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Some comments on the method of complex scaling to find physical resonance states
Author(s) -
Löwdin PerOlov
Publication year - 1986
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560300763
Subject(s) - eigenfunction , eigenvalues and eigenvectors , scaling , hamiltonian (control theory) , hilbert space , operator (biology) , complex plane , spectrum (functional analysis) , mathematics , connection (principal bundle) , mathematical analysis , pure mathematics , mathematical physics , physics , quantum mechanics , geometry , mathematical optimization , biochemistry , chemistry , repressor , transcription factor , gene
Some basic features of the method of complex scaling are briefly reviewed. This method is associated with a similarity transformation H = UHU −1 , in which the many‐particle Hamiltonian H loses its self‐adjoint character. In connection with the eigenvalue problems for H and H, one has formally the relations ψ = Uψ and E = E. However, since the proper boundary conditions have to be satisfied, the spectrum { E } may still be subject to change: even if some eigenvalues are persistent ( E = E), others may be lost, and new eigenvalues may occur also in the complex plane. It is pointed out that these changes are related to the fact that the “dilatation operator” U is an unbounded operator, and that the eigenfunctions involved are transformed not only within the ordinary L 2 Hilbert space, but also out of and into this space. Reference to a more complete treatment is given.