M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).

[CrossRef]

K. B. Möller, T. G. Jörgensen, G. Torres-Vega, “On coherent-state representations of quantum mechanics: wave mechanics in phase space,” J. Chem. Phys. 106, 7228–7240 (1997).

[CrossRef]

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B 38, 209–219 (1996).

[CrossRef]

In quantum optics and in the study of the quantum harmonic oscillator, there is a natural scaling factor between the position and the momentum axes. A Gaussian wave function that preserves its width is known as a coherent state. If, on the other hand, the width of the Gaussian changes in time, “breathing” in a periodic fashion, the wave function corresponds to a squeezed state. In the particular case when the WDF of a coherent state is used in the blurring of W(x, p), the resulting Husimi function is known as the Q function. See, for example, H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).

[CrossRef]

J. E. Harriman, “A quantum state vector phase space representation,” J. Chem. Phys. 100, 3651–3661 (1994).

[CrossRef]

G. Torres-Vega, J. H. Frederick, “Quantum mechanics in phase space: new approaches to the correspondence principle,” J. Chem. Phys. 93, 8862–8874 (1990).

[CrossRef]

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219–231 (1989).

[CrossRef]

One such method, based on the Bargmann representation defined in Ref. 17, is given in A. Voros, “Wentzel–Kramers–Brillouin method in the Bargmann representation,” Phys. Rev. A 40, 6814–6825 (1989).

[CrossRef]
[PubMed]

Fields of this form, corresponding to energy eigenstates of quantum-mechanical bound systems, were considered by Berry in Ref. 4. There he recognized what is referred to here as the ghost curves. A generalization of Berry’s work for the case of two-dimensional bound states is given inA. M. Ozorio de Almeida, J. H. Hannay, “Geometry of two dimensional tori in phase space: projections, sections and the Wigner function,” Ann. Phys. (N.Y.) 138, 115–154 (1982).

[CrossRef]

H. J. Korsch, M. V. Berry, “Evolution of Wigner’s phase space density under a nonintegrable quantum map,” Physica D 3D, 627–636 (1981).

[CrossRef]

When the phase of the wave field is unimportant, the projection property in Eq. (2.3a) can be used instead of the problematic inversion formula in Eq. (2.5). In this case, a useful property regarding the projection of Wigner functions containing Airy forms is given in Ref. 4 and is generalized for higher-order catastrophes inM. V. Berry, F. J. Wright, “Phase space projection identities for diffraction catastrophes,” J. Phys. A 13, 149–160 (1980).

[CrossRef]

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).

[CrossRef]

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).

[CrossRef]

M. V. Berry, “Semi-classical mechanics in phase space: a study of Wigner’s function,” Philos. Trans. R. Soc. London 287, 237–271 (1977).

[CrossRef]

A. Voros, “Semiclassical approximations,” Ann. Inst. Henri Poincare 14, 31–90 (1976).

In fact, Eq. (3.12) can easily be written in terms of an Airy function, such that the matching with Eq. (3.9) is explicit. See, for example, M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog. (Oxford) 57, 43–64 (1969). However, the form used in Eq. (3.12) is more convenient for the purposes of this paper.

A phase-space representation that is closely related to the GWFT was proposed in V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961). The connection between this transformation and the GWFT is discussed in Ref. 15.

[CrossRef]

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).

[CrossRef]

To our knowledge, the earliest reference to the FrFT is E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).

[CrossRef]
[PubMed]

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).

[CrossRef]

In fact, Airy defined the function that bears his name within this context. See G. B. Airy, “On the intensity of light in a neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–403 (1838).

This result follows from
∫ exp[i(xt+at3/3)]dt=2π|a|-1/3Ai[xa-1/3]. In the argument to the Airy function, the third power is always chosen to be the real root. For a discussion of Airy functions, see, e.g., M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Sec. 10.4.

In fact, Airy defined the function that bears his name within this context. See G. B. Airy, “On the intensity of light in a neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–403 (1838).

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).

[CrossRef]

G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A (to be published).

M. A. Alonso, G. W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” submitted to J. Opt. Soc. Am. A (to be published).

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).

[CrossRef]

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).

[CrossRef]

A phase-space representation that is closely related to the GWFT was proposed in V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961). The connection between this transformation and the GWFT is discussed in Ref. 15.

[CrossRef]

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219–231 (1989).

[CrossRef]

H. J. Korsch, M. V. Berry, “Evolution of Wigner’s phase space density under a nonintegrable quantum map,” Physica D 3D, 627–636 (1981).

