Abstract

We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of points and satisfy a physically important difference equation. The fractional finite Fourier–Kravchuk transform is defined to self-reproduce these functions. The analysis of finite signal processing uses the representations of the ordinary rotation group SO(3). This leads naturally to a phase space for finite optics such that the continuum limit (N) reproduces Fourier paraxial optics.

© 1997 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  3. M. Krawtchouk, “Sur une généralisation des polynômes d’Hermite,” C. R. Acad. Sci. Paris 189, 620–622 (1929).
  4. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).
  5. N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, R.I., 1968).
  6. A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).
  7. M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
    [Crossref]
  8. K. B. Wolf, Integral Transforms in Science and Engineering, Vol. 11 of Mathematical Concepts and Methods in Science and Engineering Series, A. Miele, ed. (Plenum, New York, 1979).
  9. L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981).
  10. P. Feinsilver, R. Schott, Algebraic Structures and Operator Calculus. Vol. I: Representations and Probability Theory (Kluwer, Dordrecht, The Netherlands, 1993).
  11. N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991).
    [Crossref]
  12. N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fı́s. 40, 366–377 (1994).
  13. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Applic. 25, 241–265 (1980).
    [Crossref]
  14. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995);D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [Crossref]
  15. V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
    [Crossref]
  16. R. Askey, N. M. Atakishiyev, S. K. Suslov, “Fourier transformations for difference analogs of the harmonic oscillator,” in Proceedings of the XV Workshop on Problems on High Energy Physics and Field Theory, A. P. Samokhin, G. L. Rcheulishvili, eds. (Institute for High Energy Physics, Protvino, Russia, 1995), pp. 140–144.
  17. J. H. McClellan, T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Trans. Audio Electroacoustics AU-20, 66–74 (1972);T. S. Santhanam, A. R. Tekumalla, “Quantum mechanics in finite dimensions,” Found. Phys. 6, 583–289 (1976);M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
    [Crossref]
  18. A. Aguilar, K. B. Wolf, “Symmetries of the second-difference matrix and the finite Fourier transform,” Kinam 1, 387–405 (1979); K. B. Wolf, “Travelling waves, symmetries and invariant quadratic forms in discrete systems,” Kinam 2, 223–272 (1981).
  19. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965);J. W. Cooley, P. A. W. Lewis, P. D. Welch, “Historical notes on the FFT,” IEEE Trans. Audio Electroacoust.AU-15, 76–79 (1967).
    [Crossref]

1995 (1)

1994 (1)

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fı́s. 40, 366–377 (1994).

1991 (1)

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991).
[Crossref]

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Applic. 25, 241–265 (1980).
[Crossref]

1979 (1)

A. Aguilar, K. B. Wolf, “Symmetries of the second-difference matrix and the finite Fourier transform,” Kinam 1, 387–405 (1979); K. B. Wolf, “Travelling waves, symmetries and invariant quadratic forms in discrete systems,” Kinam 2, 223–272 (1981).

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

1972 (1)

J. H. McClellan, T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Trans. Audio Electroacoustics AU-20, 66–74 (1972);T. S. Santhanam, A. R. Tekumalla, “Quantum mechanics in finite dimensions,” Found. Phys. 6, 583–289 (1976);M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
[Crossref]

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965);J. W. Cooley, P. A. W. Lewis, P. D. Welch, “Historical notes on the FFT,” IEEE Trans. Audio Electroacoust.AU-15, 76–79 (1967).
[Crossref]

1947 (1)

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[Crossref]

1929 (1)

M. Krawtchouk, “Sur une généralisation des polynômes d’Hermite,” C. R. Acad. Sci. Paris 189, 620–622 (1929).

Aguilar, A.

A. Aguilar, K. B. Wolf, “Symmetries of the second-difference matrix and the finite Fourier transform,” Kinam 1, 387–405 (1979); K. B. Wolf, “Travelling waves, symmetries and invariant quadratic forms in discrete systems,” Kinam 2, 223–272 (1981).

Askey, R.

R. Askey, N. M. Atakishiyev, S. K. Suslov, “Fourier transformations for difference analogs of the harmonic oscillator,” in Proceedings of the XV Workshop on Problems on High Energy Physics and Field Theory, A. P. Samokhin, G. L. Rcheulishvili, eds. (Institute for High Energy Physics, Protvino, Russia, 1995), pp. 140–144.

Atakishiyev, N. M.

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fı́s. 40, 366–377 (1994).

