The Budapest Quantum Optics Group

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Variational properties of the discrete variable representation: Discrete variable representation via effective operators. Part 2.

Viktor Szalay


Time: Tue Apr 17 13:00:00 2012
Location: Building 4, room 116/A

The variational basis representation (VBR), a matrix representation, of a Hamiltonian operator H gives a variational approximation to the eigenvalues of H. That is, if N basis functions are employed the resulting N approximate eigenvalues approach the corresponding exact ones from above with increasing basis size. The N basis functions span a Hilbert space {$\mathcal{S}$} of dimension N. The discrete variable representation (DVR) of H is constructed by using a set of grid points and basis functions. The same number of grid points as basis functions, say N, are employed. In one dimension (1D), the grid points are the eigenvalues of the VBR of the coordinate operator. In more dimensions, direct product bases are employed and the grid is the direct product of the corresponding 1D sets of grid points. The DVR is nonvariational, that is some of the N approximate eigenvalues are bigger, whereas others are smaller, than the exact ones. They converge by oscillating around the exact values with smalller and smaller amplitude, as the basis size and the grid size are increased simultaneously. To restore the variational property one increases the number of grid points or calculates suitable correction terms. This restores the variational property with respect to the basis set. That is the expansion coefficients of the wave function with respect to the basis set become variational parameters. Nevertheless, the DVR still remains nonvariational with respect to the grid. The grid points are not variational parameters. I will show that, by removing just a single grid point while retaining the N basis functions, in other words, by deleting a row and a column from the original DVR set up in {$\mathcal{S}$}, a DVR is obtained which is variational with the eigenvalues of H calculated in {$\mathcal{S}$} taking the role of the exact eigenvalues and it is variational with respect to both the basis set and the grid. More generally, it is shown that the DVR is variational over well specified subsets of the N-point grid. The variational property implies that an optimal O-point grid (0<O<N), i.e., the O-point grid giving eigenvalues with the possible highest accuracy, is the one which gives the (pruned) DVR of the smallest trace. Numerical examples are given to illustrate the theoretical results. The use of variational effective Hamiltonian and coordinate operators has been instrumental in this study. They have been introduced in a novel way by exploiting quasi-Hermiticity.