Resonance eigenfunction

In one way or another, the discrete-basis diagonalization methods of the 1960s attempt to provide an understanding of the behavior of roots of Hamiltonian matrices, in view of the picture of quasi-localization of resonance eigenfunctions. For example. Pels and Hazi [50] write ... 

This is in harmony with the notion that, in configuration space, for all practical purposes represents the resonance eigenfunction in the inner region, beyond which the components of, connected to the open channels describe the state. 

THE FORM OF THE RESONANCE EIGENFUNCTION AND THE COMPLEX EIGENVALUE SCHRODINGER EQUATION... 

If the solution for the decaying state is used, then the particle-number probability is not conserved since decay takes place (i.e., the system is open). This is seen by assuming the form of the time-dependent resonance eigenfunction to be defined in terms of its complex pole (24a) and of the requirement of satisfying the TDSE ... 

In the CESE approach, the coordinates of fhe Hamiltonian are real. Special attention is given to the accurate, state-specific, single- or multi-state calculation of ho. Complex coordinates are used only in certain orbitals of fhe function space, which is chosen appropriately so as to follow the two-part form of fhe resonance eigenfunction discussed above. We have called these orbitals "Gamow orbitals." They may be chosen to have particular forms in harmony wifh fhe corresponding open channels, or fhey can be expanded in standard forms, such as fhe one of Slater orbitals, albeit with a complex coordinate. Specifically, in fhe case of one-electron decay, the simplest version of fhe SSA square-infegrable resonance wavefuncfion for an N-electron atom is (r symbolizes collectively the real coordinates of fhe bound orbitals). 

As regards the theoretical framework, the form of the resonance eigenfunction according to Eq. (1) constitutes a reference point. Eq. (1) is related to time-dependent analysis (Eq. (6) and subsequent discussion) or to Hermitian K-matrix computations that depend on real values of the energy (Sections 3.1.1 and 3.1.2 and Chapter 6) or to computations that depend on complex energies and non-Hermitian constructions (Sections 3-7, 11). As regards the computational framework, Eq. (1), is implemented in terms of wavefunctions of the form of Eqs. (32,35,37, and 48). 

The full one-dimensional description introduces the well-known parity symmetry that permits to classify the resonant eigenfunctions as even or odd under the change x for —x. 

In general, one may write the full expression for the resonant eigenfunction as... 

The large amplitude of the continuum resonance states is a direct result of the non Hermitian properties of Hamiltonian (i.e. the resonance eigenfunctions which are associated with complex eigenvalues are not in the Hermitian domain of the molecular Hamiltonian). Let us explain this point in some more detail. As was mentioned above Moiseyev and Priedland [7] have proved that if two N x N real symmetric matrices H and H2 do not commute, there exists at least one value of parameter A = such that matrix H + XH2 possesses incomplete spectrum. That is at A A there are at least two specific eigenstates i and j for which ]imx j ei — ej) =0 and also lim ( i j) — 0- Since and ipj are orthogonal (within the general inner product definition i.e., i/ il i/ j) = = 0 not in the... 

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