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991).
[Crossref]

R. Askey, N. M. Atakishiyev, S. K. Suslov, “Fourier transformations for difference analogs of the harmonic oscillator,” in Proceedings of the XV Workshop on Problems on High Energy Physics and Field Theory, A. P. Samokhin, G. L. Rcheulishvili, eds. (Institute for High Energy Physics, Protvino, Russia, 1995), pp. 140–144.

Bargmann, V.

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[Crossref]

Biedenharn, L. C.

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965);J. W. Cooley, P. A. W. Lewis, P. D. Welch, “Historical notes on the FFT,” IEEE Trans. Audio Electroacoust.AU-15, 76–79 (1967).
[Crossref]

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).

Feinsilver, P.

P. Feinsilver, R. Schott, Algebraic Structures and Operator Calculus. Vol. I: Representations and Probability Theory (Kluwer, Dordrecht, The Netherlands, 1993).

Krawtchouk, M.

M. Krawtchouk, “Sur une généralisation des polynômes d’Hermite,” C. R. Acad. Sci. Paris 189, 620–622 (1929).

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Louck, J. D.

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981).

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

McClellan, J. H.

J. H. McClellan, T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Trans. Audio Electroacoustics AU-20, 66–74 (1972);T. S. Santhanam, A. R. Tekumalla, “Quantum mechanics in finite dimensions,” Found. Phys. 6, 583–289 (1976);M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
[Crossref]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Mendlovic, D.

Moshinsky, M.

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[Crossref]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Applic. 25, 241–265 (1980).
[Crossref]

Nikiforov, A. F.

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).

Ozaktas, H. M.

Parks, T. W.

J. H. McClellan, T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Trans. Audio Electroacoustics AU-20, 66–74 (1972);T. S. Santhanam, A. R. Tekumalla, “Quantum mechanics in finite dimensions,” Found. Phys. 6, 583–289 (1976);M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
[Crossref]

Quesne, C.

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[Crossref]

Schott, R.

P. Feinsilver, R. Schott, Algebraic Structures and Operator Calculus. Vol. I: Representations and Probability Theory (Kluwer, Dordrecht, The Netherlands, 1993).

Suslov, S. K.

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991).
[Crossref]

R. Askey, N. M. Atakishiyev, S. K. Suslov, “Fourier transformations for difference analogs of the harmonic oscillator,” in Proceedings of the XV Workshop on Problems on High Energy Physics and Field Theory, A. P. Samokhin, G. L. Rcheulishvili, eds. (Institute for High Energy Physics, Protvino, Russia, 1995), pp. 140–144.

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965);J. W. Cooley, P. A. W. Lewis, P. D. Welch, “Historical notes on the FFT,” IEEE Trans. Audio Electroacoust.AU-15, 76–79 (1967).
[Crossref]

Uvarov, V. B.

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

Vilenkin, N. Ya.

N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, R.I., 1968).

Wolf, K. B.

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fı́s. 40, 366–377 (1994).

A. Aguilar, K. B. Wolf, “Symmetries of the second-difference matrix and the finite Fourier transform,” Kinam 1, 387–405 (1979); K. B. Wolf, “Travelling waves, symmetries and invariant quadratic forms in discrete systems,” Kinam 2, 223–272 (1981).

K. B. Wolf, Integral Transforms in Science and Engineering, Vol. 11 of Mathematical Concepts and Methods in Science and Engineering Series, A. Miele, ed. (Plenum, New York, 1979).

Ann. Math. (1)

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[Crossref]

C. R. Acad. Sci. Paris (1)

M. Krawtchouk, “Sur une généralisation des polynômes d’Hermite,” C. R. Acad. Sci. Paris 189, 620–622 (1929).

IEEE Trans. Audio Electroacoustics (1)

J. H. McClellan, T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Trans. Audio Electroacoustics AU-20, 66–74 (1972);T. S. Santhanam, A. R. Tekumalla, “Quantum mechanics in finite dimensions,” Found. Phys. 6, 583–289 (1976);M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
[Crossref]

J. Inst. Math. Applic. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Applic. 25, 241–265 (1980).
[Crossref]

J. Opt. Soc. Am. A (1)

Kinam (1)

A. Aguilar, K. B. Wolf, “Symmetries of the second-difference matrix and the finite Fourier transform,” Kinam 1, 387–405 (1979); K. B. Wolf, “Travelling waves, symmetries and invariant quadratic forms in discrete systems,” Kinam 2, 223–272 (1981).

