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Finite-difference approximation for inverse scattering problem
REPORTS
Vol. 18 (1980)
FINITE-DIFFERENCE
ON MATHEMATICAL
APPROXIMATION PROBLEM
V. N. MELNIKOV
No. 3
PHYSICS
FOR INVERSE
SCATTERING
and B. N. ZAKHARIEV
Joint Institute for Nuclear Research, Moscow, U.S.S.R. (Received January
22, 1979)
For the first time potentials are reconstructed in a finite-difference approximation using a genuine inverse scattering method instead of multiple repeated solutions of a direct problem with iterative fitting of scattering data. Up to now a fundamental difference between spectral properties of the Schriidinger operator and its discrete analog hindered from doing this.
I. Introduction At present there is a deep asymmetry in the development of quantum theory of direct and inverse problems (IP). In contrast to numerous effective algorithms for solving the SchrGdinger equation, there are only a few examples of potential r,econstruction from scattering data by genuine IP methods [l]. Up to now IP has mainly been solved by using the direct problem [2]. Most methods for solving the direct problem are based on the finite-difference (fd) approximation to differential equations of motion. But there are no such methods for the IP. Although the IP can be solved in the framework of fd formalism itself [3-Q no algorithms of fd approximate reconstruction of potentials for real SchrGdinger equation: -+Y”(r)
+ V(r)Y(r) = EY(r)
(1)
with continuous coordinate dependence have been suggested until now. The point is that the solutions of eq. (1) and its fd analog: - [Y(n + 1) - 2Y(n) + Y(n - 1)-J/242 + v(n)Y(n) = EY(n)
(2)
are close to one another only at comparatively low energies, and in the IP solution the whole spectrum is involved (in the derivation of IP equations the completeness relation for eigenfunctions of the SchrCidinger operator is of fundamental importance). This contradiction cannot be removed by diminishing the mesh A of fd differentiation. Although the energy interval with similar properties to (1) and (2) increases with A- ‘, 13531
V. N. MELNIKOV
354
and B. N. ZAKHARIEV
its relative weight in the whole spectrum does not increase (the upper bound of fd spectrum increases with A- ‘). The way out of this situation can be found due to the fact that the upper bound of a “good” spectral part (where the solutions of (1) and (2) are similar) rises to the asymptotic region where perturbation theory is valid. It is also important for IP that the first order perturbation corrections depend mainly on the mean value Pof potential V(r). But, unlike in the continuous case, where perturbation theory works better with increasing energy, in the fd model the highest states are as sensitive to the interaction as the lowest ones. This difficulty can be avoided by using the property of weakly violated symmetry for scattering parameters with respect to the midpoint of the fd’spectrum. In the next section we briefly recall how to solve the IP for (2) in order to use this fd model in sect. III as a basis for our approximate method. II. Exact fd model of the inverse scattering problem [5] For a finite range interaction V(r > a) = 0 it is convenient to use the R-matrix formalism where we have a discrete set of scattering parameters (resonance positions E, and reduced widths y,‘). In the fd approach the R-matrix (or the scattering matrix S(E)) is determined even by a finite set {E,,yv}fd, v = 1,2,..., N = a/A:
R(E)= where E, are the eigenvalues
&E, - E)
corresponding
to (2) with the boundary
conditions:
Y&N + 1) - Y&V - 1) = BY”(N) = B~J%.
Y,(O) = 0,
Here B is an arbitrary constant, and eigenfunctions orthonormalized and complete set i
(3)
3
a, Y&)Y,(n)
(4)
Y,(n) = Y(E = EV,n) form an
= 6,,/A; c+ = l/2; CX”<~= 1;
n=l
N
an.;, yv(4yvb4 = L/A .
