A function (, ) is called harmonic if it is twice continuously dierentiable and satises the following partial dierential equation. Our results pave the way to understand and control the HHG in quasicrystals and shed light on a potential shorter and stronger attosecond light source based on compact solids. Harmonic functions We start by dening harmonic functions and looking at some of their properties. (iv) Compared with crystals, multichannel HHG in FQ has a higher yield and wider spectral range. (iii) The broken symmetry results in more frequent backscattering of electrons. (ii) Fractal bands lead to more excitation channels, analogous to the forbidden nonvertical electron transitions in crystals. Our results reveal that (i) the acceleration theorem is approximately applicable in FQ, which provides us a valuable tool to analyze the electron trajectories. We simulate the Fibonacci quasicrystal (FQ) HHG for the first time and investigate the electron dynamics on the attosecond timescale. Quasicrystals with fractal band structures could solve this problem and produce multichannel HHG emissions, which has been rarely studied. A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the. (15) Br (0) Br(0) for all functions u that are nonnegative and harmonic on B 2r(0). There is a constant C depending only on the dimension n such that sup u C inf u.
Theorem 5.1 Let B 2r(0) be an open ball in Rn.
However, the strict transition laws in crystals restrict the number of HHG channels. Another useful property of harmonic functions is the Harnack inequality. We show, under a natural sufficient condition for the weights, that the spaces of harmonic functions on D are isomorphic to corresponding spaces of continuous or bounded functions on D. High-order harmonic generation (HHG) in solids was expected to be efficient due to their high density. We also study the spaces of harmonic functions for certain non-radial weights on D.