dorsal/arxiv
View SchemaRemoval of the Energy Dependence from the Resolvent-like Energy-Dependent Interactions
| Authors | A. K. Motovilov |
|---|---|
| Categories | |
| ArXiv ID | nucl-th/9505030 |
| URL | https://arxiv.org/abs/nucl-th/9505030 |
| DOI | 10.1007/BF02065979 |
| Journal | Theor.Math.Phys. 104 (1996) 989-1007; Teor.Mat.Fiz. 104N2 (1995) 281-303 |
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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"abstract": "The spectral problem $(A + V(z))\\psi=z\\psi$ is considered with $A$, a\nself-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation\n$V(z)$ is assumed to depend on the energy $z$ as resolvent of another\nself-adjoint operator $A\u0027:$ $V(z)=-B(A\u0027-z)^{-1}B^{*}$. It is supposed that\noperator $B$ has a finite Hilbert-Schmidt norm and spectra of operators $A$ and\n$A\u0027$ are separated. The conditions are formulated when the perturbation $V(z)$\nmay be replaced with an energy-independent ``potential\u0027\u0027 $W$ such that the\nHamiltonian $H=A +W$ has the same spectrum (more exactly a part of spectrum)\nand the same eigenfunctions as the initial spectral problem. The orthogonality\nand expansion theorems are proved for eigenfunction systems of the Hamiltonian\n$ H=A + W $. Scattering theory is developed for $H$ in the case when operator\n$A$ has continuous spectrum. Applications of the results obtained to few-body\nproblems are discussed.",
"arxiv_id": "nucl-th/9505030",
"authors": [
"A. K. Motovilov"
],
"categories": [
"nucl-th",
"funct-an",
"math.FA"
],
"doi": "10.1007/BF02065979",
"journal_ref": "Theor.Math.Phys. 104 (1996) 989-1007; Teor.Mat.Fiz. 104N2 (1995)\n 281-303",
"title": "Removal of the Energy Dependence from the Resolvent-like Energy-Dependent Interactions",
"url": "https://arxiv.org/abs/nucl-th/9505030"
},
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