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965);J. W. Cooley, P. A. W. Lewis, P. D. Welch, “Historical notes on the FFT,” IEEE Trans. Audio Electroacoust.AU-15, 76–79 (1967).
[Crossref]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Rev. Mex. Fi´s. (1)

N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fı́s. 40, 366–377 (1994).

Theor. Math. Phys. (1)

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991).
[Crossref]

Other (9)

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

R. Askey, N. M. Atakishiyev, S. K. Suslov, “Fourier transformations for difference analogs of the harmonic oscillator,” in Proceedings of the XV Workshop on Problems on High Energy Physics and Field Theory, A. P. Samokhin, G. L. Rcheulishvili, eds. (Institute for High Energy Physics, Protvino, Russia, 1995), pp. 140–144.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).

N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, R.I., 1968).

A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[Crossref]

K. B. Wolf, Integral Transforms in Science and Engineering, Vol. 11 of Mathematical Concepts and Methods in Science and Engineering Series, A. Miele, ed. (Plenum, New York, 1979).

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981).

P. Feinsilver, R. Schott, Algebraic Structures and Operator Calculus. Vol. I: Representations and Probability Theory (Kluwer, Dordrecht, The Netherlands, 1993).

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Figures (3)

Fig. 1
Fig. 1

Planar multimodal waveguide model with sensor points. The guide can carry only a finite number of modes; the waveforms are sampled at the same number of equidistant sensor points (five in this figure).

Fig. 2
Fig. 2

Kravchuk functions h-1/2ϕn(ξ, N) for n=0, 1, 2, 3, 4, and 8, for the sensor spacing h=2/N. Each figure plots the values of N{1, 2, 4, 8, 16, , } compatible with n. There are N+1 sensor points, spaced by h=2/N between -N/2 and N/2. The end-point zeros occur one h outside this interval. The figures show that the N limit of the Kravchuk functions as the number and density of sensors increases is the (infinite) set of harmonic oscillator wave functions.

Fig. 3
Fig. 3

The Fourier transform is a rotation by 12π generated by the oscillator Hamiltonian. It transforms the position operator ξ into the momentum operator p^ξ=-iξ in the quantum mechanical case (plane, top); both operators have spectrum ℜ, and each generates (noncommuting) translations of phase space in L2(R). The finite Fourier–Kravchuk transform similarly rotates the position s onto the momentum Ps(N) operators, whose spectrum is finite and equally spaced. The latter two close with the Hamiltonian into the Lie algebra of the rotation group.

Equations (95)

Equations on this page are rendered with MathJax. Learn more.

