Abstract

Straightforward discretization of the equations of motion of a quasi-continuum interacting with an electromagnetic field often leads to physical and numerical difficulties. We derive two distinct methods for reducing the number of energy levels that must be treated explicitly in such calculations. One of these applies to bands of quasi-continua with slowly decreasing shoulders; the other, to bands with rapidly decreasing shoulders.

© 1984 Optical Society of America

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References

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  1. B. W. Shore, "Coherence in the quasi-continuum model," Chem. Phys. Lett. 99, 240–243 (1983).
    [CrossRef]
  2. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. Press, Oxford, 1965); E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, Cambridge, 1935), pp. 40–41.
  3. A. A. Makarov, V. T. Platonenko, and V. V. Tyakht, "Interaction of a 'level-band' quantum system with a quasiresonant monochromatic field," Zh. Eksp. Teor. Fiz. 75, 2075–2091 (1978) [Sov. Phys. JETP 48, 1044–1051 (1978)].
  4. J. J. Yeh, C. M. Bowden, and J. H. Eberly, "Interrupted coarsegrained theory of'unimolecular relaxation and stimulated recurrences in photo-excitation of a quasi-continuum," J. Chem. Phys. 76, 5936–5946 (1982); see also J. H. Eberly, J. J. Yeh, and C. M. Bowden, "Interrupted coarse-grained theory of quasicontinuum photo-excitation," Chem. Phys. Lett. 86, 76–80 (1982).
    [CrossRef]
  5. G. C. Stey and R. W. Gibberd, "Decay of quantum states in some exactly soluble models," Physica 60, 1–26 (1972).
    [CrossRef]
  6. P. W. Milonni, J. R. Ackerhalt, H. W. Galbraith, and M. L. Shih, "Exponential decay, recurrences, and quantum-mechanical spreading in a quasi-continuum model," Phys. Rev. A 28, 32–39 (1983); R. Lefebvre and J. Savolainen, "Memory functions and recurrences in intramolecular processes," J. Chem. Phys. 60, 2509–2555 (1974).
    [CrossRef]
  7. C. D. Cantrell, V. S. Letokhov, and A. A. Makarov, "Coherent excitation of multilevel systems by laser light" in Coherent Nonlinear Optics: Recent Advances, M. S. Feld and V. S. Letokhov, eds. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  8. G. L. Peterson, C. D. Cantrell, and R. S. Burkey, "Adiabatic excitation of multilevel band systems," Opt. Commun. 43, 123–127 (1982).
    [CrossRef]
  9. N. M. Witriol, A. J. Galli, W. H. Brumage, and C. M. Bowden, "Criteria for the reduction of the effective manifold of states in models of laser-induced dissociation and chemistry," Opt. Lett. 5, 24–26 (1980).
    [CrossRef] [PubMed]
  10. R. S. Burkey and C. D. Cantrell, "Solution of the Schrödinger equation for systems driven by an exponential or a semiexponential pulse," Opt. Commun. 43, 64–68 (1982).
    [CrossRef]
  11. U. W. Hochstrasser, "Orthogonal polynomials" in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964), p. 773.
  12. D. Jackson, Fourier Series and Orthogonal Polynomials (Collegiate, Menasha, Wise, 1941), Chap. 7.
  13. R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York 1962), pp. 118–164.
  14. R. Haydock, "The recursive solution of the Schrödinger equation," Comput. Phys. Commun. 20, 11–16 (1980); R. Haydock, "The recursive solution of the Schrödinger equation," in Solid State Physics, H. Ehrenreich, F. Seitz, and D. Turnbull, eds. (Academic, New York, 1980), Vol. 35, p. 215.
    [CrossRef]
  15. Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore, "Coherent dynamics of N-level atoms and molecules. II. Analytic solutions," Phys. Rev. A 16, 2048–2051 (1977).
    [CrossRef]
  16. N. M. Witriol, "Including the continuum in the N-level molecule model," Chem. Phys. Lett. 98, 77–80 (1983).
    [CrossRef]

