176
K. B. Wolf et al.
6 Concluding Remarks
We have here abstracted some of the results in Ref. [7] to exhibit the Zernike system
as a projection of the evidently highly symmetrical system of free motion on a conic.
In the case of the 3-sphere as homogeneous space for so(4), we have the benefit of
additional Lie-theoretical properties, such as Schrödinger equations with potentials
of Pöschl–Teller type.
The algebra so(4) and its special property of splitting (3) also serves for
finite and discrete image analysis, between Cartesian- and polar-pixellated arrays
[16]. In polar pixellation, the normal modes factorize and the radial functions are
also Clebsch–Gordan coefficients, although of the more general type (16) rather
than (25). Under various guises, the so(4) is an algebra that may contain other
physical or optical systems in their various realizations.
In this report we have used the Schrödinger representation of the Zernike
“wavefunctions” Ψ
I
n,m (x, y) and Ψ
II
m 1 ,m 2
(x, y) in their separated bases. It is then
natural to label kets |n, m I and |m 1 , m 2 I I as a short and equally good realization
for the states of the system, and useful for computations. Following common
Dirac notation, we could bind the two realizations through stating Ψ
I
n,m (x, y) =
(x, y|n, m I and Ψ
II
m 1 ,m 2
(x, y) = (x, y|m 1 , m 2 I I , provided a proper definition exists
for a Dirac basis {|x, y)} x,y∈D over a finite disk. This subject has been addressed in a
recent paper by Celeghini et al. [17], through the construction of a Hilbert space on
a closed subset of the R
2 plane. We have been accustomed to use Hilbert spaces
and Gel’fand triplets for functions over the whole plane R
2 ; the Zernike model
necessitates also function-theoretic analyses. Finally, we are aware that similar
constructions and projections can also be done with planes and hyperbolas—not
only spheres, and that a Lie algebra can project out a superintegrable Higgs algebra
[18, 19].
Acknowledgments N.M.A. and K.B.W. thanks project AG-100119 awarded by the Dirección
General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México. A.Y.
thanks the support of project PRO-SNI-2019 (Universidad de Guadalajara).
References
1. F. Zernike, Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form der
Phasenkontrastmethode. Physica 1, 689–704 (1934)
2. F. Zernike, H.C. Brinkman, Hypersphärische Funktionen und die in sphärischen Bereichen
orthogonalen Polynome. Verh. Akad. Wet. Amst. (Proc. Sec. Sci.) 38, 161–170 (1935)
3. G.S. Pogosyan, K.B. Wolf, A. Yakhno, Superintegrable classical Zernike system. J. Math.
Phys. 58, 072901 (2017)
4. G.S. Pogosyan, C. Salto-Alegre, K.B. Wolf, A. Yakhno A, Quantum superintegrable Zernike
system. J. Math. Phys. 58, 072101 (2017)
5. G.S. Pogosyan, K.B. Wolf, A. Yakhno, New separated polynomial solutions to the Zernike
system on the unit disk and interbasis expansion. J. Opt. Soc. Am. A 34, 1844–1848 (2017)
K. B. Wolf et al.
6 Concluding Remarks
We have here abstracted some of the results in Ref. [7] to exhibit the Zernike system
as a projection of the evidently highly symmetrical system of free motion on a conic.
In the case of the 3-sphere as homogeneous space for so(4), we have the benefit of
additional Lie-theoretical properties, such as Schrödinger equations with potentials
of Pöschl–Teller type.
The algebra so(4) and its special property of splitting (3) also serves for
finite and discrete image analysis, between Cartesian- and polar-pixellated arrays
[16]. In polar pixellation, the normal modes factorize and the radial functions are
also Clebsch–Gordan coefficients, although of the more general type (16) rather
than (25). Under various guises, the so(4) is an algebra that may contain other
physical or optical systems in their various realizations.
In this report we have used the Schrödinger representation of the Zernike
“wavefunctions” Ψ
I
n,m (x, y) and Ψ
II
m 1 ,m 2
(x, y) in their separated bases. It is then
natural to label kets |n, m I and |m 1 , m 2 I I as a short and equally good realization
for the states of the system, and useful for computations. Following common
Dirac notation, we could bind the two realizations through stating Ψ
I
n,m (x, y) =
(x, y|n, m I and Ψ
II
m 1 ,m 2
(x, y) = (x, y|m 1 , m 2 I I , provided a proper definition exists
for a Dirac basis {|x, y)} x,y∈D over a finite disk. This subject has been addressed in a
recent paper by Celeghini et al. [17], through the construction of a Hilbert space on
a closed subset of the R
2 plane. We have been accustomed to use Hilbert spaces
and Gel’fand triplets for functions over the whole plane R
2 ; the Zernike model
necessitates also function-theoretic analyses. Finally, we are aware that similar
constructions and projections can also be done with planes and hyperbolas—not
only spheres, and that a Lie algebra can project out a superintegrable Higgs algebra
[18, 19].
Acknowledgments N.M.A. and K.B.W. thanks project AG-100119 awarded by the Dirección
General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México. A.Y.
thanks the support of project PRO-SNI-2019 (Universidad de Guadalajara).
References
1. F. Zernike, Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form der
Phasenkontrastmethode. Physica 1, 689–704 (1934)
2. F. Zernike, H.C. Brinkman, Hypersphärische Funktionen und die in sphärischen Bereichen
orthogonalen Polynome. Verh. Akad. Wet. Amst. (Proc. Sec. Sci.) 38, 161–170 (1935)
3. G.S. Pogosyan, K.B. Wolf, A. Yakhno, Superintegrable classical Zernike system. J. Math.
Phys. 58, 072901 (2017)
4. G.S. Pogosyan, C. Salto-Alegre, K.B. Wolf, A. Yakhno A, Quantum superintegrable Zernike
system. J. Math. Phys. 58, 072101 (2017)
5. G.S. Pogosyan, K.B. Wolf, A. Yakhno, New separated polynomial solutions to the Zernike
system on the unit disk and interbasis expansion. J. Opt. Soc. Am. A 34, 1844–1848 (2017)