Matrix approach to Schrödinger-type equations initiated by perturbational invariants

Abstract

This research is motivated by results obtained by Vesić et al. about scalar perturbational invariants caused by scalar perturbations of a gravitational field. Five linearly independent scalar perturbational invariants are obtained there. After some algebraic computing, these five perturbational invariants are transformed to new five linearly independent scalar perturbational invariants. From the set of last mentioned five perturbational invariants, two pairs of them are selected such that the second invariant in this pair is equal to the partial derivative by conformal time of the first one. The pair invariant for the fifth one, analogous to the second invariants in the two pairs, was not obtained. In this manuscript, these existing results are expanded with respect to the basics of linear algebra and quantum mechanics. We obtained the corresponding Hermitian time-dependent matrices which transform three of obtained scalar perturbational invariants to their partial derivatives by conformal time in here. These matrices are the corresponding Hamiltonians. Their eigenvalues (energy levels) and eigenstates (energy functions) are determined. After that, the expectation values of Hamiltonians and their squares, together with the corresponding uncertainties of these Hamiltonians in the states of scalar perturbational invariants are obtained.