Geodesic
Removal of the dependence on energy from interactions depending on it as a resolvent
Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 2, pp. 281-303 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     author = {A. K. Motovilov},
     title = {Removal of the dependence on energy from interactions depending on it as a resolvent},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {281--303},
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     volume = {104},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_104_2_a6/}
}
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A. K. Motovilov. Removal of the dependence on energy from interactions depending on it as a resolvent. Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 2, pp. 281-303. http://geodesic.mathdoc.fr/item/TMF_1995_104_2_a6/

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