Eigenvalue-eigenfunction Problem for Steklov’s Smoothing Operator and Differential-difference Equations of Mixed Type
نویسندگان
چکیده
It is shown that any μ ∈ C is an infinite multiplicity eigenvalue of the Steklov smoothing operator Sh acting on the space Lloc(R). For μ 6= 0 the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.
منابع مشابه
Eigenfunction Expansions for Second-Order Boundary Value Problems with Separated Boundary Conditions
In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem....
متن کاملar X iv : m at h - ph / 0 31 10 29 v 1 2 0 N ov 2 00 3 Group theoretical approach to the intertwined Hamiltonians
We show that the finite difference Bäcklund formula for the Schrödinger Hamiltonians is a particular element of the transformation group on the set of Riccati equations considered by two of us in a previous paper. Then, we give a group theoretical explanation to the problem of Hamiltonians related by a first order differential operator. A generalization of the finite difference algorithm relati...
متن کاملEquivalence between the eigenvalue problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun’s differential equations, eigenstates degeneration, and Rabi’s model
The initial aim of the present paper is to provide a complete description for the eigenvalue problem of the non-commutative harmonic oscillator (NcHO), which is given by a two-by-two system of paritypreserving ordinary differential operator [19], in terms of Heun’s ordinary differential equations, the second order Fuchsian differential equations with four regular singularities in a complex doma...
متن کاملA Parameter Uniform Numerical Scheme for Singularly Perturbed Differential-difference Equations with Mixed Shifts
In this paper, we consider a second-order singularly perturbed differential-difference equations with mixed delay and advance parameters. At first, we approximate the model problem by an upwind finite difference scheme on a Shishkin mesh. We know that the upwind scheme is stable and its solution is oscillation free, but it gives lower order of accuracy. So, to increase the convergence, we propo...
متن کاملA numerical technique for solving a class of 2D variational problems using Legendre spectral method
An effective numerical method based on Legendre polynomials is proposed for the solution of a class of variational problems with suitable boundary conditions. The Ritz spectral method is used for finding the approximate solution of the problem. By utilizing the Ritz method, the given nonlinear variational problem reduces to the problem of solving a system of algebraic equations. The advantage o...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012