Related papers: Wigner's theorem revisited
The core of this article is a general theorem with a large number of specializations. Given a manifold $N$ and a finite number of one-parameter groups of point transformations on $N$ with generators $Y, X_{(1)}, \cdots, X_{(d)} $, we…
The equation $f^n+g^n=1$, $n\in\mathbb{N}$ can be regarded as the Fermat Diophantine equation over the function field. In this paper we study the characterization of entire solutions of some system of Fermat type functional equations by…
Making use of nonabelian harmonic analysis and representation theory, we solve the functional equation $$f_1(xy)+f_2(yx)+f_3(xy^{-1})+f_4(y^{-1}x)=f_5(x)f_6(y)$$ on arbitrary compact groups. The structure of its general solution is…
In this paper, we prove well-posedness in $C^1(\mathbb{R})$ (a.k.a. classical solutions) of the Fornberg-Whitham equation. To achieve this objective, we study its weak formulation under a Lagrangian framework. Applying the fundamental…
A general functional definition of the infinite dimensional quantum $R$-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional $R$-matrix from the general definition is demonstrated in a…
We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all…
We show that Fueter's theorem holds for a more general class of quaternionic functions than those constructed by the Fueter's method.
Let $H$ be a complex Hilbert space and let ${\mathcal F}_{s}(H)$ be the real vector space of all self-adjoint finite rank operators on $H$. We prove the following non-injective version of Wigner's theorem: every linear operator on…
We extend almost everywhere convergence in Wiener-Wintner ergodic theorem for $\sigma$-finite measure to a generally stronger almost uniform convergence and present a larger, universal, space for which this convergence holds. We then extend…
Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…
We determine the form of the Wigner functional for several types of quantum free field theories in order to analyze the representation of QFT in phase space, as well as to compare it to other mainstream formulations. We use Jackiw's…
In this paper, we obtain the general solution and the generalized Hyers-Ulam Rassias stability of the functional equation $$3(f(x+2y)+f(x-2y))=12(f(x+y)+f(x-y))+4f(3y)-18f(2y)+36f(y)-18f(x).$$
In this paper we define a new space, $LH(X,Y)$, consisting of functions $f\in X \subset Y$ (with $X,Y$ normed spaces) such that $\| f \|_X \equiv \| f \|_Y$ (where $\| \cdot \|_X$ is any norm on $X$, in general not the norm induced by $\|…
We obtain a general expression for a Wigner transform (Wigner function) on symmetric spaces of non-compact type and study the Weyl calculus of pseudodifferential operators on them.
Using the fixed point theorem we establish the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation $$\frac{1}{\mid K\mid}\sum_{k\in K}f(x+k\cdot y)=g(x)+h(y),\;\;x,y\in E$$ from a normed space $E$ into a complete…
In this paper we study a mixed problem for the nonlinear Schr\"odinger equation globally that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a global solvability theorem of the considered problem…
We calculate the Wigner distribution function for the Calogero-Sutherland system which consists of harmonic and inverse-square interactions. The Wigner distribution function is separated out into two parts corresponding to the relative and…
In view of the usefulness and importance of the kinetic equation in certain physical problems, the authors derive the explicit solution of a fractional kinetic equation of general character, that unifies and extends earlier results.…
In this paper, strongly $(\alpha,T)$-convex functions, i.e., functions $f:D\to \R$ satisfying the functional inequality $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-t\alpha\big((1-t)(x-y)\big)-(1-t)\alpha\big(t(y-x)\big)$$ for $x,y\in D$ and $t\in…
The Wigner function shares several properties with classical distribution functions on phase space, but is not positive-definite. The integral of the Wigner function over a given region of phase space can therefore lie outside the interval…