Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field

Ovidiu Costin, Rodica Costin, Ian Jauslin, Joel L. Lebowitz

Research output: Contribution to journalArticlepeer-review


We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space x⩽ 0 , the Schrödinger equation of the system is i∂tψ=-12∂x2ψ+Θ(x)(U-Excosωt)ψ , t> 0 , x∈ R , where Θ (x) is the Heaviside function and U> 0 is the effective confining potential (we choose units so that m= e= ħ= 1). The amplitude E of the external electric field and the frequency ω are arbitrary. We prove existence and uniqueness of classical solutions of the Schrödinger equation for general initial conditions ψ(x, 0) = f(x) , x∈ R . When the initial condition is in L2 the evolution is unitary and the wave function goes to zero at any fixed x as t→ ∞ . To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non- L2 initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all t> 0 . For these initial conditions we show that the solution approaches in the large t limit a periodic state that satisfies an infinite set of equations formally derived, under the assumption that the solution is periodic by Faisal et al. (Phys Rev A 72:023412, 2005). Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior, as seen both analytically and numerically. It shows a steep increase in the current as the frequency passes a threshold value ω= ωc , with ωc depending on the strength of the electric field. For small E, ωc represents the threshold in the classical photoelectric effect, as described by Einstein’s theory.

Original languageEnglish (US)
Pages (from-to)2031-2078
Number of pages48
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Sep 2023

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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