[CrossRef]

When the phase of the wave field is unimportant, the projection property in Eq. (2.3a) can be used instead of the problematic inversion formula in Eq. (2.5). In this case, a useful property regarding the projection of Wigner functions containing Airy forms is given in Ref. 4 and is generalized for higher-order catastrophes inM. V. Berry, F. J. Wright, “Phase space projection identities for diffraction catastrophes,” J. Phys. A 13, 149–160 (1980).

[CrossRef]

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).

[CrossRef]

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).

[CrossRef]

M. V. Berry, “Semi-classical mechanics in phase space: a study of Wigner’s function,” Philos. Trans. R. Soc. London 287, 237–271 (1977).

[CrossRef]

In fact, Eq. (3.12) can easily be written in terms of an Airy function, such that the matching with Eq. (3.9) is explicit. See, for example, M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog. (Oxford) 57, 43–64 (1969). However, the form used in Eq. (3.12) is more convenient for the purposes of this paper.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986) pp. 252–320.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).

[CrossRef]

L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995), pp. 93–112.

To our knowledge, the earliest reference to the FrFT is E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).

[CrossRef]
[PubMed]

N. G. De Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).

[CrossRef]

G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A (to be published).

M. A. Alonso, G. W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” submitted to J. Opt. Soc. Am. A (to be published).

G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993).

[CrossRef]

G. Torres-Vega, J. H. Frederick, “Quantum mechanics in phase space: new approaches to the correspondence principle,” J. Chem. Phys. 93, 8862–8874 (1990).

[CrossRef]

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).

[CrossRef]

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986) pp. 252–320.

Fields of this form, corresponding to energy eigenstates of quantum-mechanical bound systems, were considered by Berry in Ref. 4. There he recognized what is referred to here as the ghost curves. A generalization of Berry’s work for the case of two-dimensional bound states is given inA. M. Ozorio de Almeida, J. H. Hannay, “Geometry of two dimensional tori in phase space: projections, sections and the Wigner function,” Ann. Phys. (N.Y.) 138, 115–154 (1982).

[CrossRef]

J. E. Harriman, “A quantum state vector phase space representation,” J. Chem. Phys. 100, 3651–3661 (1994).

[CrossRef]

K. B. Möller, T. G. Jörgensen, G. Torres-Vega, “On coherent-state representations of quantum mechanics: wave mechanics in phase space,” J. Chem. Phys. 106, 7228–7240 (1997).

[CrossRef]

This distribution is sometimes referred to as the coherent state representation. For an excellent overview and collection of key references, see J. R. Klauder, B. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).

H. J. Korsch, M. V. Berry, “Evolution of Wigner’s phase space density under a nonintegrable quantum map,” Physica D 3D, 627–636 (1981).

[CrossRef]

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993).

This function is a solution of the Hamilton–Jacobi (or eikonal) equation. See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 3–9.

A general overview on this subject is given by H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to thefractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

In quantum optics and in the study of the quantum harmonic oscillator, there is a natural scaling factor between the position and the momentum axes. A Gaussian wave function that preserves its width is known as a coherent state. If, on the other hand, the width of the Gaussian changes in time, “breathing” in a periodic fashion, the wave function corresponds to a squeezed state. In the particular case when the WDF of a coherent state is used in the blurring of W(x, p), the resulting Husimi function is known as the Q function. See, for example, H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).

[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.

A general overview on this subject is given by H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to thefractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

K. B. Möller, T. G. Jörgensen, G. Torres-Vega, “On coherent-state representations of quantum mechanics: wave mechanics in phase space,” J. Chem. Phys. 106, 7228–7240 (1997).

[CrossRef]

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B 38, 209–219 (1996).

[CrossRef]

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993).

This function is a solution of the Hamilton–Jacobi (or eikonal) equation. See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 3–9.

A general overview on this subject is given by H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to thefractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

Fields of this form, corresponding to energy eigenstates of quantum-mechanical bound systems, were considered by Berry in Ref. 4. There he recognized what is referred to here as the ghost curves. A generalization of Berry’s work for the case of two-dimensional bound states is given inA. M. Ozorio de Almeida, J. H. Hannay, “Geometry of two dimensional tori in phase space: projections, sections and the Wigner function,” Ann. Phys. (N.Y.) 138, 115–154 (1982).

[CrossRef]

J. F. Price, “Uncertainty principles and sampling theorems,” in Fourier Techniques and Applications, J. F. Price, ed. (Plenum, New York, 1985), pp. 25–44.

This distribution is sometimes referred to as the coherent state representation. For an excellent overview and collection of key references, see J. R. Klauder, B. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).

