Ivancevic_Applied-Diff-Geom

1160 Applied Differential Geometry: A Modern IntroductionUsing the conditions for P i (σ) and ¯P i (σ) in (6.260), we can reduce theconditions for Q j (σ) and ¯Q j (σ) to those for ∆Q j (σ) and ∆ ¯Q j (σ):{ i∆Q∆Q j j (π − σ) (0 ≤ σ ≤ π(σ) =2 )−i∆Q j (π − σ) ( π 2{ ≤ σ ≤ π),∆ ¯Q −i∆ j (σ) =¯Qj (π − σ) (0 ≤ σ ≤ π 2 )i∆ ¯Q j (π − σ) ( π 2 ≤ σ ≤ π).These formulas are translated to the relations between the modes via theFourier transformation. The result is expressed in the vector notation as(1 − X)|Q i )| ˆV 4 〉 X = (1 + X)|Q i )| ˆV 4 〉 X = 0, (6.264)where the vectors |Q i ) and |Q i ) stand for⎡Q 0,i +|Q i ) = ⎢⎣i4αG ′ ik θ kj ∑ ∞Q 1,iQ 2,in=0 X 0nP n,j⎤⎥⎦ ,.⎡¯Q 0,i +i4αG ′ ik θ kj ∑ ∞n=0 X ¯P ⎤0n n,j¯Q 1,i|Q i ) = ⎢⎣¯Q 2,i⎥⎦ .In (6.264), passing the vectors through the phase factor of the | ˆV 4 〉 andusing the continuity conditions in the Neumann case.(1 + X)|P i )|V 4 〉 X = (1 − X)| ¯P i )|V 4 〉 X = 0, (6.265)(1 − X)|Q i )|V 4 〉 X = (1 + X)| ¯Q i )|V 4 〉 X = 0,we get the equations, which the coefficients Z nm ’s should satisfy,∑∞[(1 − X) m0 ( ¯Z 0n + i π 2 ¯X∞∑∑∞0n )P n,j + (1 − X) mn¯Z nn ′P n ′ ,j]|V 4 〉 X = 0n=0n=0n=1n ′ =0∑∞[(1 + X) m0 (Z 0n − i π 2 X 0n) ¯P∞∑∑∞n,j + (1 + X) mn Z nn ′¯Pn′ ,j]|V 4 〉 X = 0n=1n ′ =0for m ≥ 0. Now all our remaining task is to solve these equations. It is