Abstract
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.
Highlights
In this paper we consider the spectral and oscillation theory for the linear Hamiltonian systems = y 'JH (t, λ) y, t ∈[a,b],=y x(t, u(t, λ λ ) ), λ ∈ R (1)with the Dirichlet boundary conditions (1) are not “degenerate” with respect to change in λ ∈ R, i.e., if y(t,λ) solves system (1) for different λ1,λ2 ∈ R on some non-degenerate interval of [a,b], necessary y(t, λ=) 0, t ∈[a,b].For a special case of the second order Sturm– Liouville differential equation (r(t, λ)x ') '+ q(t, λ)x = 0, t ∈[a,b]x(a, λ) = x(b, λ), (2)
Under the strict normality assumption the matrix X (t,λ) is invertible except at isolated values of λ ∈ R and left finite eigenvalues reduce to the classical ones which are determined by the condition det X (b, λ0 ) = 0 with the multiplicities θ − (λ0 )= θ + (λ0 )= n − rank X (b, λ0 )
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) in [α, β ),α < β
Summary
In this paper we consider the spectral and oscillation theory for the linear Hamiltonian systems. Under the strict normality assumption the matrix X (t,λ) is invertible except at isolated values of λ ∈ R and left (right) finite eigenvalues reduce to the classical ones which are determined by the condition det X (b, λ0 ) = 0 with the multiplicities θ − (λ0 )= θ + (λ0 )= n − rank X (b, λ0 ). The global oscillation theorem (see Theorem 3.5 in [5]) relates the number of left finite eigenvalues of (1),(2) in the interval (−∞, β ] with the number of the so-called proper focal points of the principal solution of system (1) in This result is derived under the additional monotonicity assumption. 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)
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