XVIth International Workshop on
Quantum Systems in
Chemistry and Physics
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Application of Variational Principle to the Dirac Equation
Grzegorz Pestka
Institute of Physics, Nicolaus Copernicus University, Torun, Poland
In the variational determination of the Dirac Hamiltonian eigenstates a major problem which has to be taken into consideration is the "variational collapse". Commonly by the "variational collapse" we understand unlimited decrease of some variational eigenvalues, but in general under this abbreviated name we can understand a broader set of problems related to an improper choice of the variational space. Thus, in general, "variational collapse" also contains such effects in the calculated spectrum as a wrong order of eigenstates, convergence to wrong eigenstates, or appearance of spurious states. All these effects can appear also if the basis set approaches completeness, i.e. it would be correct if we aimed at solving a bounded from below eigenvalue problem.

In this talk a survey of problems related to the correct construction of the basis sets in unbounded from below eigenproblems involving multicomponent eigenfunctions will be presented. In particular, the importance of the exact balance between the large and small component variational spaces in the Dirac-Pauli representation of the Dirac equation will be explained. As a consequence of the way the balance between the spaces has been formulated, the problem of the boundary conditions will be elucidated.


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