  We propose a new concept for the dynamics of a quantum bath, the Chebyshev
space, and a new method based on this concept, the Chebyshev space method. The
Chebyshev space is an abstract vector space that exactly represents the
fermionic or bosonic bath degrees of freedom, without a discretization of the
bath density of states. Relying on Chebyshev expansions the Chebyshev space
representation of a bath has very favorable properties with respect to
extremely precise and efficient calculations of groundstate properties, static
and dynamical correlations, and time-evolution for a great variety of quantum
systems. The aim of the present work is to introduce the Chebyshev space in
detail and to demonstrate the capabilities of the Chebyshev space method.
Although the central idea is derived in full generality the focus is on model
systems coupled to fermionic baths. In particular we address quantum impurity
problems, such as an impurity in a host or a bosonic impurity with a static
barrier, and the motion of a wave packet on a chain coupled to leads. For the
bosonic impurity, the phase transition from a delocalized electron to a
localized polaron in arbitrary dimension is detected. For the wave packet on a
chain, we show how the Chebyshev space method implements different boundary
conditions, including transparent boundary conditions replacing infinite leads.
Furthermore the self-consistent solution of the Holstein model in infinite
dimension is calculated. With the examples we demonstrate how highly accurate
results for system energies, correlation and spectral functions, and
time-dependence of observables are obtained with modest computational effort.
