Computable upper bounds for the adiabatic approximation errors

For a given Hermitian Hamiltonian H(s) (s is an element of [0, 1]) with eigenvalues E-k(s) and the corresponding eigenstates vertical bar E-k(s)> (1 <= k <= N), adiabatic evolution described by the dilated Hamiltonian H-T(t) := H(t/T) (t is an element of [0, T]) starting from any fixed eigenstate vertical bar E-n(0)> is discussed in this paper. Under the gap-condition that vertical bar E-k(s) - E-n(s)vertical bar >= lambda > 0 for all s is an element of [0, 1] and all k not equal n, computable upper bounds for the adiabatic approximation errors between the exact solution vertical bar psi(T)(t)> and the adiabatic approximation solution vertical bar psi(adi)(T)(t)> to the Schrodinger equation (1) over dot vertical bar(psi) over dot(T)(t)> = H-T(t)vertical bar psi(T)(t)> with the initial condition vertical bar psi T(0)> = vertical bar E-n(0)> are given in terms of fidelity and distance, respectively. As an application, it is proved that when the total evolving time T goes to infinity, parallel to vertical bar psi(T)(t)> - vertical bar psi(adi)(T)(t)>parallel to converges uniformly to zero, which implies that vertical bar psi(T)(t)> approximate to vertical bar psi(adi)(T)(t)> for all t is an element of [0, T] provided that T is large enough.