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We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form where is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schrödinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schrödinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from Damanik and Goldstein [J. Amer. Math. Soc. 29 (2016), pp. 825–856], Damanik et al. [J. Math Pures Appl. (9) 186 (2024), pp. 251–302], and Damanik et al. [J. Funct. Anal. 286 (2024), No. 110238] by shorter arguments.
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Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field
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∂t ψ = − 2 1 ∂x 2 ψ + (x)(U − E x cos ωt)ψ, t > 0, x ∈ R, where (x) is the Heaviside function and U > 0 is the effective confining potential (we choose units so that m = e = = 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 2the initial condition is in L 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-L 2 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.
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- Award ID(s):
- 2206241
- PAR ID:
- 10427996
- Editor(s):
- Springer
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 402
- Issue:
- 2
- ISSN:
- 0010-3616
- Subject(s) / Keyword(s):
- time dependent Schrodinger equation asymptotic analysis resurgence
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1007/s00220-023-04771-0
