The Eigenfunction Expansion for a Dirichlet Problem with Explosive Factor

Abstract

We prove the eigenfunction expansion formula for a Dirichlet problem with explosive factor by two ways, first by standard method and second by proving a convergence in some metric space .

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

“…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.…”

“…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.…”

“…Following [2], we state some basic asymptotic relations that are useful in the discussion. The solutions (x, λ) and ψ(x, λ) of the Dirichlet problem (1.1)-(1.2) have the following asymptotic formula…”

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

“…Following [14] the function rðx; 0; kÞ, on the contour C n , takes the form rðx; 0; kÞ 6 B jgj e Àjsj a ; g 2 C n ; 8n; ð2:10Þ…”

“…Following [14], we state the basic results that are needed in the subsequent investigation, where, the authors proved that the Dirichlet problem (1.1) and (1.2) has a countable number of eigenvalues k AE n ; n ¼ 0; 1; 2; . .…”