(6)
In fd R-matrix IP we have to reconstruct the values V(n) in (2) from {E,, y,)fd; IZ,V = 1,2 )...)N. This can be done by solving an algebraic analog [4, 51 of the Gel’fand-Levitan equation or more simply by using the recurrence relations [5,6]: V(n) = Aa, ;
E,Y;
- l/A’,
v=l
Y,(n) = 2A2[J’(n + 1) - E, + 1/A2]Y,(n
+ 1) - Y,(n + 2).
(8)
APPROXIMATION
FOR INVERSE SCATTERING
PROBLEM
355
Although the continuous IP is incorrectly posed, a numerical solution [5] of the fd IP is relatively stable due to the so called “parametric regularization”. Only a finite number of purely algebraic operations are needed to reconstruct V(n). III. Solution of continuous IP in fd approximation The fd formalism described can serve as a basis for reconstructing V(r) in (l), provided an algorithm is given for calculating {E,,~,,})fd for V(n) in (2) from the corresponding scattering data of eq. (1). As mentioned in the introduction, this is not trivial because of a significant difference between spectral properties of discrete and continuous problems. This can be illustrated by a simple example of the square well potential (I/(0 < r < a) = 0; Y,.(O) = Y,,(a) = 0): EC,““’ = v27c2/2a2; \
Efd = [l - cos(vnA/a)]/A2.
(9)
E Fnt and the corresponding upper fd eigenvalue Ei differ by about a factor of 2.5(!), whereas in IP the initial spectral (scattering) data should be given very carefully. In order to go over from (E,., yVjcont to {E,,, y,,jfdperturbation theory can be used for those values v where 1 <
(10)
Therefore these E,, with a good accuracy, are,equal to the corresponding values for the constant potential V(r) = F If the same were valid for the fd equation, then {EVjfd could be easily determined from {E,,}cont.But the distanced,, in the fd case increases with v only up to the midpoint of the spectral interval. Unlike the continuous case, for the upper eigenvalues d, becomes as small as for the lowest E, (see (9)), what may preclude the applicability of perturbation theory ar the top of the fd spectrum. Fortunately, the fact that the unperturbed (or in the case I/ const) spectrum possesses such a symmetry has not only shortcomings. It hints at a relevant algorithm of transformation {E, jcont-+ (EVjfd.Although for arbitrary V(n) the spectral symmetry with respect to the midpoint of the fd eigenvalue interval is violated, we may assume that the upper half of the set (E,,} rdis, within satisfactory accuracy, a mirror reflection of the lower one. Indeed the equation [4] [cp(n + 1) + cp(n - 1)]/2 = (1 - A2E)e-d2”‘“‘cp(n)
(11)
356
V. N. MELNIKOV
and B. N. ZAKHARIEV
has an exactly symmetrical spectrum and differs from (2) by the terms of an order of 4’V(n) which are small for short enough mesh d even if V(n) itself is not small. In addition, in the first approximation the difference between eigenvalues corresponding to (2) and (11) possesses the above-mentioned symmetry (eq. (2) can be regarded as a perturbation of (11)). So, the rule for constructing {Ey}fd is as follows. The lowest eigenvalues (up to the energies where perturbation theory is valid) are chosen to be equal for (2) and (1). Using the symmetry just discussed, the same number of E, at the top of the fd spectrum is also determined. All other E, for (2), (4) are taken to be equal to eigenvalues for V(n) E l? In order to calculate the reduced widths {y,}“, one can use their connection with two sets of eigenvalues (Ev}fd, {E:}fd corresponding to two different constants B,B’ in the boundary condition (4) [8]: Y,”= 10; (E, - EL)/ fi (E, - &,).
(12)
a=1 flfv
Thus, the set of scattering parameters for fd IP is determined, and V(n) can be reconstructed as described in sect. II. The results of numerical calculation for two different shapes of potential V(r) are shown in the figure below. v
P
E -2
4 f r/A
2
2
--I
1
0 r/A -1
-2
-3
APPROXIMATION
FOR INVERSE
SCATTERING
PROBLEM
357
The solid lines represent the potentials V(r). The dotted lines correspond to the values v(n) reconstructed from the scattering parameters (E,,,Y~}~~determined as described above from the first three (N = 50) resonance positions calculated by solving (1). Acknowledgments We are grateful to B. V. Rudyak A. A. Suzko, Ya. A. Smorodinsky Zhigunov for useful discussions on the IP solution.
and V. P.
