Self-Adjointness and Discreteness of the Spectrum of Block Jacobi Matrices
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Self-Adjointness and Discreteness of the Spectrum of Block Jacobi Matrices V. S. Budyka1, 2* and M. M. Malamud2** 1
Donetsk State Academy of Management and Government Service, Donetsk, 83015 Ukraine 2 Peoples’ Friendship University of Russia (RUDN University), Moscow, 117198 Russia Received February 18, 2020; in final form, February 18, 2020; accepted February 19, 2020
DOI: 10.1134/S000143462009014X Keywords: Jacobi matrix, discrete spectrum, self-adjointness, deficiency indices.
1. INTRODUCTION After Krein [1], block Jacobi matrices ⎛ A0 B0 Op Op Op . . . Op ⎜ B∗ A B1 Op Op . . . Op 1 ⎜ 0 ⎜O ∗ B A B2 Op . . . Op ⎜ p 2 1 J=⎜ . . . .. .. .. ⎜ .. .. .. . . ... . ⎜ ⎜ ∗ ⎝Op Op Op Op Op . . . Bn−1 .. .. .. .. .. .. .. . . . . . . .
Op Op Op .. .
Op Op Op .. .
Op Op Op .. .
An .. .
Bn .. .
Op .. .
⎞ ... . . .⎟ ⎟ . . .⎟ ⎟ ⎟ . . .⎟ ⎟ ⎟ . . .⎠ .. .
(1)
were studied in numerous papers (see [2]–[14] and the references therein). Here An = A∗n , Bn ∈ Cp×p , det Bn = 0, n ∈ N0 := N ∪ {0}, Op is the zero matrix, and Cp×p is the set of all p × p matrices with entries from C. With each matrix J is associated a minimal Jacobi operator in l2 (N; Cp ) (see [2], [3]). The operator J is symmetric, but is not necessarily self-adjoint. It is known that 0 ≤ n± (J) ≤ p, as well as that the conditions n+ (J) = p and n− (J) = p are equivalent (see [1] and [3]). Note that if n± (J) = p, then all the self-adjoint extensions of J have a discrete spectrum. The simplest and best-known condition for the self-adjointness of the operator J is the Carleman matrix test (see [2], [3, Theorem VII.2.9] as well as [4]): ∞ Bj −1 = +∞. j=0
The maximality conditions for the deficiency indices of the matrix (1) were obtained in [4]–[6], [13], [14]. Jacobi matrices are closely related to the matrix moment problem (see [1]–[3]). Here self-adjointness guarantees the uniqueness of the matrix measure Σ solving the moment problem. With the Jacobi matrix J is associated the difference matrix expression ∗ Un−1 + Bn Un+1 + An Un , (LU )n = Bn−1
U0 = Ip ,
U−1 = Op ,
Un ∈ Cp×p,
n ∈ N0 ,
2 defining the sequence of matrix polynomials {Pn (z)}∞ 0 , which are orthogonal in the space L (R; Σ). We note that, in the recent papers [9], [10], and [12], a close relationship was found between the ¨ spectral properties of the Schrodinger operators HX,α (see Sec. 4) and the Dirac operators DX,α with point interactions, on the one hand, and special Jacobi matrices on the other. This relationship was generalized and studied in [7], [8], [11], and [13]–[15], which led to a sufficiently complete spectral analysis of the operators HX,α and DX,α . * **
E-mail: [email protected] E-mail: [email protected]
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2. SELF-ADJOINTNESS CONDITIONS FOR THE JACOBI MATRICES Definition 1 ([16], [17]). Let K and T be densely defined linear operators in the Hilbert space H. An operator K is said to be subordinate to an operator T if dom T ⊂ dom K and the following inequality holds: K ≤ aT u + bu,
a > 0,
b ≥ 0,
u ∈ dom T.
(2)
If, in (2
