Lie Algebraic Solution of Linear Differential Equations

Abstract

The solution U(t) to the linear differential equation dU/dt = h(t)U can be represented by a finite product of exponential operators; In many interesting cases the representation is global. U(t) = exp[g1(t)H1] exp [g2(t)H2] … exp[gn(t)Hn] where gi(t) are scalar functions and Hi are constant operators. The number, n, of terms in this expansion is equal to the dimension of the Lie algebra generated by H(t). Each term in this product has time-independent eigenvectors. Some applications of this solution to physical problems are given.

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

“…The key element behind the Lie algebraic approach is that the general form of the evolution operator of such type of Hamiltonian can be expressed as [55][56][57][58][59] …”

Applications of Lie systems in quantum mechanics and control theory

“…If instead, the Lie algebra of G is semi-simple, then the integrability by quadratures is not assured [6,8,12,23,24].…”

“…Thus, after a brief account of a generalization of the method proposed by Wei and Norman [6,8,12,23,24], to be used later, we will study the particular case where the Lie systems of interest are Hamiltonian systems as well, both in the classical and quantum frameworks. The theory is illustrated through the particularly interesting example of generic classical and quantum quadratic time-dependent Hamiltonians.…”

Quasi-Lie schemes: theory and applications

“…, in terms of a set of solutions of the second-order linear system (21). Summing up, the application of our scheme to the family of dissipative Milne-Pinney equationsẍ = a(t)ẋ + b(t) x + exp 2 a(t) dt k x 3 shows that it admits a time-dependent superposition principle:…”

“…A generalisation of the method used by Wei and Norman for linear systems [21] is very useful in solving such equations and furthermore there exist reduction techniques that can also be used [11]. Finally, as right-invariant vector fields X R project onto the fundamental vector fields in each homogeneous space of G, the solution of (10) allows us to find the general solution for the corresponding Lie system in each homogeneous space.…”