REFERENCES [l]
[2]
[3]
[4] [5] [6]
[7] [8]
Chadan K., Sabatier P. S.: Inoerse problems in quantum scattering theory, Springer, Berlin-HeidelbergNew York, 1977. Malyarov V. V. et ul.: Sot. J. NW/. Phys. 27, No. 4 (1978). Melnikov V. N., Rudyak B. V.. Zakhariev B. N.: Preprint E4-10429 Dubna, 1977. Denisov A. M.: Ser. J. Camp. Math. 17 (1977) 753. Zhigunov V. P.: Preprint OMBT 777123, Serpukhov 1977. Wizner et al.: Part.und Nucl. 9, No. 3 (1977). Case K. M., Kac M.: J. M&h. PAys. 14 (1973),594. Case K. M.: J. Math. P1zy.s. 14 (1974),916; 16 (1974),2166. Zakhariev B. N. et al.: Sot. J. Nurl. Phq’s. 25, No 2 (1977). Zakhariev B. N. rt aI.: (a review article) PurLand Nucl. 8, No 2 (1977). Case K. M.: Phy.s. Fluids 20 (1977),2031. Kato T.: Perturhution theory,for linear operators, Springer, BerlinHeidelberg~New York, 1966. Levitan B. M., Gasymov M. G.: Usp. Mat. Nauk 19, No. 2, 3 (1964).
Added in proof: The approximate IP solutions are reviewed by B. N. Zakhariev er al. in Part. und Nucl. 13, No 6 (1982) 1284, and Proceedings qf‘ Rencontre interdisciplinaire - prohldmes inverses, december 1982, Montpellier, France.
Vol. 18 (1980)
FINITE-DIFFERENCE
ON MATHEMATICAL
APPROXIMATION PROBLEM
V. N. MELNIKOV
No. 3
PHYSICS
FOR INVERSE
SCATTERING
and B. N. ZAKHARIEV
Joint Institute for Nuclear Research, Moscow, U.S.S.R. (Received January
22, 1979)
For the first time potentials are reconstructed in a finite-difference approximation using a genuine inverse scattering method instead of multiple repeated solutions of a direct problem with iterative fitting of scattering data. Up to now a fundamental difference between spectral properties of the Schriidinger operator and its discrete analog hindered from doing this.
I. Introduction At present there is a deep asymmetry in the development of quantum theory of direct and inverse problems (IP). In contrast to numerous effective algorithms for solving the SchrGdinger equation, there are only a few examples of potential r,econstruction from scattering data by genuine IP methods [l]. Up to now IP has mainly been solved by using the direct problem [2]. Most methods for solving the direct problem are based on the finite-difference (fd) approximation to differential equations of motion. But there are no such methods for the IP. Although the IP can be solved in the framework of fd formalism itself [3-Q no algorithms of fd approximate reconstruction of potentials for real SchrGdinger equation: -+Y”(r)
+ V(r)Y(r) = EY(r)
(1)
with continuous coordinate dependence have been suggested until now. The point is that the solutions of eq. (1) and its fd analog: - [Y(n + 1) - 2Y(n) + Y(n - 1)-J/242 + v(n)Y(n) = EY(n)
(2)
are close to one another only at comparatively low energies, and in the IP solution the whole spectrum is involved (in the derivation of IP equations the completeness relation for eigenfunctions of the SchrCidinger operator is of fundamental importance). This contradiction cannot be removed by diminishing the mesh A of fd differentiation. Although the energy interval with similar properties to (1) and (2) increases with A- ‘, 13531