1983 (3)

B. W. Shore, "Coherence in the quasi-continuum model," Chem. Phys. Lett. 99, 240–243 (1983).
[CrossRef]

P. W. Milonni, J. R. Ackerhalt, H. W. Galbraith, and M. L. Shih, "Exponential decay, recurrences, and quantum-mechanical spreading in a quasi-continuum model," Phys. Rev. A 28, 32–39 (1983); R. Lefebvre and J. Savolainen, "Memory functions and recurrences in intramolecular processes," J. Chem. Phys. 60, 2509–2555 (1974).
[CrossRef]

N. M. Witriol, "Including the continuum in the N-level molecule model," Chem. Phys. Lett. 98, 77–80 (1983).
[CrossRef]

1982 (3)

R. S. Burkey and C. D. Cantrell, "Solution of the Schrödinger equation for systems driven by an exponential or a semiexponential pulse," Opt. Commun. 43, 64–68 (1982).
[CrossRef]

G. L. Peterson, C. D. Cantrell, and R. S. Burkey, "Adiabatic excitation of multilevel band systems," Opt. Commun. 43, 123–127 (1982).
[CrossRef]

J. J. Yeh, C. M. Bowden, and J. H. Eberly, "Interrupted coarsegrained theory of'unimolecular relaxation and stimulated recurrences in photo-excitation of a quasi-continuum," J. Chem. Phys. 76, 5936–5946 (1982); see also J. H. Eberly, J. J. Yeh, and C. M. Bowden, "Interrupted coarse-grained theory of quasicontinuum photo-excitation," Chem. Phys. Lett. 86, 76–80 (1982).
[CrossRef]

1980 (3)

N. M. Witriol, A. J. Galli, W. H. Brumage, and C. M. Bowden, "Criteria for the reduction of the effective manifold of states in models of laser-induced dissociation and chemistry," Opt. Lett. 5, 24–26 (1980).
[CrossRef] [PubMed]

C. D. Cantrell, V. S. Letokhov, and A. A. Makarov, "Coherent excitation of multilevel systems by laser light" in Coherent Nonlinear Optics: Recent Advances, M. S. Feld and V. S. Letokhov, eds. (Springer-Verlag, Berlin, 1980).
[CrossRef]

R. Haydock, "The recursive solution of the Schrödinger equation," Comput. Phys. Commun. 20, 11–16 (1980); R. Haydock, "The recursive solution of the Schrödinger equation," in Solid State Physics, H. Ehrenreich, F. Seitz, and D. Turnbull, eds. (Academic, New York, 1980), Vol. 35, p. 215.
[CrossRef]

1978 (1)

A. A. Makarov, V. T. Platonenko, and V. V. Tyakht, "Interaction of a 'level-band' quantum system with a quasiresonant monochromatic field," Zh. Eksp. Teor. Fiz. 75, 2075–2091 (1978) [Sov. Phys. JETP 48, 1044–1051 (1978)].

1977 (1)

Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore, "Coherent dynamics of N-level atoms and molecules. II. Analytic solutions," Phys. Rev. A 16, 2048–2051 (1977).
[CrossRef]

1972 (1)

G. C. Stey and R. W. Gibberd, "Decay of quantum states in some exactly soluble models," Physica 60, 1–26 (1972).
[CrossRef]

1964 (1)

U. W. Hochstrasser, "Orthogonal polynomials" in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964), p. 773.

Ackerhalt, J. R.

P. W. Milonni, J. R. Ackerhalt, H. W. Galbraith, and M. L. Shih, "Exponential decay, recurrences, and quantum-mechanical spreading in a quasi-continuum model," Phys. Rev. A 28, 32–39 (1983); R. Lefebvre and J. Savolainen, "Memory functions and recurrences in intramolecular processes," J. Chem. Phys. 60, 2509–2555 (1974).
[CrossRef]

Bialynicka-Birula, I.

Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore, "Coherent dynamics of N-level atoms and molecules. II. Analytic solutions," Phys. Rev. A 16, 2048–2051 (1977).
[CrossRef]

Bialynicka-Birula, Z.

Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore, "Coherent dynamics of N-level atoms and molecules. II. Analytic solutions," Phys. Rev. A 16, 2048–2051 (1977).
[CrossRef]

Bowden, C. M.

J. J. Yeh, C. M. Bowden, and J. H. Eberly, "Interrupted coarsegrained theory of'unimolecular relaxation and stimulated recurrences in photo-excitation of a quasi-continuum," J. Chem. Phys. 76, 5936–5946 (1982); see also J. H. Eberly, J. J. Yeh, and C. M. Bowden, "Interrupted coarse-grained theory of quasicontinuum photo-excitation," Chem. Phys. Lett. 86, 76–80 (1982).
[CrossRef]

N. M. Witriol, A. J. Galli, W. H. Brumage, and C. M. Bowden, "Criteria for the reduction of the effective manifold of states in models of laser-induced dissociation and chemistry," Opt. Lett. 5, 24–26 (1980).
[CrossRef] [PubMed]

Brumage, W. H.

Burkey, R. S.

R. S. Burkey and C. D. Cantrell, "Solution of the Schrödinger equation for systems driven by an exponential or a semiexponential pulse," Opt. Commun. 43, 64–68 (1982).
[CrossRef]

G. L. Peterson, C. D. Cantrell, and R. S. Burkey, "Adiabatic excitation of multilevel band systems," Opt. Commun. 43, 123–127 (1982).
[CrossRef]

Cantrell, C. D.

G. L. Peterson, C. D. Cantrell, and R. S. Burkey, "Adiabatic excitation of multilevel band systems," Opt. Commun. 43, 123–127 (1982).
[CrossRef]

R. S. Burkey and C. D. Cantrell, "Solution of the Schrödinger equation for systems driven by an exponential or a semiexponential pulse," Opt. Commun. 43, 64–68 (1982).
[CrossRef]

C. D. Cantrell, V. S. Letokhov, and A. A. Makarov, "Coherent excitation of multilevel systems by laser light" in Coherent Nonlinear Optics: Recent Advances, M. S. Feld and V. S. Letokhov, eds. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Eberly, J. H.

J. J. Yeh, C. M. Bowden, and J. H. Eberly, "Interrupted coarsegrained theory of'unimolecular relaxation and stimulated recurrences in photo-excitation of a quasi-continuum," J. Chem. Phys. 76, 5936–5946 (1982); see also J. H. Eberly, J. J. Yeh, and C. M. Bowden, "Interrupted coarse-grained theory of quasicontinuum photo-excitation," Chem. Phys. Lett. 86, 76–80 (1982).
[CrossRef]

Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore, "Coherent dynamics of N-level atoms and molecules. II. Analytic solutions," Phys. Rev. A 16, 2048–2051 (1977).
[CrossRef]

Galbraith, H. W.

P. W. Milonni, J. R. Ackerhalt, H. W. Galbraith, and M. L. Shih, "Exponential decay, recurrences, and quantum-mechanical spreading in a quasi-continuum model," Phys. Rev. A 28, 32–39 (1983); R. Lefebvre and J. Savolainen, "Memory functions and recurrences in intramolecular processes," J. Chem. Phys. 60, 2509–2555 (1974).
[CrossRef]

Galli, A. J.

Gibberd, R. W.

G. C. Stey and R. W. Gibberd, "Decay of quantum states in some exactly soluble models," Physica 60, 1–26 (1972).
[CrossRef]

Hamming, R. W.

R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York 1962), pp. 118–164.

Haydock, R.

R. Haydock, "The recursive solution of the Schrödinger equation," Comput. Phys. Commun. 20, 11–16 (1980); R. Haydock, "The recursive solution of the Schrödinger equation," in Solid State Physics, H. Ehrenreich, F. Seitz, and D. Turnbull, eds. (Academic, New York, 1980), Vol. 35, p. 215.
[CrossRef]

Hochstrasser, U. W.

