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## Some parabolic equations for measures and Gaussian semigroups

Cornell University
,
2020.
No. arXiv:2012.07174.

This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their properties. The results are given without proofs but the proofs will appear in the longer text which is now under preparation.
In this paper, infinite-dimensional generalizations of the Euclidean analogue of the Schrödinger equation for anharmonic oscillator are considered in the class of measures. The Cauchy problem for these equations is solved. In particular cases, explicit formulas for fundamental solutions are obtained, which are a generalization of the Mehler formula, and the uniqueness of the solution with certain properties is proved. An analogue of the Ornstein-Uhlenbeck measure is constructed. The definition of Gaussian semigroups is given and their connection with the considered parabolic equations is described.

Keywords: Schrodinger equationparabolic equations for measures exact solutionCauchy problemlinear evolution equationGaussian semigroupanharmonic oscillator

Publication based on the results of:

Remizov I., Potential analysis 2020 Vol. 52 P. 339-370

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients ...

Added: September 30, 2018

Vybornyi E., Theoretical and Mathematical Physics 2014 Vol. 181 No. 2 P. 1418-1427

We propose an operator method for calculating the semiclassical asymptotic form of the energy splitting value in the general case of tunneling between symmetric orbits in the phase space. We use this approach in the case of a particle on a circle to obtain the asymptotic form of the energy tunneling splitting related to the ...

Added: August 5, 2014

Vybornyi E., Наноструктуры. Математическая физика и моделирование 2015 Т. 12 № 1 С. 5-84

We consider the problem of constructing semiclassical asymptotic expansions of discrete spectrum and the corresponding stationary states of one-dimensional Schrödinger operator in the case of resonance tunneling. We consider two basic models: tunneling in an asymmetric double-well potential on a line and momentum tunneling of a particle in a potential field on a circle. For ...

Added: February 12, 2016

Beklaryan A., Beklaryan L., Дифференциальные уравнения 2017 Т. 53 № 2 С. 148-159

The Cauchy problem for the homogeneous linear functional-differential equation of a pointwise type, defined on the line, is considered. In the case of one-dimensional equation we formulated the theorem of existence and uniqueness of solutions with estimating of its order of growth. This research is carried out within the formalism based on group peculiarities of ...

Added: February 12, 2017

Remizov I., Journal of Functional Analysis 2016 Vol. 270 No. 12 P. 4540-4557

For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space $\mathcal{F}$ the operator $\exp(-it\mathcal{H})$ is the evolution operator for the Schr\"odinger equation $i\psi'_t=\mathcal{H}\psi$, i.e. if $\psi(0,x)=\psi_0(x)$ then $\psi(t,x)=(\exp(-it\mathcal{H})\psi_0)(x)$ for $x\in Q.$ The space $\mathcal{F}$ here is the space of wave functions $\psi$ defined on an abstract space $Q$, the configuration space of a quantum system, ...

Added: March 3, 2018

Nazaikinskii V., BEDRIKOVETSKY P., LIUDMILA I. KUZMINA et al., SIAM Journal on Applied Mathematics 2020 Vol. 80 No. 5 P. 2120-2143

An initial-boundary value problem for a quasilinear system describing deep bed ltration of a monodisperse suspension in a medium with pores of various sizes is investigated analytically.
The ltration function is assumed to have power-law type while tending to zero with the power index
lower than one. We found that this assumption has two consequences: (i) the ...

Added: January 7, 2021

Vedenin A., Galkin V., Karatetskaia E. et al., / Cornell University. Series arXiv "math". 2020.

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations. ...

Added: October 12, 2019

Chernyshev V.L., Russian Journal of Mathematical Physics 2016 Vol. 23 No. 3 P. 348-354

On a two-dimensional surface, a Schrödinger operator is considered with a potential whose critical points form a closed curve. We pose the problem of describing the semiclassical spectral series corresponding to this curve. The standard construction for describing the spectral series corresponding to isolated nondegenerate equilibria or to periodic trajectories of Hamiltonian systems is not ...

Added: October 25, 2014

Remizov I., Applied Mathematics and Computation 2018 Vol. 328 P. 243-246

We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The method is based on the Chernoff approximation procedure applied to a specially constructed shift operator. It is ...

Added: May 25, 2018

Vedenin A., Remizov I., / Cornell University. Series math "arxiv.org". 2020.

Abstract. The method of Chernoff approximation was discovered by Paul Chernoff in 1968 and now is a powerful and flexible tool of contemporary functional analysis. This method is different from grid-based approach and helps to solve numerically the Cauchy problem for evolution equations, e.g., for heat equation and for more general parabolic second-order partial differential equations ...

