arXiv subjects
Publications and source records attributed to N. Chernyavskaya.
We study the problem of correct solvability in the space $L_p(\mathbb R),$ $p\in[1,\infty)$ of the equation $$ -(r(x) y'(x))'+q(x)y(x)=f(x),\quad x\in \mathbb R $$ under the conditions $$r>0,\quad q\ge 0,\quad \frac{1}{r}\in L_1(\mathbb R),\quad q\in L_1(\mathbb R).$$
This document summarises the current theoretical and experimental status of the di-Higgs boson production searches, and of the direct and indirect constraints on the Higgs boson self-coupling, with the wish to serve as a useful guide for the next years. The document discusses the theoretical status, including state-of-the-art predictions for di-Higgs cross sections, developments on the effective field theory approach, and studies on specific new physics scenarios that can show up in the di-Higgs final state. The status of di-Higgs searches and the direct and indirect constraints on the Higgs self-coupling at the LHC are presented, with an overview of the relevant experimental techniques, and covering all the variety of relevant signatures. Finally, the capabilities of future colliders in determining the Higgs self-coupling are addressed, comparing the projected precision that can be obtained in such facilities. The work has started as the proceedings of the Di-Higgs workshop at Colliders, held at Fermilab from the 4th to the 9th of September 2018, but it went beyond the topics discussed at that workshop and included further developments.
We consider the differential equation \begin{align}\label{ab} -y'(x)+q(x)y(x)=f(x), \quad x \in \mathbb R, \end{align} where $f \in L_{p}(\mathbb R)$, $p\in [1,\infty)$, and $0\leq q \in L_{1}^{\rm loc}(\mathbb R)$, $\int\limits_{-\infty}^{0}q(t)\,dt=\int\limits_{0}^{\infty}q(t)\,dt=\infty,$ \begin{align*} q_{0}(a)=\inf_{x\in \mathbb R}\int_{x-a}^{x+a}q(t)\,dt=0 \quad{\rm \ for ~ any }\quad a\in (0,\infty). \end{align*} Under these conditions, the equation ({\rm \ref{ab}}) is not correctly solvable in $L_{p}(\mathbb R)$ for any $p \in [1, \infty) $. Let $q^{*}(x)$ be the Otelbaev-type average of the function $q(t), t\in \mathbb{R}$, at the point $t=x$; $θ(x)$ be a continuous positive function for $x \in \mathbb R$, and \begin{align*} L_{p,θ}(\mathbb R) = \{f\in L_{p}^{\rm loc}(\mathbb R):\, \int_{-\infty}^{\infty}|θ(x)f(x)|^{p}\,dx<\infty \}, \end{align*} \begin{align*} \|f\|_{L_{p,θ}(\mathbb R)}=\left(\int_{-\infty}^{\infty}|θ(x)f(x)|^{p}\,dx\right)^{1/p}\ \end{align*} We show that if there exists a constant $c\in [1, \infty)$, such that the inequality $$c^{-1}q^{*}(x)\leq θ(x)\leq cq^{*}(x)$$ holds for all $x \in \mathbb{R}$, then under some additional conditions for $q$ the pair of spaces $\{L_{p, θ}(\mathbb R); L_{p}(\mathbb R)\}$ is admissible for the equation ({\rm \ref{ab}}).
In this paper the simplest singular boundary problem of Dirichlet type for linear differential equation of the first order of general form is considered. The main result of this paper is criterion of correct solvability of above problem in the space L_p(R).