The Oscillation Numbers and the Abramov Method of Spectral Counting for Linear Hamiltonian Systems

In this paper we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. For the Hamiltonian problems we do not assume any controllability and strict normality assumptions which guarantee that the classical eigenvalues of the problems are isolated. We also omit the Legendre condition for their Hamiltonians. We show that the Abramov method of spectral counting can be modified for the more general case of finite eigenvalues of the Hamiltonian problems and then the constructive ideas of the Abramov method can be used for stable calculations of the oscillation numbers and finite eigenvalues of the Hamiltonian problems.


Introduction
In this paper we consider the spectral and oscillation theory for the linear Hamiltonian systems where 0 ( , ) ( , ), , ( , ) 0 0 Here 0 A ≥ means that the symmetric matrix A is nonnegative definite, and ,0 I denote the identity and zero matrices of appropriate dimensions. The Hamiltonian [ , ] t a b ∈ and R λ ∈ . The linear Hamiltonian system (1) covers as a special case scalar and matrix second order Sturm-Liouville differential equations, as well as Sturm-Liouville differential equations of arbitrary even order, other selfconjugate differential equations and systems [3][4].
Based on the consideration in [5] we do not suppose any conditions connected with strict monotonicity of some matrix functions associated with (1). Such conditions will guarantee that the classical eigenvalues of (1), (2) are isolated (see [1][2][3]). For example, in [3] problem (1), (2) is considered under the strict normality assumptions with respect to [ , ] t a b ∈ and R λ ∈ . In particular, the strict normality supposes that solutions of (1) are not "degenerate" with respect to change in R λ ∈ , i.e., if ( , ) y t λ solves system (1) for different 1 2 , R λ λ ∈ on some non-degenerate interval of [ , ] a b , then necessary ( , ) 0, [ , ]. y t t a b λ = ∈ For a special case of the second order Sturm-Liouville differential equation q t λ is nondecreasing in λ, but strict monotonicity of ( , ) q t λ is not required in this paper. This corresponds to removing the strict normality assumption compared to [3,Assumption (8.3.7), p. 245].
The spectral and oscillation theory for the Hamiltonian systems without the strict normality and the controllability assumptions are developed in [5][6][7] (see also the references given therein). In [5] the notion of a finite eigenvalue of (1), (2) was introduced. Let (the so-called conjoined bases). Consider a conjoined basis ( , ) Y t λ with the initial condition ( , ) (0 ) T Y a I λ = (the principal solution of (1) at t a = ). Then, under assumption (3) ( , ) rank X t λ is piecewise constant in R λ ∈ (see [3]) and 0 R λ ∈ is called a left (right) finite eigenvalue of (1),(2) with the multiplicity Here 0 ( , ) rank X b λ ± denote the left-hand (the righthand) limits of ( , ) rank X b λ at 0 λ . Under the strict normality assumption the matrix ( , ) X t λ is invertible except at isolated values of R λ ∈ and then left (right) finite eigenvalues reduce to the classical ones which are determined by the condition 0 det ( , ) 0 In general the multiplicities The global oscillation theorem (see Theorem 3.5 in [5]) relates the number of left finite eigenvalues of for the Hamiltonian ( , ) H t λ (the Legendre condition).
In [6][7], following the main ideas in [1][2] we generalized Theorem 3.5 in [3] introducing the so-called oscillation numbers which calculate the number of left finite eigenvalues in the interval ( ] , , α β α β < without assumption (7). In the recent paper (J. Elyseeva, Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter, submitted) we introduced the dual oscillation numbers which make possible to evaluate right finite eigenvalues of (1), (2) The main results of this paper (see Theorems 2,4) relate the spectral count introduced by A. Abramov in [1] with the oscillation numbers in [6][7]. The Abramov method of spectral counting [1][2] is developed for the case when classical eigenvalues are isolated and their multiplicities are defined by (6). In this paper we show how the approach in [1] can be modified for the more general case of finite eigenvalues with the multiplicities (5). The practical value of these results is in possible implementations of the outstanding ideas of [1][2] for stable calculations of the oscillation numbers and finite eigenvalues of (1), (2).

Oscillation numbers and the Abramov spectral count
In this section we recall the main notions used in the paper such that the comparative index (see [8] and Chapter 3 in [9]), the oscillation numbers [6][7], as well as the Abramov method of spectral counting in [1].

The comparative index
According to the definition of the comparative index in [8][9] (see also the references given therein) we consider 2n n × matrices ˆˆ( ) , ( ) condition (4) using the notation †( ) , where the symmetric matrices , Q Q solve the matrix equations ˆˆˆ, .
The comparative index is defined by Here ind P denotes the number of negative eigenvalues of the symmetric matrix P . For the case We also define the dual comparative index We also have (combining Properties 5,6 of the comparative index in [8]) where we use the Wronskian according to

The oscillation numbers
Consider the definition of the oscillation numbers for conjoined bases ( , ) Y t λ of (1) according to [6][7], where R λ ∈ is a fixed number. Introduce a partition of the interval [ , ] a b : that for any 1 [ , ] [ , ], 0,1, , 1 Then we define the oscillation number where we use the notation for the substitution . According to Section 2.1 and (13) takes the form .
According to the main results in [6][7] where the multiplicity of a left eigenvalue of (1), (see Theorem 3.2 in [6]). Remark that the sum in the right-hand side of (17) is finite because ( , ) rank X t λ is piecewise constant in R λ ∈ (see [5]).

The Abramov spectral count
Here we recall the construction of the spectral count in [1] for problem (1),(2) when the left boundary condition ( , ) 0 x a λ = is transferred to the right point t b = . The general case described in [1] can be considered analogously.
For a conjoined basis ( , ) Y t λ of (1) with a fixed λ consider a partition .
Then there exist the symmetric matrices Remark that the symmetric matrices ( , ) (15) and (18)

Main results
In this section we prove main connections between the oscillation numbers (14) and the Abramov spectral function (20). Lemma 1. For arbitrary conjoined basis ( , ) Y t λ of (1) we have the following representation for oscillation number (14)  Proof. The proof is based on the identity which follows from (10), (11). Indeed, by the definition of ( , ) Y Y µ and *( , ) Y Y µ we have in the left-hand side of (11) ( , ) ( ( , ))),  .
The proof of (23) is completed.

Modifications of the oscillation numbers
In this section we present different forms of representation of the oscillation numbers (14). The theoretical ground of these representations is the following separation theorem for the oscillation numbers (see Theorem 4.1 in [5]).
Remark that in (24) ( , ) Y t λ and ˆ( , ) Y t λ are considered for the same fixed value of λ . We present the following modification of the oscillation number which can be used as a spectral count for (1), (2).