ISSN:
1572-929X
Keywords:
Singular perturbations
;
Krein's resolvents formula
;
self-adjoint extensions.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Gesztesy and Simon recently have proven the existence of the strong resolvent limit A∞,ω for Aα,ω = A + α (·ω)ω,α→∞ where A is a self-adjoint positive operator, ω∈ $$\mathcal{H}_{ - 1} (\mathcal{H}_s ,\;s \in R^1 $$ being the ‘A-scale’). In the present note it is remarked that the operator A∞,ω also appears directly as the Friedrichs extension of the symmetric operator $$\dot A$$ :=A⌈ \{f∈ $$\mathcal{D}$$ (A)| 〈f,ω〉=0\}. It is also shown that Krein's resolvents formula: (A_b,ω-z)-1 =(A-z)-1+ $$b_z^{ - 1} $$ (·, $$\eta _{\bar z} $$ ) ηz, with b=b-(1+z) (ηz,η-1),ηz= (A-z)-1ω defines a self-adjoint operator Ab,ω for each ω∈ $$\mathcal{H}_{ - 2} $$ and b∈ R1. Moreover it is proven that for any sequence ωn∈ $$\mathcal{H}_{ - 1} $$ which goes to ω in $$\mathcal{H}_{ - 2} $$ there exists a sequence αn→0 such that $$A_{\alpha _n ,\omega _n } $$ → Ab,ω in the strong resolvent sense.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008651918800
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