Self-adjoint operators and solving the Schrödinger equation

In this tutorial we collect facts from the theory of self-adjoint operators, mostly with a view of what is relevant for applications in mathematical quantum mechanics, in particular for solving the Schrödinger equation. Specific topics include the spectral theorem and functional calculus for self-adjoint operators, Stone’s Theorem, Laplacians and Fourier transform, Duhamel’s formula and Dyson series, time-evolution of non-interacting quantum systems, as well as the Baker-Campbell-Hausdorff fomula and Trotter product formula. We assume that the user of these notes is familiar with basic functional analysis and linear operator theory in Hilbert spaces. In most instances we don’t provide proofs. Details on both, basic background as well as proofs, can be found in numerous books, such as [1, 3, 4, 5, 6]. 1 Self-Adjoint Operators Let H be a separable complex Hilbert space with inner product 〈·, ·〉, which we choose linear in the second argument and conjugate-linear in the first argument. Let T be a densely defined linear operator in H, i.e. T : D(T )→ H linear with D(T ) dense in H. The adjoint T ∗ of T is the linear operator in H defined by D(T ∗) = {g ∈ H : ∃h ∈ H such that 〈h, f〉 = 〈g, Tf〉 for all f ∈ D(T )} (1) T ∗g = h . (2) T is called self-adjoint if T ∗ = T (which includes the requirement D(T ∗) = D(T )). Self-adjoint operators T are hermitean (defined as 〈Tg, f〉 = 〈g, Tf〉 for all f, g ∈ D(T )) and symmetric (meaning that T ∗ is an extension of T ). For unbounded operators these three properties are not equivalent. However, if T ∈ B(H), the bounded and everywhere defined operators on H, then hermitean, symmetric and selfadjoint all have the same meaning. The resolvent set of a linear operator T in H is given by ρ(T ) = {z ∈ C : T − z is injective and (T − z)−1 ∈ B(H)} (3)