U. W. Hochstrasser, "Orthogonal polynomials" in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964), p. 773.

Jackson, D.

D. Jackson, Fourier Series and Orthogonal Polynomials (Collegiate, Menasha, Wise, 1941), Chap. 7.

Letokhov, V. S.

C. D. Cantrell, V. S. Letokhov, and A. A. Makarov, "Coherent excitation of multilevel systems by laser light" in Coherent Nonlinear Optics: Recent Advances, M. S. Feld and V. S. Letokhov, eds. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Makarov, A. A.

C. D. Cantrell, V. S. Letokhov, and A. A. Makarov, "Coherent excitation of multilevel systems by laser light" in Coherent Nonlinear Optics: Recent Advances, M. S. Feld and V. S. Letokhov, eds. (Springer-Verlag, Berlin, 1980).
[CrossRef]

A. A. Makarov, V. T. Platonenko, and V. V. Tyakht, "Interaction of a 'level-band' quantum system with a quasiresonant monochromatic field," Zh. Eksp. Teor. Fiz. 75, 2075–2091 (1978) [Sov. Phys. JETP 48, 1044–1051 (1978)].

Milonni, P. W.

P. W. Milonni, J. R. Ackerhalt, H. W. Galbraith, and M. L. Shih, "Exponential decay, recurrences, and quantum-mechanical spreading in a quasi-continuum model," Phys. Rev. A 28, 32–39 (1983); R. Lefebvre and J. Savolainen, "Memory functions and recurrences in intramolecular processes," J. Chem. Phys. 60, 2509–2555 (1974).
[CrossRef]

Peterson, G. L.

G. L. Peterson, C. D. Cantrell, and R. S. Burkey, "Adiabatic excitation of multilevel band systems," Opt. Commun. 43, 123–127 (1982).
[CrossRef]

Platonenko, V. T.

A. A. Makarov, V. T. Platonenko, and V. V. Tyakht, "Interaction of a 'level-band' quantum system with a quasiresonant monochromatic field," Zh. Eksp. Teor. Fiz. 75, 2075–2091 (1978) [Sov. Phys. JETP 48, 1044–1051 (1978)].

Shih, M. L.

P. W. Milonni, J. R. Ackerhalt, H. W. Galbraith, and M. L. Shih, "Exponential decay, recurrences, and quantum-mechanical spreading in a quasi-continuum model," Phys. Rev. A 28, 32–39 (1983); R. Lefebvre and J. Savolainen, "Memory functions and recurrences in intramolecular processes," J. Chem. Phys. 60, 2509–2555 (1974).
[CrossRef]

Shore, B. W.

B. W. Shore, "Coherence in the quasi-continuum model," Chem. Phys. Lett. 99, 240–243 (1983).
[CrossRef]

Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore, "Coherent dynamics of N-level atoms and molecules. II. Analytic solutions," Phys. Rev. A 16, 2048–2051 (1977).
[CrossRef]

Stey, G. C.

G. C. Stey and R. W. Gibberd, "Decay of quantum states in some exactly soluble models," Physica 60, 1–26 (1972).
[CrossRef]

Tyakht, V. V.

A. A. Makarov, V. T. Platonenko, and V. V. Tyakht, "Interaction of a 'level-band' quantum system with a quasiresonant monochromatic field," Zh. Eksp. Teor. Fiz. 75, 2075–2091 (1978) [Sov. Phys. JETP 48, 1044–1051 (1978)].

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. Press, Oxford, 1965); E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, Cambridge, 1935), pp. 40–41.

Witriol, N. M.

Yeh, J. J.