This result follows from
∫ exp[i(xt+at3/3)]dt=2π|a|-1/3Ai[xa-1/3]. In the argument to the Airy function, the third power is always chosen to be the real root. For a discussion of Airy functions, see, e.g., M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Sec. 10.4.

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).

[CrossRef]

K. B. Möller, T. G. Jörgensen, G. Torres-Vega, “On coherent-state representations of quantum mechanics: wave mechanics in phase space,” J. Chem. Phys. 106, 7228–7240 (1997).

[CrossRef]

G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993).

[CrossRef]

G. Torres-Vega, J. H. Frederick, “Quantum mechanics in phase space: new approaches to the correspondence principle,” J. Chem. Phys. 93, 8862–8874 (1990).

[CrossRef]

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).

[CrossRef]

One such method, based on the Bargmann representation defined in Ref. 17, is given in A. Voros, “Wentzel–Kramers–Brillouin method in the Bargmann representation,” Phys. Rev. A 40, 6814–6825 (1989).

[CrossRef]
[PubMed]

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).

[CrossRef]

A. Voros, “Semiclassical approximations,” Ann. Inst. Henri Poincare 14, 31–90 (1976).

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).

[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.

When the phase of the wave field is unimportant, the projection property in Eq. (2.3a) can be used instead of the problematic inversion formula in Eq. (2.5). In this case, a useful property regarding the projection of Wigner functions containing Airy forms is given in Ref. 4 and is generalized for higher-order catastrophes inM. V. Berry, F. J. Wright, “Phase space projection identities for diffraction catastrophes,” J. Phys. A 13, 149–160 (1980).

[CrossRef]

A. Voros, “Semiclassical approximations,” Ann. Inst. Henri Poincare 14, 31–90 (1976).

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros, “Quantum maps,” Ann. Phys. (N.Y.) 122, 26–63 (1979).

[CrossRef]

Fields of this form, corresponding to energy eigenstates of quantum-mechanical bound systems, were considered by Berry in Ref. 4. There he recognized what is referred to here as the ghost curves. A generalization of Berry’s work for the case of two-dimensional bound states is given inA. M. Ozorio de Almeida, J. H. Hannay, “Geometry of two dimensional tori in phase space: projections, sections and the Wigner function,” Ann. Phys. (N.Y.) 138, 115–154 (1982).

[CrossRef]

A phase-space representation that is closely related to the GWFT was proposed in V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961). The connection between this transformation and the GWFT is discussed in Ref. 15.

[CrossRef]

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B 38, 209–219 (1996).

[CrossRef]

G. Torres-Vega, J. H. Frederick, “Quantum mechanics in phase space: new approaches to the correspondence principle,” J. Chem. Phys. 93, 8862–8874 (1990).

[CrossRef]

G. Torres-Vega, J. H. Frederick, “A quantum mechanical representation in phase space,” J. Chem. Phys. 98, 3103–3120 (1993).

[CrossRef]

J. E. Harriman, “A quantum state vector phase space representation,” J. Chem. Phys. 100, 3651–3661 (1994).

[CrossRef]

K. B. Möller, T. G. Jörgensen, G. Torres-Vega, “On coherent-state representations of quantum mechanics: wave mechanics in phase space,” J. Chem. Phys. 106, 7228–7240 (1997).

[CrossRef]

M. A. Alonso, G. W. Forbes, “New approach to semiclassical analysis in mechanics,” J. Math. Phys. 40, 1699–1718 (1999).

[CrossRef]

When the phase of the wave field is unimportant, the projection property in Eq. (2.3a) can be used instead of the problematic inversion formula in Eq. (2.5). In this case, a useful property regarding the projection of Wigner functions containing Airy forms is given in Ref. 4 and is generalized for higher-order catastrophes inM. V. Berry, F. J. Wright, “Phase space projection identities for diffraction catastrophes,” J. Phys. A 13, 149–160 (1980).

[CrossRef]

M. V. Berry, N. L. Balasz, “Evolution of semiclassical quantum states in phase space,” J. Phys. A 12, 625–642 (1979).

[CrossRef]

M. V. Berry, “Semi-classical mechanics in phase space: a study of Wigner’s function,” Philos. Trans. R. Soc. London 287, 237–271 (1977).

[CrossRef]

In quantum optics and in the study of the quantum harmonic oscillator, there is a natural scaling factor between the position and the momentum axes. A Gaussian wave function that preserves its width is known as a coherent state. If, on the other hand, the width of the Gaussian changes in time, “breathing” in a periodic fashion, the wave function corresponds to a squeezed state. In the particular case when the WDF of a coherent state is used in the blurring of W(x, p), the resulting Husimi function is known as the Q function. See, for example, H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).

