Dipartimento di Scienze Fisiche,

Universita di Napoli Federico II,

C.U. di Monte S.Angelo,

via Cintia,

80126 Napoli,

ITALY

and

Istituto Nazionale di Fisica Nucleare,

Sez. di Napoli,

ITALY

E-mail: Paolo.Aniello@na.infn.it

**A new perturbative expansion of the time evolution
operator associated with a quantum system**

**Abstract:**

In two fundamental papers published in 1949, Dyson developed a perturbative
expansion of the time evolution operator that has been adopted extensively
in any field of physics. Dyson expansion has a transparent physical interpretation
in terms of time ordered elementary processes which makes its application
particularly appealing, especially in quantum field theory. On the other
hand, for many applications, Dyson expansion has severe drawbacks, as a
low convergence rate and the lack of unitarity of its truncations. Later,
Magnus introduced an expansion of the evolution operator such that each
of its truncations retains the property of being unitary. We will present
a new perturbative expansion of the evolution operator associated with
a (in general, time-dependent) quantum Hamiltonian that enjoys remarkable
properties. In particular, each of its truncations has property of being
unitary and it generalizes Magnus expansion, so opening the possibility
of achieving computational advantages and of satisfying specific requirements.
In the case of a time-independent Hamiltonian, the link with standard perturbation
theory for linear operators will be discussed.