Added: December 14, 2020

Bruning J., Grushin V. V., Dobrokhotov S. Y., Математические заметки 2012 Т. 92 № 2 С. 163-180

An example of Schrodinger and Klein-Gordon equations with fast oscillating coefficients is used to show that they can be averaged by an adiabatic approximation based on V.P. Maslov's operator method. ...

Added: December 24, 2012

Abrashkin A. A., Discrete and Continuous Dynamical Systems 2019 Vol. 39 No. 8 P. 4443-4453

A class of non-stationary surface gravity waves propagating in the
zonal direction in the equatorial region is described in the f -plane approx
imation. These waves are described by exact solutions of the equations of
hydrodynamics in Lagrangian formulation and are generalizations of Gerstner
waves. The wave shape and non-uniform pressure distribution on a free sur
face depend on two ...

Added: June 19, 2019

Rabinowitch A. S., Journal of Mathematical Physics 2016 Vol. 57 P. 083103-1-083103-6

Added: October 5, 2019

Shaposhnikov S., Bogachev V., Roeckner M., Journal of Mathematical Sciences 2011 Vol. 179 No. 1 P. 1-41

We survey recent results related to uniqueness problems for parabolic equations for measures. Our study are motivated by equations of such a type, namely, the FokkerPlanckKolmogorov equations for transition probabilities of diffusion processes. Solutions are considered in the class of probability measures and in the class of signed measures with integrable densities. We present some ...

Added: October 14, 2014

Florido Calvo F. A., Remizov I., / Cornell University. Series arXiv "math". 2021.

Dynamics of closed quantum systems on curves, surfaces and more general manifolds is governed by the Schroedinger equation with time-independent Hamiltonian. Solving Cauchy problem for this equation provides full information on the future and the past of the system if we know the state of the system at the initial moment of time t=0. However, ...

Added: December 16, 2021

Shaposhnikov S., Теория вероятностей и ее применения 2011 Т. 56 № 2 С. 318-350

Получены локальные и глобальные оценки решений параболических уравнений для мер. Исследовано поведение решения на бесконечности и при подходе к начальному условию. ...

Added: October 14, 2014

Remizov I., Journal of Mathematical Physics 2019 Vol. 60 No. 7 P. 1-8

We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the C0-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and ...

Added: June 28, 2019

Anikin A. Y., Brüning J., Dobrokhotov S. et al., Russian Journal of Mathematical Physics 2019 Vol. 26 No. 3 P. 265-276

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can ...

Added: September 18, 2019

Vedenin A., Воеводкин В. С., Galkin V. et al., Математические заметки 2020 Т. 108 № 3 С. 463-468

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations. ...

Added: October 21, 2019

Solvability Problems for a Linear Homogeneous Functional-Differential Equation of the Pointwise Type

Beklaryan A., Beklaryan L. A., Differential Equations 2017 Vol. 53 No. 2 P. 145-156

The Cauchy problem for a linear homogeneous functional-differential equation of the pointwise type defined on a straight line is considered. Theorems on the existence and uniqueness of the solution in the class of functions with a given growth are formulated for the case of the one-dimensional equation. The study is performed using the group peculiarities ...

Added: March 6, 2017

Karasev M., Vybornyi E., Journal of Mathematical Physics 2016

We consider the one-dimensional Schrodinger operator in the semiclassical regime assuming that its double-well potential is the sum of a finite "physically given" well and a square shape probing well whose width or depth can be varied (tuned). We study the dynamics of an initial state localized in the physical well. It is shown that ...

Added: October 23, 2015

Ludmila I. Kuzmina, Osipov Y., International Journal for Computational Civil and Structural Engineering 2016 Vol. 12 No. 3 P. 145-150

Filtration problem of suspension with identical particles in a porous medium is considered. Physical and mathematical models for size-exclusion capture mechanism of suspended particles at the inlet of filter pores are presented. An exact solution is constructed for the forward and back flow of suspension in a porous medium in case of linear filtration coefficient. ...

Added: December 26, 2016

Vybornyi E., Наноструктуры. Математическая физика и моделирование 2015 Т. 13 № 2 С. 43-54

We conceder semiclassical asymptotics of the energy levels shift of the Schrödinger operator discrete spectrum with a one-dimensional single-well potential that appears due to a deformation of the potential in the classically forbidden region. Since such a deformation of the potential effects on the quantum particle only due to the tunneling effects, then the corresponding ...

Added: February 18, 2016

L.I. Kuzmina, Nazaikinskii V. E., Osipov Y. V., Russian Journal of Mathematical Physics 2019 Vol. 26 No. 1 P. 130-134

We consider an initial{boundary value problem for a simple semilinear ltration
equation with nonunique characteristics and prove that uniqueness nevertheless holds for the
solution of this problem. The solution is then constructed by quadratures. ...

Added: February 14, 2019