NDLI: On the deficiency index of the vector-valued Sturm

Content Provider	Springer Nature Link
Author	Mirzoev, K. A. Safova, T. A.
Copyright Year	2016
Abstract	Let R$_{+}$:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R$_{+}$ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R$_{+}$ and the elements of the matrix functions P $^{−1}$, Q, and R are measurable on R$_{+}$ and summable on each of its closed finite subintervals. We study the operators generated in the space L n 2 (R$_{+}$) by formal expressions of the form l[f] = −(P(f' − Rf))' − R*P(f' − Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = −(P $_{0}$ f')' + i((Q $_{0}$ f)' + Q $_{0}$ f') + P'$_{1}$ f, where everywhere the derivatives are understood in the sense of distributions and P $_{0}$, Q $_{0}$, and P $_{1}$ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P 0 −1 exists and ǁP0ǁ, ǁP 0 −1 ǁ, ǁP 0 −1 ǁǁP $_{1}$ǁ$^{2}$, ǁP 0 −1 ǁǁQ $_{0}$ǁ$^{2}$ ∈ L loc 1 (R$_{+}$). Themain goal in this paper is to study of the deficiency index of the minimal operator L $_{0}$ generated by expression l[f] in L n 2 (R$_{+}$) in terms of the matrix functions P, Q, and R (P $_{0}$, Q $_{0}$, and P $_{1}$). The obtained results are applied to differential operators generated by expressions of the form $l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f$ , where x $_{ k }$, k = 1, 2,..., is an increasing sequence of positive numbers, with lim$_{ k→+∞}$ x $_{ k }$ = +∞, H $_{ k }$ is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.
Starting Page	290
Ending Page	303
Page Count	14
File Format	PDF
ISSN	00014346
Journal	Mathematical Notes
Volume Number	99
Issue Number	1-2
e-ISSN	15738876
Language	English
Publisher	Pleiades Publishing
Publisher Date	2016-03-31
Publisher Place	Moscow
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Sturm–Liouville operator deficiency index Hermitian matrix-function Jacobi matrix Cauchy–Bunyakovskii inequality quasiderivative quasidifferential equation Mathematics
Content Type	Text
Resource Type	Article

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