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The dynamics at the critical-point of a general first-order quantum phase transition in a finite system is examined, from an algebraic perspective. Suitable Hamiltonians are constructed whose spectra exhibit coexistence of states…
We give simple numerical bounds for $\zeta(s)$, $\vartheta(s)$, $\mathop{\mathcal R}(s)$, $Z(t)$, for use in the numerical computation of these functions. The purpose of the paper is to give bounds for several functions needed in the…
Let ${\bf{w}}(t,x):=(u,v)(t,x),\ t>0,\ x\in \mathbb{R}^{n},$ be the $\mathbb{R}^2$-valued spatial-temporal random field ${\bf{w}}=(u, v)$ arising from a certain two-equation system of fractional kinetic equations of reaction-diffusion type,…
We investigate systems of the form $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$ where $A \in B(\mathcal{H})$ is a normal operator in a separable Hilbert space $\mathcal{H}$, $\mathcal{G}\subset \mathcal{H}$ is a countable set, and $L$ is a positive…
We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. This estimate is crucial in the integrability analysis of a tensor-valued bilinear…
We discuss transformations generated by dynamical quantum systems which are bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an infinite-dimensional complex Hilbert space. We introduce the notion of Hermitian…
Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework…
The dynamics of spinorial wave functions in a causal fermion system is studied. A so-called dynamical wave equation is derived. Its solutions form a Hilbert space, whose scalar product is represented by a conserved surface layer integral.…
In this paper, we present more regularity conditions which ensure the boundedness of dilation operators on Besov and Triebel-Lizorkin spaces equiped with general weights.
Many examples of zeta functions in number theory and combinatorics are special cases of a construction in homotopy theory known as a decomposition space. This article aims to introduce number theorists to the relevant concepts in homotopy…
Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity…
We propose the construction of equations of motion based on symmetries in quantum-mechanical systems, using Heisenberg's uncertainty principle as a minimal foundation. From canonical operators, two spaces of conjugate operators are…
We use the functorial properties of Rieffel's pseudodifferential calculus to study families of operators associated to topological dynamical systems acted by a symplectic space. Information about the spectra and the essential spectra are…
In this work we prove analogues of Bessel inequality and Riesz-Fisher theorem in Hilbert spaces with respect to sequences. We apply our generalized Bessel inequality to the Hilbert spaces associated with the Normal, Beta, Gamma and certain…
In this article, we prove Herglotz's theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz's theorem for functional time series allows to…
A class of Riemann-Hilbert problems corresponding to quasi-permutation monodromy matrices is solved in terms of Szeg\"o kernel on auxiliary Riemann surfaces. The tau-function of Schlesinger system turns out to be closely related to…
We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally,…
This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems…
The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in…