Abstract
The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L2(Rm) for m⩾3. The asymptotic formula
is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = Vo + V1; Vo < 0, ¦Vo =o (¦Vo¦3/2) as ¦x¦ → ∞; σ (t/2, Vo) ⩽cσ (t. Vo) and V1∈Lm/2,loc, σ(t, V1) =o (σ (t, Vo)), where σ (t,f)= mes {x:¦f (x) ¦ > t).
Similar content being viewed by others
Literature cited
B. M. Levitan, “On the asymptotic behavior of the Green function and the expansion of the Schrö'dinger operator in characteristic functions,” Mat. Sb.,41, No. 4, 439–458 (1957).
A. G. Kostyuchenko, “Asymptotic distribution of characteristic values of elliptic operators,” Dokl. Akad. Nauk SSSR,158, No. 1, 41–44 (1964).
G. V. Rozenblyum, “An asymptotic of characteristic numbers of the Schrödinger operator,” Mat. Sb.,93, No. 3, 347–367 (1974).
F. H. Brownell and C. W. Clark, “Asymptotic distribution of eigenvalues of the lower part of the Schrödinger operator spectrum,” J. Math. Mech.,10, No. 1, 31–70 (1961).
B. Ya. Skachek, “Distribution of discrete spectrum of some singular differential operators,” Sb. Teor. Funkts., Funkts. Analiz. Prilozh., Khar'kov State Univ., No. 3, 110–116 (1966).
G. V. Rozenblyum, “On estimates of the spectrum of the Schrödinger operator,” Sb. Probl. Mat. Analiz., Leningrad State Univ., No. 5, 154–166 (1975).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 399–407, March, 1977.
Rights and permissions
About this article
Cite this article
Rozenblyum, G.V. An asymptotic of the negative discrete spectrum of the Schrödinger operator. Mathematical Notes of the Academy of Sciences of the USSR 21, 222–227 (1977). https://doi.org/10.1007/BF01106748
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01106748