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 TMF, 1995, Volume 104, Number 2, Pages 281–303 (Mi tmf1338)

Removal of the dependence on energy from interactions depending on it as a resolvent

A. K. Motovilov

Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics

Abstract: The spectral problem $(A + V(z))\psi =z\psi$ is considered with $A$, a self-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation $V(z)$ is assumed to depend on the energy $z$ as resolvent of another self-adjoint operator $A':$ $V(z)=-B(A'-z)^{-1}B^{*}$. It is supposed that operator $B$ has a finite Hilbert–Schmidt norm and spectra of operators $A$ and $A'$ are separated. The conditions are formulated when the perturbation $V(z)$ may be replaced with an energy-independent “potential” $W$ such that the Hamiltonian $H=A +W$ has the same spectrum (more exactly a part of spectrum) and the same eigenfunctions as the initial spectral problem. The orthogonality and expansion theorems are proved for eigenfunction systems of the Hamiltonian $H=A + W$. Scattering theory is developed for $H$ in the case when operator $A$ has continuous spectrum. Applications of the results obtained to few-body problems are discussed.

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English version:
Theoretical and Mathematical Physics, 1995, 104:2, 989–1007

Bibliographic databases:

Citation: A. K. Motovilov, “Removal of the dependence on energy from interactions depending on it as a resolvent”, TMF, 104:2 (1995), 281–303; Theoret. and Math. Phys., 104:2 (1995), 989–1007

Citation in format AMSBIB
\Bibitem{Mot95} \by A.~K.~Motovilov \paper Removal of the dependence on energy from interactions depending on it as a resolvent \jour TMF \yr 1995 \vol 104 \issue 2 \pages 281--303 \mathnet{http://mi.mathnet.ru/tmf1338} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1488675} \zmath{https://zbmath.org/?q=an:0856.47044} \transl \jour Theoret. and Math. Phys. \yr 1995 \vol 104 \issue 2 \pages 989--1007 \crossref{https://doi.org/10.1007/BF02065979} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995UD33400007} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. Yu. Konstantinov, “Spectral Analysis of One Class of Matrix Differential Operators”, Funct. Anal. Appl., 31:3 (1997), 209–211
2. Hardt, V, “A factorization theorem for the transfer function associated with a 2 x 2 operator matrix having unbounded couplings”, Journal of Operator Theory, 48:1 (2002), 187
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