Singular perturbation series in quantum mechanics (Q1284172)

The author investigates singular perturbation series on a simple solvable example of a harmonic oscillator described by the Hamiltonian \[ H_\varepsilon =\frac{1}{8\pi^2}\partial^2_x +\frac{1}{2}\varepsilon x^2 \] with the term \(\frac{1}{2}\varepsilon x^2\) playing the role of perturbation. He proves rigorously that the transition amplitude \(\langle\psi ,U(t, \varepsilon)\varphi\rangle\), where \(U(t,\varepsilon)=\exp (-itH_\varepsilon)\), possesses for any positive integer \(m\) an asymptotic expansion of the form \[ \langle\psi ,U(t,\varepsilon)\varphi\rangle = \sum_{k}^{m-1} [a_k (t) \varepsilon^{k/2} \log\varepsilon +b_k (t)\varepsilon^{k/2} ] . \] The proof is constructive (by taking \(\varphi\) and \(\psi\) as linear combinations of functions \(x^j (1+x^2)^{-k/2}\)). This is a phenomenon of a general nature - for a singular perturbation, in general, there are no exclusively power series expansions. Standard perturbation expansions used in QFT (Dyson series, Feynman diagrams, path integrals, etc.) are all structured so as to produce only power series and thus fail to detect the logarithmic terms.