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Minimal spectral functions of an ordinary differential operator

Part of: Operator theory: Ordinary differential operators Boundary value problems

Published online by Cambridge University Press:  30 August 2012

Vadim Mogilevskii
Vadim Mogilevskii
Affiliation:
Department of Calculus, Lugans′k National University, 2 Oboronna, Lugans′k 91011, Ukraine ([email protected])
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Abstract

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Let l[y] be a formally self-adjoint differential expression of an even order on the interval [0, b〉(b ≤ ∞) and let L0 be the corresponding minimal operator. By using the concept of a decomposing boundary triplet, we consider the boundary problem formed by the equation l[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 the m-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 indices n±(L0) and non-separated boundary conditions) the known estimate of the spectral multiplicity of the (exit space) self-adjoint extension à ⊃ L0. Results are obtained for expressions l[y] with operator-valued coefficients and arbitrary (equal or unequal) deficiency indices n±(L0).

Keywords

differential operatordecomposing D-boundary tripletboundary conditionsminimal spectral functionspectral multiplicity

MSC classification

Secondary: 34B05: Linear boundary value problems 34B20: Weyl theory and its generalizations 34B40: Boundary value problems on infinite intervals 47E05: Ordinary differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 731 - 769
Copyright
Copyright © Edinburgh Mathematical Society 2012

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