Vol 85, No 1 (2021)

We consider the Cauchy problem for a Schrödinger equation whose Hamiltonian is the difference of the operatorof multiplication by the potential and the operator of taking the second derivative. Here the potential is a realdifferentiable function of a real variable such that this function and its derivative are bounded. This equationhas been studied since the advent of quantum mechanics and is still a good model case for variousmethods of solving partial differential equations. We find solutions of the Cauchy problem in the form of quasi-Feynman formulae by using Remizov's theorem. Quasi-Feynman formulae are relatives of Feynmanformulae containing multiple integrals of infinite multiplicity. Their proof is easier than that of Feynman formulae butthey give longer expressions for the solutions. We provide detailed proofs of all theorems and deliberately restrict thespectrum of our results to the domain of classical mathematical analysis and elements of real analysis trying to avoidgeneral methods of functional analysis. As a result, the paper is long but accessible to readers whoare not experts in the field of functional analysis.

We study the wedge of solutions of the inequality $A(u) \ge 0$, where $A$ is a linear elliptic operator of order $2m$ acting on functions \linebreak of $n$ variables. We establish interior estimates of the form $\|u; W_p^{2m-1}(\omega)\| \le C(\omega,\Omega) \|u;L(\Omega)\|$ for the elements of this wedge, where $\omega$ is a compact subdomain of $\Omega$, $W_p^{2 m-1}(\omega)$ is the Sobolev space, $p (n-1) < n$, $L(\Omega)$ is the Lebesgue space of integrable functions, and the constant $C(\omega,\Omega)$ is independent of $u$.

We prove that the Grothendieck standard conjecture of Lefschetz type holdsfor a smooth complex projective $4$-dimensional variety $X$fibred by Abelian varieties (possibly, with degeneracies)over a smooth projective curve if the endomorphism ring $\operatorname{End}_{\overline{\kappa(\eta)}} (X_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})$ of the genericgeometric fibre is not an orderof an imaginary quadratic field. This conditionholds automatically in the cases when the reduction of the generic scheme fibre $X_\eta$ at someplace of the curve is semistable in the sense of Grothendieck and hasodd toric rank or the generic geometric fibre is not a simple Abelian variety.