Related papers: Feynman's linear divergence problem
This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the…
On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the…
The acoustic scattering operator on the real line is mapped to a Schr\"odinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse…
We present a perturbation theory by extending a prescription due to Feynman for computing the probability density function for the random flight motion. The method can be applied to a wide variety of otherwise difficult circumstances. The…
The scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation we prove the existence of the scattering operator.
In this paper, we consider two types of the scattering problems (relativistic case), namely, the stationary scattering problem, where the distance $r$ tends to infinity, and the dynamical scattering problem, where the time $t$ tends to…
This paper is concerned with the final value problem for a system of nonlinear wave equations. The main issue is to solve the problem for the case where the nonlinearity is of a long range type. By assuming that the solution is spherically…
We consider second order differential operators with coefficients which are Gaussian random fields. When the covariance becomes singular at short distances then the propagators of the Schr\"odinger equation as well as of the wave equation…
We consider a long-range scattering theory for discrete Schr\"odinger operators on the hexagonal lattice, which describe tight-binding Hamiltonians on the graphene sheet. We construct Isozaki-Kitada modifiers for a pair of the difference…
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the…
We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption…
In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the…
We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator…
Recent advances in machine learning establish the ability of certain neural-network architectures called neural operators to approximate maps between function spaces. Motivated by a prospect of employing them in fundamental physics, we…
In this paper, we define the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We…
We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems involving systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of…
In a thin multidimensional layer we consider a second order differential PT-symmetric operator. The operator is of rather general form and its coefficients are arbitrary functions depending both on slow and fast variables. The PT-symmetry…
We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of…
Motivated by recent results concerning the asymptotic behaviour of differential operators with highly contrasting coefficients, which have involved effective descriptions involving generalised resolvents, we construct the functional model…
Extensions of standard one-dimensional supersymmetric quantum mechanics are discussed. Supercharges involving higher order derivatives are introduced leading to an algebra which incorporates a higher order polynomial in the Hamiltonian. We…