Generalized Jacobi Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients
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GENERALIZED JACOBI MATRICES AND SPECTRAL ANALYSIS OF DIFFERENTIAL OPERATORS WITH POLYNOMIAL COEFFICIENTS K. A. Mirzoev, N. N. Konechnaya, T. A. Safonova, and R. N. Tagirova
UDC 517.984, 517.929
Abstract. This paper is devoted to the matrix representation of ordinary symmetric differential operators with polynomial coefficients on the whole axis. We prove that in this case, generalized Jacobi matrices appear. We examine the problem of defect indexes for ordinary differential operators and generalized Jacobi matrices corresponding to them in the spaces L2 (−∞, +∞) and l2 , respectively, and analyze the spectra of self-adjoint extensions of these operators (if they exist). This method allows one to detect new classes of entire differential operators of minimal type (in the sense of M. G. Krein) with certain defect numbers. In this case, the defect numbers of these operators can be not only less than or equal, but also greater than the order of the corresponding differential expressions. In particular, we construct examples of entire differential operators of minimal type that are generated by irregular differential expressions. Keywords and phrases: regular differential expression, irregular differential expression, differential operator, generalized Jacobi matrix, defect index, integer operators of minimal type. AMS Subject Classification: 47E05, 39A10
CONTENTS 1. 2. 3. 4.
Introduction. Preliminary Information . . . . . . Condition of Self-Adjointness for the Operator L Conditions for Nonzero Defect Numbers . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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Introduction. Preliminary Information
1.1. Normal forms of differential operators with polynomial coefficients. On the real line R, let us consider a formally self-adjoint differential expression l of even or odd order r of the form r dj qj (x) j , (1) l= dx j=0
where qj , j = 0, 1, . . ., are given polynomials in the variable x. It is well known that there exist polynomials ak , k = 0, 1, . . . , [r/2], and bk , k = 0, 1, . . . , [(r − 1)/2], such that the expression l is represented in the form [r/2]
ly =
k=0
ak y (k)
(k)
[(r−1)/2]
+i
bk y (k+1)
(k)
(k+1) + bk y (k) ,
k=0
where y is an admissible function and [x] denotes the integer part of the number x. This representation is called the divergent form of the expression l. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018. c 2021 Springer Science+Business Media, LLC 1072–3374/21/2522–0213
213
The operator generated by the expression l on indefinitely differentiable finite functions on R in the Hilbert space L2 (−∞, +∞)(=: L2 ) of square integrable func
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