We consider a higher order functional-differential operator with an integrable potential and regular separated boundary conditions. obtain asymptotic formulas estimates for solutions to the corresponding equation asymptotics of eigenvalues differential operator.

The monodromy operator of a linear delay differential equation with periodic coefficients is formulated as an integral operator. The kernel of this operator includes a factor formed from the fundamental solution of the linear delay differential equation. Although the properties of the fundamental solutions are known, in general there is no closed form for the fundamental solution. This paper de...

If we are given a smooth differential operator in the variable x ∈ R/2πZ, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S)-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Di...

Krein [1] mentioned that for each PD equation we have two extreme operators, one is the minimal in which solution and its derivatives on the boundary are zero, the other one is the maximal operator in which there is no prescribed boundary conditions. They claim it is not possible to have a related boundary value problem for an arbitrarily chosen operator in between. They have only considered lo...

We present a method for determining the stability of a class of stochastically forced ordinary differential equations, where the forcing term can be obtained by passing white noise through a filter of arbitrarily high degree. We use the Fokker-Planck equation to write a partial differential equation for the second moments, which we turn into an eigenvalue problem for a second-order differential...

Let M be a complete Riemannian manifold and let Ω•(M) denote the space of differential forms on M . Let d : Ω(M) → Ω(M) be the exterior differential operator and let ∆ = dd + dd be the Laplacian. We establish a sufficient condition for the Schrödinger operator H = ∆ + V (x) (where the potential V (x) : Ω(M) → Ω(M) is a zero order differential operator) to be self-adjoint. Our result generalizes...

We present a conjecture that the universal enveloping algebra of differential operators ∂ ∂tk over C coincides with the universal enveloping algebra of the (Borel subalgebra of) Virasoro generators from the Kontsevich model. Thus, we can decompose any (pseudo)differential operator to some combination of the Virasoro operators. We use this decomposition to present the r.h.s. of the Givental form...