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By Berger Marcel

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Id^ix) + g5Al)[Ya)f8 - ig6a{x)Al\Jb, Ja]fs = 0. 26) and i^„ = -[D„DV\ *£„ = 3 ^ - d ^ = F^Ya, + z^afcc^^. e # = "*i 0~ where \I>i and ^ 2 are 2 x 2 wavefunction. 2. 1) where 5 = \ and B = B(£) is magnetic field. The Hamiltonian can easily be diagonalized for any background angular momentum (or spin) K. The S stands for spin of electron and for simplicity K = Si(5i = 1/2) is an average background spin contributed by other source, say, control spin. Denoting by H = H0 + Hi(t), H0 = a S i • S 2 , ffi(i) = /*B(t) • S 2 .

8) so that as b —> +oo, r t ^ ^ r l i 1 ^\\pf T LYn. 9), there is essentially the Lie derivative operator ^LYn. This should convince the reader that as when b —> +oo, the trace of the heat kernel exp (—tA^nc J should localize near the closed geodesies in X. Prom the above, we find that up to scaling, 2A2, no interpolates in a proper sense between the Hodge Laplacian and the geodesic flow. 4. A self-adjointness property The operator A^nc is certainly not self-adjoint in the classical sense. However it is shown in B U 5 that it is self-adjoint with respect to a nondegenerate Hermitian form of signature (oo,oo), which we now describe.

Quantum Groups and Quantum Integrable Systems, World Scientific, Singapore, 63-88 (1991). 13. L. Ge, K. M. Cho, Phys. Lett. A 260(1999) 484. 14. L. Ge, K. M. Cho, Phys. Lett. A 249(1998) 358. 15. B. L. Ge, K. Xue, J. Phys. A. 33 (2000) L345. 16. B. L. Ge, K. Xue, J. Phys. A. 34 (2001) 919. 17. B. L. Ge, K. Xue, J. Phys. A. 35 (2001) L7. 18. D. Yu. A. Takhtajan, Algebraic Analysis, 1 (1988) 178 (in Russian). 19. M. Haldane, Phys. Rev. Lett. 60 (1988) 635. 20. Shastry, Phys. Rev. Lett. 60 (1988) 639.

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