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1900 Graphene – Synthesis, Characterization, Properties andGrAapphepnelicSyanthieosnis s Thus, one should seek solution in the form ψ0 = σ(0)(s, ν)e1. Solvability of the problem (H0 − EI)ψ1 = −vF H1ψ0 leads to the orthogonality condition < e1, H1ψ0 >= 0, (16) which can be simplified as follows U1 − U0 − E = α. (17) ρ This relation is well-known in the asymptotic theory of Gaussian beams (see (20), (21), (17),(18) (19)). It is equivalent to the requirement that everything is being constructed in a asymptotically small neighbourhood of the fixed curve which is a classical trajectory - solution to hamiltonian system. Solvability of the problem requires the orthogonality condition (H0−EI)ψ2 =−vFH1ψ1−vFH2ψ0 which leads to transport equation for the unknown σ(0)(s, ν) 2ia(s)σ(0) + σ(0) − a(s)(ν2a(s)d(s) + θ ̇)σ(0) = 0, (18) s νν d(s)= 1 2(U2−U1)+α. a(s) ρ ρ This is so-called parabolic equation of Gaussian beam boundary layer (see (20), (21), (17),(18) (19)). It has a solution (0) eiΓν2/2 −iθ/2 z ̇ σ=√ze ,Γ=az, (19) ( 2 0 ) z ̇ = p/a(s), p ̇ = −a(s)d(s)z (21) p2 a(s)d(s)z2 H(z, p) = 2a(s) + 2 . (see (20), (21), (18) (19)). Here Γ satisfies the Ricatti equation , Γ ̇ + 1 Γ 2 + a d = 0 . a S = S0(s)+S1(s)n+ 1Γ(s)n2 +... The expression gives the approximate solution to the Hamilton-Jacobi equation (8). 2 The function z(s) satisfies the system of equations in the hamiltonian form with the hamiltonianPDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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