expHt

expHt denotes the matrix exponential of the product H t, where H is a square matrix (often a Hamiltonian in physics) and t is a scalar time parameter. It is defined by the convergent power series exp(H t) = sum_{n=0}^∞ (H t)^n / n!. This operator is central to solving linear systems and linear differential equations.

Notation and interpretation: exp(H t) is often written e^{H t}. In differential form, it is the unique solution operator to dX/dt = H X with X(0) = I, giving X(t) = exp(H t). In physics, exp(-i H t / ħ) is the time-evolution operator for a time-independent Hamiltonian.

Key properties: If H is diagonalizable as H = P D P^{-1}, then exp(H t) = P exp(D t) P^{-1}, with exp(D t) being diag(e^{λ_i t}). If H is Hermitian and t is real, exp(H t) is Hermitian and positive-definite; if H is skew-Hermitian, exp(H t) is unitary. The eigenvalues of exp(H t) are e^{λ_i t}, where λ_i are the eigenvalues of H.

Computational approaches: For small matrices, direct series or diagonalization suffices. For general or large matrices, diagonalization or Jordan form can be used to compute exp(H t); for sparse or large systems, scaling and squaring combined with Padé approximants or Krylov subspace methods are common.

Applications: Widely used in solving linear ODEs, control theory, and quantum mechanics. In quantum physics, the time-evolution operator is exp(-i H t / ħ). In dynamical systems, it propagates state vectors in time.

Example: If H = [[a, 0], [0, b]], then exp(H t) = diag(e^{a t}, e^{b t}).