J. J. Yeh, C. M. Bowden, and J. H. Eberly, "Interrupted coarsegrained theory of'unimolecular relaxation and stimulated recurrences in photo-excitation of a quasi-continuum," J. Chem. Phys. 76, 5936–5946 (1982); see also J. H. Eberly, J. J. Yeh, and C. M. Bowden, "Interrupted coarse-grained theory of quasicontinuum photo-excitation," Chem. Phys. Lett. 86, 76–80 (1982).
[CrossRef]

Chem. Phys. Lett. (2)

B. W. Shore, "Coherence in the quasi-continuum model," Chem. Phys. Lett. 99, 240–243 (1983).
[CrossRef]

N. M. Witriol, "Including the continuum in the N-level molecule model," Chem. Phys. Lett. 98, 77–80 (1983).
[CrossRef]

Comput. Phys. Commun. (1)

R. Haydock, "The recursive solution of the Schrödinger equation," Comput. Phys. Commun. 20, 11–16 (1980); R. Haydock, "The recursive solution of the Schrödinger equation," in Solid State Physics, H. Ehrenreich, F. Seitz, and D. Turnbull, eds. (Academic, New York, 1980), Vol. 35, p. 215.
[CrossRef]

J. Chem. Phys. (1)

J. J. Yeh, C. M. Bowden, and J. H. Eberly, "Interrupted coarsegrained theory of'unimolecular relaxation and stimulated recurrences in photo-excitation of a quasi-continuum," J. Chem. Phys. 76, 5936–5946 (1982); see also J. H. Eberly, J. J. Yeh, and C. M. Bowden, "Interrupted coarse-grained theory of quasicontinuum photo-excitation," Chem. Phys. Lett. 86, 76–80 (1982).
[CrossRef]

Opt. Commun. (2)

G. L. Peterson, C. D. Cantrell, and R. S. Burkey, "Adiabatic excitation of multilevel band systems," Opt. Commun. 43, 123–127 (1982).
[CrossRef]

R. S. Burkey and C. D. Cantrell, "Solution of the Schrödinger equation for systems driven by an exponential or a semiexponential pulse," Opt. Commun. 43, 64–68 (1982).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

P. W. Milonni, J. R. Ackerhalt, H. W. Galbraith, and M. L. Shih, "Exponential decay, recurrences, and quantum-mechanical spreading in a quasi-continuum model," Phys. Rev. A 28, 32–39 (1983); R. Lefebvre and J. Savolainen, "Memory functions and recurrences in intramolecular processes," J. Chem. Phys. 60, 2509–2555 (1974).
[CrossRef]

Z. Bialynicka-Birula, I. Bialynicka-Birula, J. H. Eberly, and B. W. Shore, "Coherent dynamics of N-level atoms and molecules. II. Analytic solutions," Phys. Rev. A 16, 2048–2051 (1977).
[CrossRef]

Physica (1)

G. C. Stey and R. W. Gibberd, "Decay of quantum states in some exactly soluble models," Physica 60, 1–26 (1972).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

A. A. Makarov, V. T. Platonenko, and V. V. Tyakht, "Interaction of a 'level-band' quantum system with a quasiresonant monochromatic field," Zh. Eksp. Teor. Fiz. 75, 2075–2091 (1978) [Sov. Phys. JETP 48, 1044–1051 (1978)].

Other (5)

U. W. Hochstrasser, "Orthogonal polynomials" in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964), p. 773.

D. Jackson, Fourier Series and Orthogonal Polynomials (Collegiate, Menasha, Wise, 1941), Chap. 7.

R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York 1962), pp. 118–164.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. Press, Oxford, 1965); E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge U. Press, Cambridge, 1935), pp. 40–41.

C. D. Cantrell, V. S. Letokhov, and A. A. Makarov, "Coherent excitation of multilevel systems by laser light" in Coherent Nonlinear Optics: Recent Advances, M. S. Feld and V. S. Letokhov, eds. (Springer-Verlag, Berlin, 1980).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

A typical (1, N) system. Transitions are allowed from the ground state to the upper band, not within the upper band itself.

Fig. 2
Fig. 2

Form of the semiexponential pulse. Initially (t = −∞) the field envelope increases as eλt, but eventually it goes to a constant value.