κ0, κ1, , κN.
ξn=(n-N/2)h,n=0, 1, , N.
f0=f(ξ0),f1=f(ξ1), , fN=f(ξN).
n(ξ, η)=no-(ν12ξ2+ν22η2)+ ,
Hˆϕ := 12 -2ξ2-2η2+ν12ξ2+ν22η2ϕ=i ζ ϕ.
[Hˆ, [Hˆ, ξ]]=ν12ξ,[Hˆ, [Hˆ, η]]=ν22η,
ψn(ξ)=1π2nn! Hn(ξ)exp(-ξ2/2),
ξ=mω x,n=0, 1, 2, ,
(ψn, ψm)R=-dξψn(ξ)*ψm(ξ)=δn,m,
f(ξ)=n=0cnψn(ξ).
f(ξj)=n=0Ncn(N)ψn(ξj),
f(ξ0)f(ξ1)f(ξN)=ψ0(ξ0)ψ0(ξ1)ψ0(ξN)ψ1(ξ0)ψ1(ξ1)ψ1(ξN)ψN(ξ0)ψN(ξ1)ψN(ξN)c0(N)c1(N)cN(N),
j=0Nϕn(sj, N)ϕn(sj, N)=δn,n,
sj=j-N/2,j=0, 1, , N.
ϕn(s, N)=2n-N/2kn(s+N/2, N)×n!(N-n)!Γ(N/2+s+1)Γ(N/2-s+1)1/2.
f(ξj)=(N/2)1/4n=0Nκn(N)ϕn(N/2ξj, N),
j=0, 1, , N.
κn(N)=(N/2)-1/4j=0Nϕn(N/2ξj, N)f(ξj).
fA(ξ, N)=(N/2)1/4n=0Nκn(N)ϕn(N/2ξ, N).
H(N)(s)ϕn(s, N)=(n+12)ϕn(s, N),
n=0, 1, , N,
[s, H(N)(s)]=12[α(-s)exp(s)-α(s)exp(-s)]=iPs(N),
[H(N)(s), Ps(N)]=is.
H(N)(s)=-12 [α(s)exp(-s)+α(-s)exp(s)]+12 (N+1),
α(s)=[(N/2+s)(N/2-s+1)]1/2.
[H(N1)(s1)+H(N2)(s2)]ϕ(s1, s2)=(n1+n2+1)ϕ(s1, s2)
x=ƛξ,ξ=h1s1=2/N1s1,
y=ƛη,η=h2s2=2/N2s2.
ϕn(s1, s2; N)=2n-N/2kn(s1+N/2, N)×n!(N-n)! sin πs2πs2Γ(N/2+s1+1)Γ(N/2-s1+1)1/2
H(N)(s1, s2)=H(N)(s1)+H(0)(s2),
exp[iζH(N)(s1, s2)]ϕn(s1, s2; N)=exp[iζ(n+1)]ϕn(s1, s2; N),
f(x)f˜(x)=(Ff )(x)=12π Rdx exp(-ixx)f(x).
Fψn(x)=exp(-iπn/2)ψn(x).
Ccos t-sin tsin tcos tf(x)=exp-it2 -d2dx2+x2f(x)=-dxCF(x, x; t)f(x),
CF(x, x; t)=exp(-i4 π sgn sin t)2π|sin t|×expi2 sin t (x2 cos t-2xx+x2 cos t)=n=0ψn(x)exp[-i(n+1/2)t]ψn(x).
C0-110=exp(-iπ/4)F.
-dxCF(x, x; t1)CF(x, x; t2)=σ(t1, t2; t1+t2)CF(x, x; t1+t2),
σ(t1, t2; t1+t2)=1-1,sgn sin(t1+t2)+sgn sin t1+sgn sin t2=positivenegative.
F^αϕn(s, N)=exp(iαπ/4)exp-iπ2 αH(N)(s)ϕn(s, N)=exp(-iαπn/2)ϕn(s, N),
F^αf(ξn)=n=0NFn,nαf(ξn)=(N/2)1/4n=0Nκn(N)F^αϕn(hξn, N)=n=0Nj=0Nϕn(hξj, N)f(ξj)×exp(-iαπn/2)ϕn(hξn, N),
Fn,nα=j=0Nϕj(hξn, N)exp(-ijαπ/2)ϕj(hξn, N)
=exp[i(π/2)(n+n-Nα/2)]×CNnCNn cosN(πα/4)tann+n×(πα/4)2F1[-n, -n;-N;sin-2(πα/4)]
=exp[i(π/2)(n-n-Nα/2)]×n!(N-n)!n!(N-n)! sinn-n(πα/4)×cosN-n-n(πα/4)kn[sin2(πα/4)](n, N).
Fα1Fα2f(ξj)=Fα2n=0NFj,nα1f(ξn)=n=0NFj,nα1n=0NFn,nα2f(ξn)=n,n=0NFj,nα1Fn,nα2f(ξn)=Fα1+α2f(ξj).
Fn,nα=1=exp[i(π/2)(n+n-N/2)]×2-NCNnCNn2F1(-n, -n; -N; 2)=exp[i(π/2)(n-n-N/2)]ϕn(ξn, N).
dm,ml(β)=lm|exp(-iβJ^2)|lm=Dm,ml(0, β, 0)=(-1)m-mdn-1ϱ(s)kn(p)(s, N),
dm,ml(π/2)=(-1)m-m2m (l+m)!(l-m)!(l+m)!(l-m)!×kl-m(l-m, N)=dm,ml(-π/2)=(-1)j-mϕj-m(m, N)=(-1)m-mϕj-m(-m, N).
