On the spectral property of a Dirichlet problem with explosive factor

The inverse spectral problem of some singular version of one-dimensional Schrödinger operator with explosive factor in finite interval

“…ð2:17Þ so that, the convergence of U AE n ðtÞ implies the convergence of T AE n ðtÞ From [16], a À n ¼ À ðpÀaÞ 3 p 2 n 2 e 2g À n a 1 þ 1 n À Á Â Ã and g À n ¼ p pÀa n À 1 4 À Á þ 1 n À Á , from which and a similar formula for a oÀ n and g oÀ n , we have…”

The equiconvergence of the eigenfunction expansion for a singular version of one-dimensional Schrodinger operator with explosive factor

“…In [1], the author studied the asymptotic formulas of the eigenvalues, and eigenfunctions of problem (1.1)-(1.2) and proved that the eigenfunctions are orthogonal with weight function r(x). In [2], the author also studied the eigenfunction expansion of the problem(1.1)-(1.2).…”

“…The calculation of the trace formula for the eigenvalues of the problem(1.1)-(1.2) is to appear. We mention here the basic definition and results from [1] that are needed in the progress of this work. Let (x, λ), ψ(x, λ) be the solutions of the problem (1.1)-(1.2) with the boundary conditions (0, λ) = 0, '(0, λ) = 1, ψ(π, λ) = 0, ψ'(π, λ) = 1 and let W(λ) = (x, λ)ψ'(x, λ) -ψ(x, λ)'(x, λ) be the Wronskian of the two linearly independent solutions (x, λ), ψ(x, λ).…”

“…We introduce the following notations, let Δ n,f (x) denotes the nth partial sum :20) where, from [1], a ± k = 0 . It should be noted here, from [2], that as n ∞, the series n,f (x) .…”

“…This means that the two expansions have the same condition of convergence. Following [1], the contour Γ n is defined by…”