scholarly journals Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H - \lambda W$ in a gap of $H$

1999 ◽  
Vol 351 (3) ◽  
pp. 857-899
Author(s):  
S. Z. Levendorskiĭ
Keyword(s):  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Asymptotic Formulae

DISCRETE SPECTRUM ASYMPTOTICS FOR THE SCHRÖDINGER OPERATOR WITH A SINGULAR POTENTIAL AND A MAGNETIC FIELD

1996 ◽  
Vol 08 (06) ◽  
pp. 861-903 ◽  
Author(s):  
A.V. SOBOLEV
Keyword(s):  
Magnetic Field ◽  
Schrödinger Operator ◽  
Coulomb Potential ◽  
Singular Potential ◽  
Correction Term ◽  
Schrodinger Operator ◽  
Planck Constant ◽  
Asymptotic Formulae ◽  
Leading Term ◽  
Scott Correction

Object of the study is the operator H=H0(h, µ)+V in L (Rd), d≥2, where H0(h, μ) is the Schrödinger operator with a magnetic field of intensity μ≥0 and the Planck constant h∈(0, h0]. The electric (real-valued) potential V=V(x) is assumed to be asymptotically homogeneous of order −β, β≥0 as x→0. One obtains asymptotic formulae with remainder estimates as h→0, μh≤C for the trace Ms=tr{ɸgs(H)} where [Formula: see text], s∈[0, 1]. Due to the condition μh≤C the leading term of Ms does not depend on μ. It depends on the relation between the parameters d, s and β. There are five regions, in which either leading terms or remainder estimates have different form. In one of these regions Ms admits a two-term asymptotics. In this case, for an asymptotically Coulomb potential the second term coincides with the well-known Scott correction term.

scholarly journals On the two-dimensional Schrödinger operator with an attractive potential of the Bessel-Macdonald type

2020 ◽  
pp. 168385
Author(s):  
Wellisson B. De Lima ◽  
Oswaldo M. Del Cima ◽  
Daniel H.T. Franco ◽  
Bruno C. Neves
Keyword(s):  
Schrödinger Operator ◽  
Attractive Potential ◽  
Two Dimensional ◽  
Schrodinger Operator
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