An orthomodular lattice $L$ is said to be interval homogeneous (resp. centrally interval homogeneous) if it is $\sigma$-complete and satisfies the following property: Whenever $L$ is isomorphic to an interval, $[a,b]$, in $L$ then $L$ is isomorphic to each interval $[c,d]$ with $c\le a$ and $d\ge b$ (resp. the same condition as above only under the assumption that all elements $a$, $b$, $c$, $d$ are central in $L$). Let us denote by Inthom (resp. Inthom${}_c$) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous...

The notion of a joint distribution in $\sigma$-finite measures of observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.

This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma$-finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system...

We consider a Schrödinger-type differential expression $H_V={\nabla }^{*}\nabla +V$, where $\nabla$ is a $C^{\infty }$-bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M,g)$ with metric $g$ and positive $C^{\infty }$-bounded measure $d\mu$, and $V$ is a locally integrable section of the bundle of endomorphisms of $E$. We give a sufficient condition for $m$-sectoriality of a realization of $H_V$ in $L^2\left(E\right)$. In the proof we use generalized Kato’s inequality as well as a result on the positivity of $u\in L^2\left(M\right)$ satisfying the...

The paper deals with a new mathematical model for quantum mechanics based on the fuzzy set theory [1]. The indefinite integral of observables is defined and some basic properties of the integral are examined.

Motivati dall'analisi asintotica dei vortici nella teoria di Chern-Simons-Higgs, si studia l'equazione $$-\mathrm{\Delta }u=\lambda \left(\frac{e^u}{{\int }_{\mathrm{\Omega }}{e}^{u}dx}-\frac{1}{\left|\mathrm{\Omega }\right|}\right),u\in H^1\left(\mathrm{\Omega }\right)$$ dove $\mathrm{\Omega }={\mathbb{R}}^2/{\mathbb{Z}}^2$ é il toro piatto bidimensionale. In contrasto con l'analogo problema di Dirichlet, si dimostra che per $\lambda \in \left(8\pi ,4{\pi }^2\right)$ l'equazione ammette una soluzione non banale. Tale soluzione cattura il carattere bidimensionale dell'equazione, nel senso che, per tali valori di $\lambda$, l'equazione non può ammettere soluzioni (periodiche) non banali dipendenti da una sola variabile (vedi [10]).

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on $S^2$. We also construct a solution of the equation $\Delta u=u$ in ${ℝ}^2$ that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

We develop the $\overline{\partial }$-approach to inverse scattering at zero energy in dimensions $d\ge 3$ of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction procedure, stability estimate and approximate reconstruction for the problem of finding a sufficiently small potential $v$ in the Schrödinger equation from a fixed non-overdetermined (“backscattering” type) restriction ${h|}_{\Gamma }$ of the Faddeev generalized scattering amplitude $h$ in the...