Fig. 3
Fig. 3

Ground-state probability amplitude for a uniform rectangular band of full width 0.6 cm−1, as calculated by our technique using 16 levels in the discretized band. This result is accurate to about six decimal places.

Fig. 4
Fig. 4

Error resulting from Rice discretizations of 16, 32, 64, and 128 evenly spaced levels in the calculation of the of full width 0.6 cm−1. The zero point of errornd is taken to be the amplitude calculated using our method (see Fig. 3).

Equations (30)

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H ( t ) = H 0 + E ( t ) μ ,
d d t a ( t ) = i E ( t ) - μ ( Δ ) g ( Δ ) b ( t , Δ ) d Δ ,
d d t b ( t , Δ ) = i Δ b ( t , Δ ) + i E ( t ) μ ( Δ ) a ( t ) .
d d t a ( t ) = - E ( t ) t 0 t E ( t ) a ( t ) χ ( t - t ) d t ,
χ ( t ) = - μ ( Δ ) 2 g ( Δ ) e i Δ t d Δ .
d d t a ( t ) = i E ( t ) n > 0 μ n b n ( t ) ,
d d t b n ( t ) = i Δ n b n ( t ) + i E ( t ) μ n a ( t ) .
χ ( t ) = n > 0 μ n 2 exp ( i Δ n t ) ,
Φ = 2 i N - a * ( t ) μ ( Δ ) g ( Δ ) b ( t , Δ ) d Δ ,
Φ = 2 i N n > 0 a * ( t ) μ n b n ( t ) .
w ( Δ ) p ( Δ ) q ( Δ )             [ p ( Δ ) , q ( Δ ) polynomials ] .
μ n = [ 2 π i p ( Δ n ) q ( Δ n ) ] 1 / 2 ,
χ ( t ) = Δ n exp ( i Δ n t ) p ( Δ n ) Δ n d Δ q ( Δ ) ,
w ( Δ ) = μ ( Δ ) 2 g ( Δ ) = μ 2 σ π 1 ( Δ - s ) 2 + σ 2 ,
d d t [ a ( t ) b ( t ) ] = i [ 0 E ( t ) μ E ( t ) μ s + i σ ] [ a ( t ) b ( t ) ] .
a ( τ ) = exp ( i ζ - τ ) M [ i ( s + i σ ) ζ - 2 λ α , - i s + i σ λ , 2 i α τ ] ,
ζ - = s + i σ 2 - α
α = [ ( s + i σ 2 ) 2 + ( μ 01 2 ) 2 ] 1 / 2 .
μ n 2 = σ δ π 1 ( n δ - s ) 2 + σ 2 ( tanh π σ δ ) μ 2 ,
χ ( t ) = μ 2 1 + tanh π σ δ 2 exp [ i ( s + i σ ) t ] + μ 2 1 - tanh π σ δ 2 exp [ i ( s - i σ ) t ] .
- w ( Δ ) f n ( Δ ) f m ( Δ ) d Δ = δ n m .
w ( Δ ) f ( Δ ) d Δ ,
- w ( Δ ) f ( Δ ) d Δ n > 0 w n f ( Δ n ) ,
χ ( t ) n > 0 w n exp ( i Δ n ) ,
δ χ 0 t 0 t E ( t ) E ( t ) d t d t ,
δ χ 2 2 N + 1 ( N ! ) 4 ( 2 N + 1 ) [ ( 2 N ) ! ] 3 ( Δ 0 t ) 2 N .
0 ,             - d 0 e 0 ,             - d 1 e 1 ,             - d 2 e 2 , ,
E ( t ) μ ,             1 e 0 ,             1 e 1 ,             1 e 2 , ,
f n + 1 ( Δ ) = ( d n + e n Δ ) f n ( Δ ) - c n f n - 1 ( Δ ) .
[ 1 0 ] ,             [ 0 μ ( Δ ) μ f 0 ( Δ ) ] ,             [ 0 μ ( Δ ) μ f 1 ( Δ ) ] , .

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