Spectral functions for the Schrödinger operator on R+ with a singular potential

2010 ◽  
Vol 51 (5) ◽  
pp. 053512 ◽  
Author(s):  
Klaus Kirsten ◽  
Paul Loya
Keyword(s):  
Schrödinger Operator ◽  
Singular Potential ◽  
Schrodinger Operator ◽  
Spectral Functions

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 A generalization related to Schrödinger operators with a singular potential

2002 ◽  
Vol 29 (10) ◽  
pp. 609-611 ◽  
Author(s):  
Toka Diagana
Keyword(s):  
Schrödinger Operator ◽  
Singular Potential ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Open Question

The purpose of this note is to generalize a result related to the Schrödinger operatorL=−Δ+Q, whereQis a singular potential. Indeed, we show thatD(L)={0}inL2(ℝd)ford≥4. This fact answers to an open question that we formulated.

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