Inverse Schro ̈dinger equation and the exact wave function

 Hiroshi Nakatsuji

 

 Received 17 December 2001; published 10 May 2002

 

 Using the inverse of the Hamiltonian, we introduce the inverse Schro ̈dinger equation~ISE!that is equivalentto the ordinary Schro ̈dinger equation~SE!. The ISE has the variational principle and theH-square group ofequations as the SE has. When we use a positive Hamiltonian, shifting the energy origin, the inverse energybecomes monotonic and we further have the inverse Ritz variational principle and cross-H-square equations.The concepts of the SE and the ISE are combined to generalize the theory for calculating the exact wavefunction that is a common eigenfunction of the SE and ISE. The Krylov sequence is extended to include theinverse Hamiltonian, and the complete Krylov sequence is introduced. The iterative configuration interaction~ICI!theory is generalized to cover both the SE and ISE concepts and four different computational methods ofcalculating the exact wave function are presented in both analytical and matrix representations. The exactwave-function theory based on the inverse Hamiltonian can be applied to systems that have singularities in theHamiltonian. The generalized ICI theory is applied to the hydrogen atom, giving the exact solution without anysingularity problem.

 

http://qcri.or.jp/pdfs/276-PRA.65.052122.pdf 

 

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/39822/1/PhysRevA_65_052122.pdf