On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

Abstract

We study pseudospectral and spectral functions for Hamiltonian system Jy ′ − B(t) = λ∆(t)y and differential equation l[y] = λ∆(t)y with matrix-valued coefficients defined on an interval I = [a, b) with the regular endpoint a. It is not assumed that the matrix weight ∆(t) ≥ 0 is invertible a.e. on I. In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization σ = σ τ of all pseudospectral and spectral functions σ by means of a Nevanlinna parameter τ and single out in terms of τ and boundary conditions the class of functions y for which the inverse Fourier transform y(t) = R ϕ(t, s) dσ(s) y(s) converges uniformly. We also show that for scalar equation l[y] = λ∆(t)y the set of spectral functions is not empty. This enables us to extend the Kats-Krein and Atkinson results for scalar Sturm -Liouville equation −(p(t)y ′ ) ′ + q(t)y = λ∆(t)y to such equations with arbitrary coefficients p(t) and q(t) and arbitrary non trivial weight ∆(t) ≥ 0.

On eigenfunction expansions of differential equations with degenerating weight

“…Let τ = τ C ∈ R mer (C d−r ) be given by (2.9) and let τ 0 and K be the operator and multivalued parts of τ respectively. Then by Lemma 2.8 K = ker C 1 (λ) and τ 0 (λ) = − C −1 1 (λ)C 00 (λ), λ ∈ C \ R. This implies that η C = η τ , where η τ ∈ C(C d−r ) is the linear relation defined in [20,Theorem 2.4].…”

“…19) can be defined as a unique function in N t k such that Proof. Together with (4.17 (a2) EV = {t k } ν + ν − is an infinite set without finite limit points and for any y ∈ L 2 ∆ (I) there exists a sequence { y By using (4.9) and (4.11) one can easily verify that π ∆ N t = N t , t ∈ R. Moreover, according to [20,Proposition 5.11] the equation (4.4) is definite, that is the equalities l[y] = t∆(x)y(x) and ∆(x)y(x) = 0 (a.e. on I) yields y = 0.…”

“…Below within this subsection we recall some results from our paper [20] concerning equation (4.4) with the nontrivial Weight ∆.…”

“…According to [20] boundary problem (1.1), (1.3) for equation (1.1) with the nontrivial weight ∆ generates an operator T = T * in L 2 ∆ (I) (for the case of the positive Weight see e.g. [24]).…”

“…In conclusion note that this paper is a continuation of [20], where "continuous" eigenfunction expansions in the form of the generalized Fourier transform were considered. A non-decreasing strongly left continuous operator function ξ : R → B(H) with ξ(0) = 0 is called a distribution function or shortly a distribution.…”