Spectral Analysis of Higher-Order Differential Operators with Discontinuity Conditions at Interior Points
- PDF / 222,525 Bytes
- 11 Pages / 594 x 792 pts Page_size
- 97 Downloads / 199 Views
SPECTRAL ANALYSIS OF HIGHER-ORDER DIFFERENTIAL OPERATORS WITH DISCONTINUITY CONDITIONS AT INTERIOR POINTS V. A. Yurko
UDC 517.984
Abstract. Higher-order differential operators on a finite interval with discontinuity conditions inside the interval are studied. Properties of spectral characteristics are obtained, and completeness and expansion theorems are proved for that class of operators.
CONTENTS 1. Introduction . . . . . . . . . . . . . . . 2. Properties of Spectral Characteristics 3. Expansion Theorem and Completeness References . . . . . . . . . . . . . . . .
. . . . . . . . . . . . Theorem . . . . . .
1.
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
705 706 711 714
Introduction
Consider the differential equation y(x) := y (n) (x) +
n−2
pj (x)y (j) (x) = λy(x),
0 < x < T,
(1.1)
j=0
with the boundary conditions y (ν−1) (0) = y (ν−1) (T ) = 0,
ν = 1, m,
(1.2)
and the following discontinuity conditions at an interior point a from (0, T ): y
(ν−1)
(a + 0) =
ν
aνj y (j−1) (a − 0),
ν = 1, n.
(1.3)
j=1 (ν)
Here n = 2m, pj (x) are complex-valued functions, pj (x) are absolutely continuous provided that ν = 0, j − 1 and x ∈ [0, T ], and aνj are complex numbers such that aνν = 0. Thus, the discontinuity conditions are generated by the transition matrix A = [aνj ]ν,j=1,n , where aνj = 0 provided that ν < j and det A = 0. The above problem is called problem L. Let functions ϕj (x, λ), j = 1, n, satisfy Eq. (1.1), conditions (1.3), and the initial conditions (ν−1)
ϕj
(0, λ) = δνj , ν = 1, n,
(1.4)
where δνj is the Kronecker symbol. It is clear that (ν−1)
det[ϕj
(x, λ)]ν,j=1,n = η(x),
(1.5)
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 2, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2017. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2504–0705
705
where η(x) := 1 if x < a and η(x) := det A if x > a. Introduce the notation (ν−1)
Δ(λ) := det[ϕj
(T, λ)]j=m+1,n; ν=1,m .
(1.6)
The function Δ(λ) is entire with respect to λ and its order is equal to 1/n; its zeroes {λl }l≥0 (counted with their multiplicities) coincide with the eigenvalues of problem L of kind (1.1)–(1.3). The function Δ(λ) is called the characteristic function of problem L. Let {ϕl (x)}l≥0 be the system of eigenfunctions and adjoint functions (root functions) of problem L. Boundary-value problems with discontinuity conditions inside the interval frequently occur in mathematics, mechanics, physics, geophysics, and other areas of natural sciences. As a rule, they are related to discontinuous and nonsmooth properties of media. For example, discontinuous problems arise in electronics if parameters of nonsmooth feeders with prescribed characteristics are designed (see [9, 10]). Also, discontinuous problems arise in the conductivity investigation fo
Data Loading...