Dm,ml(α, β, γ)=lm|exp(-iαJ^3)×exp(-iβJ^2)exp(-iγJ^3)|lm=exp(-imα)dm,ml(β)exp(-imγ)
Dm,ml(α, 0, 0)=lm|exp(-iαJ^3)|lm=exp(-imα)δm,m.
exp(ilαπ/2)Fn,nα=m,m=-llDl-n,ml(0, π/2, 0)Dm,ml×(-απ/2, 0, 0)Dm,l-nl(0, -π/2, 0)=l, l-n|exp[-i(π/2)J^2]×exp[i(π/2)αJ^3]exp[i(π/2)J^2]|l, l-n
=l, l-n|exp[i(π/2)αJ^1]|l, l-n
=k=0Nϕk(sn)exp[i(l-k)απ/2]ϕk(sn),
sn=n-l.
limNN/2Fn,nα=exp(iπα/4)C(ξ, ξ, πα/2).
F^αϕn(s1, s2; N)=exp(iαπ/2)exp-iπ2 αH(N)(s1, s2)ϕn(s1, s2; N)=exp(-iαπn/2)ϕn(s1, s2; N),
[s, iPs(N)]=-12 [α(s)exp(-s)+α(-s)exp(s)]=H(N)(s)-12 (N+1).
J1=s,
J2=-Ps(N),
J3=H(N)(s)-12 (N+1).
[J1, J2]=iJ3,[J2, J3]=iJ1,[J3 J1]=iJ2,
J2=J12+J22+J32=l(l+1),l=N/2.
(F^αf)j=j=02ll, l-j|exp[i(π/2)αJ^1]|l, l-jfj,
l=N/2.
Ej,k=1M exp(-2πijk/M).
Eα=14 j=14 exp[5iπ(α-j)/4] sin π(α-j)sin π(α-j)/4 Ej.
E=exp(iπH/2),
H=52 1+12 (1+i)E+12 E2+12 (1-i)E3.
j=0Nϱ(j)km(j, N)kn(j, N)=dn2δm,n,
ϱ(ξ)=12N Nξ=N!2NΓ(ξ+1)Γ(N-ξ+1),
dn2=122n Nn=N!22nn!(N-n)!.
[ξ-n-12(N-2n)]kn(ξ, N)
=(n+1)kn+1(ξ, N)+14(N-n+1)kn-1(ξ, N)
kn(ξ, N)=(-1)n2n NnF-n, -ξ-N;2.
(-1)n2n Nnkn(ξ, N)=(-1)ξ2ξ Nξkξ(n, N).
limN(12N)-n/2kn(12N+12Nξ, N)=12nn! Hn(ξ),
limN12Nϱ(12N+12Nξ)=1π exp(-ξ2),
j=0Nϕm(ξj, N)ϕn(ξj, N)=δm,n,
ϕn(ξ, N)=dn-1kn(12N+ξ, N)ϱ(12N+ξ),
0nN,-12Nξ12N.
-1-12Nξ12N+1,
limN(12N)1/4ϕn12Nξ, N=ψn(ξ).
(12N-n)ϕn(ξ, N)=12[α(ξ)ϕn(ξ-1, N)+α(ξ+1)ϕn(ξ+1, N)],
α(ξ)=[(12N+ξ)(12N-ξ+1)]1/2.
{-12[α(ξ)exp(-ξ)+α(ξ+1)exp(ξ)]+12(N+1)}ϕn(ξ, N)=(n+12)ϕn(ξ, N),
(n+1)k˜n+1(ξ, N)=ξk˜n(ξ, N)-14(N-n+1)k˜n-1(ξ, N).
k˜0(ξ, N)=1,k˜1(ξ, N)=ξ,
k˜2(ξ, N)=12[ξ2-14N],
k˜3(ξ, N)=13![ξ3-14(3N-2)ξ],
k˜4(ξ,N)=14![ξ4-12(3N-4)ξ2+316N(N-2)],
k˜5(ξ, N)=15![ξ5-52(N-2)ξ3+116(15N2-50N+24)ξ],
k˜6(ξ, N)=16![ξ6-54(3N-8)ξ4+116(45N2-210N+184)ξ2-1564N(N2-6N+8)],
k˜7(ξ, N)=17![ξ7-74(3N-10)ξ5+716(15N2-90N+112)ξ3-364(35N3-280N2+588N-240)ξ],
k˜8(ξ, N)=18![ξ8-7(N-4)ξ6+78(15N2-110N+176)ξ4-116(105N3-1050N2+2968N-2112)ξ2+105256N(N3-12N2+44N-48)],
k˜9(ξ, N)=19![ξ9-3(3N-14)ξ7+218(9N2-78N+152)ξ5-116(315N3-3780N2+13356N-13088)ξ3+9256(105N4-1540N3+7308N2-12176N+4480)ξ],
k˜10(ξ, N)=110![ξ10-154(3N-16)ξ8+218(15N2-150N+344)ξ6-532(315N3-4410N2+18648N-22976)ξ4+9256(525N4-9100N3+52780N2-115600N+72064)ξ2-9451024N(N4-20N3+140N2-400N+384)].

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