V. N. MELNIKOV
354
and B. N. ZAKHARIEV
its relative weight in the whole spectrum does not increase (the upper bound of fd spectrum increases with A- ‘). The way out of this situation can be found due to the fact that the upper bound of a “good” spectral part (where the solutions of (1) and (2) are similar) rises to the asymptotic region where perturbation theory is valid. It is also important for IP that the first order perturbation corrections depend mainly on the mean value Pof potential V(r). But, unlike in the continuous case, where perturbation theory works better with increasing energy, in the fd model the highest states are as sensitive to the interaction as the lowest ones. This difficulty can be avoided by using the property of weakly violated symmetry for scattering parameters with respect to the midpoint of the fd’spectrum. In the next section we briefly recall how to solve the IP for (2) in order to use this fd model in sect. III as a basis for our approximate method. II. Exact fd model of the inverse scattering problem [5] For a finite range interaction V(r > a) = 0 it is convenient to use the R-matrix formalism where we have a discrete set of scattering parameters (resonance positions E, and reduced widths y,‘). In the fd approach the R-matrix (or the scattering matrix S(E)) is determined even by a finite set {E,,yv}fd, v = 1,2,..., N = a/A:
R(E)= where E, are the eigenvalues
&E, - E)
corresponding
to (2) with the boundary
conditions:
Y&N + 1) - Y&V - 1) = BY”(N) = B~J%.
Y,(O) = 0,
Here B is an arbitrary constant, and eigenfunctions orthonormalized and complete set i
(3)
3
a, Y&)Y,(n)
(4)
Y,(n) = Y(E = EV,n) form an
= 6,,/A; c+ = l/2; CX”<~= 1;
n=l
N
an.;, yv(4yvb4 = L/A .
(6)
In fd R-matrix IP we have to reconstruct the values V(n) in (2) from {E,, y,)fd; IZ,V = 1,2 )...)N. This can be done by solving an algebraic analog [4, 51 of the Gel’fand-Levitan equation or more simply by using the recurrence relations [5,6]: V(n) = Aa, ;
E,Y;
- l/A’,
v=l
Y,(n) = 2A2[J’(n + 1) - E, + 1/A2]Y,(n
+ 1) - Y,(n + 2).
(8)
APPROXIMATION
FOR INVERSE SCATTERING
PROBLEM
355
Although the continuous IP is incorrectly posed, a numerical solution [5] of the fd IP is relatively stable due to the so called “parametric regularization”. Only a finite number of purely algebraic operations are needed to reconstruct V(n). III. Solution of continuous IP in fd approximation The fd formalism described can serve as a basis for reconstructing V(r) in (l), provided an algorithm is given for calculating {E,,~,,})fd for V(n) in (2) from the corresponding scattering data of eq. (1). As mentioned in the introduction, this is not trivial because of a significant difference between spectral properties of discrete and continuous problems. This can be illustrated by a simple example of the square well potential (I/(0 < r < a) = 0; Y,.(O) = Y,,(a) = 0): EC,““’ = v27c2/2a2; \
Efd = [l - cos(vnA/a)]/A2.
