2005/10/28 by Andrew Hassell, Hassell, Andrew, Simon Marshall +1
Computer Science · Mathematics · #35P20 #81Q05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math/0510617
openalex publication_date 2005/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function NL(E), the number of bound states of the operator L = Δ+V in \Rd below -E. Here V is a bounded potential behaving asymptotically like P(ω)r-2 where P is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator ΔSd-1+P on the sphere has negative eigenvalues -μ1,...,-μn less than -(d-2)2/4, we prove that NL(E) may be estimated as NL(E)) = \fraclog(E-1)2π∑i=1n √(μi-(d-2)2/4) +O(1); thus, in particular, if there are no such negative eigenvalues then L has a finite discrete spectrum. Moreover, under some additional assumptions including that d=3 and that there is exactly one eigenvalue -μ1 less than -1/4, with all others > -1/4, we show that the negative spectrum is asymptotic to a geometric progression with ratio exp(-2π/√(μ1 - \qtr)).