Truncated many-body dynamics of interacting bosons: A variational principle with error monitoring

We develop a method to describe the temporal evolution of an interacting system of bosons, for which the field operator expansion is truncated after a finite number $M$ of modes, in a rigorously controlled manner. Using McLachlan's principle of least error, we find a self-consistent set of equations for the many-body state. As a particular benefit, and in distinction to previously proposed approaches, the presently introduced method facilitates the dynamical increase of the number of orbitals during the temporal evolution, due to the fact that we can rigorously monitor the error made by increasing the truncation dimension $M$. The additional orbitals, determined by the condition of least error of the truncated evolution relative to the exact one, are obtained from an initial trial state by steepest $constrained$ descent.