[CrossRef]

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).

[CrossRef]

One such method, based on the Bargmann representation defined in Ref. 17, is given in A. Voros, “Wentzel–Kramers–Brillouin method in the Bargmann representation,” Phys. Rev. A 40, 6814–6825 (1989).

[CrossRef]
[PubMed]

H. J. Korsch, M. V. Berry, “Evolution of Wigner’s phase space density under a nonintegrable quantum map,” Physica D 3D, 627–636 (1981).

[CrossRef]

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).

[CrossRef]

To our knowledge, the earliest reference to the FrFT is E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).

[CrossRef]
[PubMed]

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219–231 (1989).

[CrossRef]

In fact, Eq. (3.12) can easily be written in terms of an Airy function, such that the matching with Eq. (3.9) is explicit. See, for example, M. V. Berry, “Uniform approximation: a new concept in wave theory,” Sci. Prog. (Oxford) 57, 43–64 (1969). However, the form used in Eq. (3.12) is more convenient for the purposes of this paper.

In fact, Airy defined the function that bears his name within this context. See G. B. Airy, “On the intensity of light in a neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–403 (1838).

The evaluation at (x, p)=(x¯+η, p¯+δ) of the exponent of the integrand in Eq. (3.1) gives ϕ(x+12x′)-ϕ(x-12x′)-px′≈Φ+[ϕ′(x2)-ϕ′(x1)]δ-(x2-x1)η-(δ-ϕ″η)s+124ϕ″′¯s3+O(s5),
where ϕ″=ϕ″(x1)=ϕ″(x2). The contribution to Wis then W(x¯+η, p¯+δ)≈8k2ϕ′′′¯1/3a(x1)a(x2)Ai-8k2ϕ′′′¯1/3sgn[ϕ′′′¯](δ-ϕ″η)×2 cos(k{Φ+[ϕ′(x2)-ϕ′(x1)]η-(x2-x1)δ}).
[Notice that these two expressions reduce to Eqs. (3.13) and (3.14) when η=0.] The argument of the cosine factor stays constant at all points that fall along a line parallel to the secant from x1 to x2.

The Pearcey function is defined as Ip(x, y)=∫ exp[i(xt+yt2+t4)]dt.See, for example, Ref. 27, pp. 60–61.

A general overview on this subject is given by H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to thefractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A (to be published).

M. A. Alonso, G. W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” submitted to J. Opt. Soc. Am. A (to be published).

See, for example, the reference in Note 7, pp. 22–26.

Y. A. Kravtsov, Y. A. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1993).

This result follows from
∫ exp[i(xt+at3/3)]dt=2π|a|-1/3Ai[xa-1/3]. In the argument to the Airy function, the third power is always chosen to be the real root. For a discussion of Airy functions, see, e.g., M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Sec. 10.4.

If we use only the real phase space, the geometric prescription gives us only the real-valued saddle points. However, through the expressions involving Airy functions, these are enough to account for those that have small imaginary parts; those with large imaginary parts turn out to be insignificant.

Note that, as x1 approaches x2, the relation p¯=ϕ′(x¯)+δ=12{ϕ′[x¯+(x2-x1)/2]+ϕ′[x¯-(x2-x1)/2]} leads to (x2-x1)2≈8δ/ϕ′′′(x¯). With this, the connection becomes straightforward since the argument of the cosine in Eq. (3.12) can be written as ϕ[x¯+(x2-x1)/2]-ϕ[x¯-(x2-x1)/2]-(x2-x1)p¯≈-23σ[8δ3/ϕ′′′(x¯)]1/2.

This function is a solution of the Hamilton–Jacobi (or eikonal) equation. See, for example, Y. A. Kravtsov, Y. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 3–9.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.

J. F. Price, “Uncertainty principles and sampling theorems,” in Fourier Techniques and Applications, J. F. Price, ed. (Plenum, New York, 1985), pp. 25–44.

N. G. De Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

The analogous result to Eq. (2.5) for momentum representation also has this property: f˜(p)=1f˜*(p0)∫Wx,p+p02×exp[-ikx(p-p0)]dx.

L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995), pp. 93–112.

This distribution is sometimes referred to as the coherent state representation. For an excellent overview and collection of key references, see J. R. Klauder, B. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).

See Ref. 11, Chap. 7, pp. 93–100.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986) pp. 252–320.