eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2021): 1.024
A practical method for constructing a reflectionless potential with a given energy spectrum; pp. 358–377
PDF | doi: 10.3176/proc.2016.4.01

Matti Selg

A fully algebraic approach to constructing one-dimensional reflectionless potentials with any number (N) of bound states is described. A simple and easily applicable general formula is derived, using the methods of the theory of determinants. In particular, useful properties of special determinants – the alternants – have been exploited. The modified determinant that uniquely fixes the potential contains only 2N−1  terms, which is a huge win compared to the N! terms of the original expansion. Moreover, the modified determinant can be very easily evaluated using the properties of alternants. To this end, two useful theorems have been proved. The main formula takes an especially simple form if one aims to reconstruct a symmetric reflectionless potential. Several examples are presented to illustrate the efficiency of the method.


1. Marchenko, V. A. Certain questions of the theory of second-order differential operators. Dokl. Akad. Nauk SSSR, 1950, 72, 457–460 (in Russian).

2. Marchenko, V. A. On the reconstruction of the potential energy from phases of the scattered waves. Dokl. Akad. Nauk SSSR, 1955, 104, 695–698 (in Russian).

3. Gelfand, I. M. and Levitan, B. M. On the determination of a differential equation from its spectral function. Izv. Akad. Nauk SSSR. Ser. Mat., 1951, 15, 309–360 (in Russian). [Am. Math. Soc. Transl. (ser. 2), 1955, 1, 253–304].

4. Krein, M. G. On the transfer function of a one-dimensional boundary value problem of the second order. Dokl. Akad. Nauk SSSR, 1953, 88, 405–408 (in Russian).

5. Krein, M. G. On integral equations generating differential equations of second order. Dokl. Akad. Nauk SSSR, 1954, 97, 21–24 (in Russian).

6. Chadan, K. and Sabatier, P. C. Inverse Problems in Quantum Scattering Theory. 2nd edn., Springer, New York, 1989.

7. Kay, I. The Inverse Scattering Problem. Institute of Mathematical Sciences, New York University, Research Report No. EM-74, 1955.

8. Kay, I. and Moses, H. E. The determination of the scattering potential from the spectral measure function. Nuovo Cimento, 1956, 3, 276–304.

9. Faddeev, L. D. On the relation between S-matrix and potential for the onedimensional Schr¨odinger operator. Dokl. Akad. Nauk SSSR, 1958, 121, 63–66 (in Russian). [Math. Rev., 1959, 20, 773].

10. Faddeev, L. D. The inverse problem in the quantum theory of scattering. Usp. Mat. Nauk, 1959, 14, 57–119 (in Russian). J. Math. Phys., 1963, 4, 72–104].

11. Faddeev, L. D. Properties of the S-matrix of the one-dimensional Schr¨odinger equation. Trudy Mat. Inst. Akad. Nauk SSSR, 1964, 73, 314–336 (in Russian). [Amer. Math. Soc. (ser. 2), 1967, 65, 139–166].

12. Faddeev, L. D. Inverse problems of quantum scattering theory, II. Itogi Nauki Tekh. Sovrem. Prob. Mat., 1974, 3, 93–180 (in Russian).

13. Marchenko, V. A. Sturm-Liouville Operators and Applications. Naukova Dumka, Kiev, 1977 (in Russian). [Birkh¨auser, Basel, 1986; AMS, 2011 (revised edn.)].

14. Thacker, H. B., Quigg, C., and Rosner, J. L. Inverse scattering problem for quarkonium systems. I. One-dimensional formalism and methodology. Phys. Rev. D, 1978, 18, 274–287.

15. Meyer, C. D. Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, 2000.

16. Muir, T. A Treatise on the Theory of Determinants. Macmillan, London, 1882.

17. Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. 2nd edn (unabridged republication), Dover, New York, 2000.

18. Cox, D., Little, J., and O’Shea, D. Ideals, Varieties and Algorithms. 2nd edn, Springer, New York, 1997.

19. Schonfeld, J. F., Kwong, W., and Rosner, J. L. On the convergence of reflectionless approximations to confining potentials. Ann. Phys., 1980, 128, 1–28.

20. Schiff, L. I. Quantum Mechanics. McGraw-Hill, New York, 1949.

21. Morse, P. M. Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev., 1929, 34, 57–64.

Back to Issue