(9)
E Fnt and the corresponding upper fd eigenvalue Ei differ by about a factor of 2.5(!), whereas in IP the initial spectral (scattering) data should be given very carefully. In order to go over from (E,., yVjcont to {E,,, y,,jfdperturbation theory can be used for those values v where 1 <
(10)
Therefore these E,, with a good accuracy, are,equal to the corresponding values for the constant potential V(r) = F If the same were valid for the fd equation, then {EVjfd could be easily determined from {E,,}cont.But the distanced,, in the fd case increases with v only up to the midpoint of the spectral interval. Unlike the continuous case, for the upper eigenvalues d, becomes as small as for the lowest E, (see (9)), what may preclude the applicability of perturbation theory ar the top of the fd spectrum. Fortunately, the fact that the unperturbed (or in the case I/ const) spectrum possesses such a symmetry has not only shortcomings. It hints at a relevant algorithm of transformation {E, jcont-+ (EVjfd.Although for arbitrary V(n) the spectral symmetry with respect to the midpoint of the fd eigenvalue interval is violated, we may assume that the upper half of the set (E,,} rdis, within satisfactory accuracy, a mirror reflection of the lower one. Indeed the equation [4] [cp(n + 1) + cp(n - 1)]/2 = (1 - A2E)e-d2”‘“‘cp(n)
(11)
356
V. N. MELNIKOV
and B. N. ZAKHARIEV
has an exactly symmetrical spectrum and differs from (2) by the terms of an order of 4’V(n) which are small for short enough mesh d even if V(n) itself is not small. In addition, in the first approximation the difference between eigenvalues corresponding to (2) and (11) possesses the above-mentioned symmetry (eq. (2) can be regarded as a perturbation of (11)). So, the rule for constructing {Ey}fd is as follows. The lowest eigenvalues (up to the energies where perturbation theory is valid) are chosen to be equal for (2) and (1). Using the symmetry just discussed, the same number of E, at the top of the fd spectrum is also determined. All other E, for (2), (4) are taken to be equal to eigenvalues for V(n) E l? In order to calculate the reduced widths {y,}“, one can use their connection with two sets of eigenvalues (Ev}fd, {E:}fd corresponding to two different constants B,B’ in the boundary condition (4) [8]: Y,”= 10; (E, - EL)/ fi (E, - &,).
(12)
a=1 flfv
Thus, the set of scattering parameters for fd IP is determined, and V(n) can be reconstructed as described in sect. II. The results of numerical calculation for two different shapes of potential V(r) are shown in the figure below. v
P
E -2
4 f r/A
2
2
--I
1
0 r/A -1
-2
-3
APPROXIMATION
FOR INVERSE
SCATTERING
PROBLEM
357
The solid lines represent the potentials V(r). The dotted lines correspond to the values v(n) reconstructed from the scattering parameters (E,,,Y~}~~determined as described above from the first three (N = 50) resonance positions calculated by solving (1). Acknowledgments We are grateful to B. V. Rudyak A. A. Suzko, Ya. A. Smorodinsky Zhigunov for useful discussions on the IP solution.
and V. P.
REFERENCES [l]
[2]
[3]
[4] [5] [6]
[7] [8]
Chadan K., Sabatier P. S.: Inoerse problems in quantum scattering theory, Springer, Berlin-HeidelbergNew York, 1977. Malyarov V. V. et ul.: Sot. J. NW/. Phys. 27, No. 4 (1978). Melnikov V. N., Rudyak B. V.. Zakhariev B. N.: Preprint E4-10429 Dubna, 1977. Denisov A. M.: Ser. J. Camp. Math. 17 (1977) 753. Zhigunov V. P.: Preprint OMBT 777123, Serpukhov 1977. Wizner et al.: Part.und Nucl. 9, No. 3 (1977). Case K. M., Kac M.: J. M&h. PAys. 14 (1973),594. Case K. M.: J. Math. P1zy.s. 14 (1974),916; 16 (1974),2166. Zakhariev B. N. et al.: Sot. J. Nurl. Phq’s. 25, No 2 (1977). Zakhariev B. N. rt aI.: (a review article) PurLand Nucl. 8, No 2 (1977). Case K. M.: Phy.s. Fluids 20 (1977),2031. Kato T.: Perturhution theory,for linear operators, Springer, BerlinHeidelberg~New York, 1966. Levitan B. M., Gasymov M. G.: Usp. Mat. Nauk 19, No. 2, 3 (1964).
Added in proof: The approximate IP solutions are reviewed by B. N. Zakhariev er al. in Part. und Nucl. 13, No 6 (1982) 1284, and Proceedings qf‘ Rencontre interdisciplinaire - prohldmes inverses, december 1982, Montpellier, France.