Description of spectral functions of differential operators with arbitrary deficiency indices

2007 ◽  
Vol 81 (3-4) ◽  
pp. 553-559 ◽  
Author(s):  
V. I. Mogilevskii
Keyword(s):  
Differential Operators ◽  
Spectral Functions ◽  
Deficiency Indices

scholarly journals Minimal spectral functions of an ordinary differential operator

2012 ◽  
Vol 55 (3) ◽  
pp. 731-769 ◽  
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Boundary Conditions ◽  
Boundary Problem ◽  
Weyl Function ◽  
Spectral Functions ◽  
Boundary Triplet ◽  
Deficiency Indices ◽  
Separated Boundary Conditions ◽  
Adjoint Extension ◽  
Even Order ◽  
Classical Characteristic

AbstractLetl[y]be a formally self-adjoint differential expression of an even order on the interval [0,b〉(b ≤ ∞) and letL0be the corresponding minimal operator. By using the concept of a decomposing boundary triplet, we consider the boundary problem formed by the equationl[y] − λy = f,f ∈ L2[0, b〉, and the Nevanlinna λ-dependent boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of them-function, which in the case of self-adjoint separated boundary conditions coincides with the classical characteristic (Titchmarsh–Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e. all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indicesn±(L0)and non-separated boundary conditions) the known estimate of the spectral multiplicity of the (exit space) self-adjoint extensionà ⊃ L0. Results are obtained for expressionsl[y]with operator-valued coefficients and arbitrary (equal or unequal) deficiency indicesn±(L0).

scholarly journals Functional Model of Dissipative Fourth Order Differential Operators

2013 ◽  
Vol 21 (2) ◽  
pp. 237-252 ◽  
Author(s):  
Hüseyin Tuna
Keyword(s):  
Characteristic Function ◽  
Fourth Order ◽  
Differential Operators ◽  
Functional Model ◽  
Dissipative Operator ◽  
Eigenvalues And Eigenvectors ◽  
Deficiency Indices ◽  
Maximal Dissipative Operator

Abstract In this paper, maximal dissipative fourth order operators with equal deficiency indices are investigated. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which based on the method of Pavlov and define its characteristic function. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative fourth order operators.

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

Reducibility of differential operators some: examples

1979 ◽  
Vol 86 (1) ◽  
pp. 29-33 ◽  
Author(s):  
B. Fishel ◽  
N. Denkel
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Volterra Operator ◽  
Second Order ◽  
Order Differential Operator ◽  
Order Operator ◽  
Minimal Operator ◽  
Deficiency Indices ◽  
First Order ◽  
Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

Sign in / Sign up

Export Citation Format

Share Document