique we are able to treat various potentials.

math.AP↗

arXiv2020

Green's Function for the Schrödinger Equation with a Generalized Point Interaction and Stability of Superoscillations

In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the $δ$ and $δ'$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.

math.AP↗