An approach for solving a variety of inverse coefficient problems for the Sturm–Liouville equation −y″ + q(x)y = ρ2y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectrum Sturm–Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

1.
K.
Chadan
,
D.
Colton
,
L.
Päivärinta
, and
W.
Rundell
,
An Introduction to Inverse Scattering and Inverse Spectral Problems
(
SIAM
,
Philadelphia
,
1997
).
2.
G.
Freiling
and
V.
Yurko
,
Inverse Sturm–Liouville Problems and Their Applications
(
Nova Science Publishers Inc.
,
Huntington, NY
,
2001
).
3.
G. M. L.
Gladwell
,
Inverse Problems in Vibration
, 2nd ed. (
Kluwer Academic
,
New York
,
2005
).
4.
K.
Gou
and
Z.
Chen
, “
Inverse Sturm–Liouville problems and their biomedical engineering applications
,”
JSM Cent. Math. Stat.
2
,
1
(
2015
).
5.
S. I.
Kabanikhin
,
Inverse and Ill-Posed Problems: Theory and Applications
(
De Gruyter
,
Berlin
,
2012
).
6.
V. V.
Kravchenko
,
Direct and Inverse Sturm–Liouville Problems: A Method of Solution
(
Birkhäuser
,
Cham
,
2020
).
7.
B. M.
Levitan
,
Inverse Sturm–Liouville Problems
(
VSP
,
Zeist
,
1987
).
8.
V. A.
Marchenko
,
Sturm–Liouville Operators and Applications: Revised Edition
(
AMS Chelsea Publishing
,
2011
).
9.
A. G.
Ramm
,
Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering
(
Springer
,
Boston
,
2005
).
10.
A. O.
Vatulyan
,
Coefficient Inverse Problems of Mechanics
(
Fizmatlit
,
Moscow
,
2019
) (in Russian).
11.
V. A.
Yurko
,
Introduction to the Theory of Inverse Spectral Problems
(
Fizmatlit
,
Moscow
,
2007
) (in Russian).
12.
S. A.
Avdonin
and
M. I.
Belishev
, “
Boundary control and dynamical inverse problem for nonselfadjoint Sturm–Liouville operator (BC-method)
,”
Control Cybern.
25
(
3
),
429
440
(
1996
).
13.
N. P.
Bondarenko
, “
Local solvability and stability of the inverse problem for the non-self-adjoint Sturm–Liouville operator
,”
Boundary Value Probl.
2020
,
123
.
14.
S. A.
Buterin
, “
On inverse spectral problem for non-selfadjoint Sturm–Liouville operator on a finite interval
,”
J. Math. Anal. Appl.
335
(
1
),
739
749
(
2007
).
15.
S. A.
Buterin
and
M.
Kuznetsova
, “
On Borg’s method for non-selfadjoint Sturm–Liouville operators
,”
Anal. Math. Phys.
9
,
2133
2150
(
2019
).
16.
M.
Horvath
and
M.
Kiss
, “
Stability of direct and inverse eigenvalue problems: The case of complex potentials
,”
Inverse Probl.
27
,
095007
(
2011
).
17.
M.
Marletta
and
R.
Weikard
, “
Weak stability for an inverse Sturm–Liouville problem with finite spectral data and complex potential
,”
Inverse Probl.
21
,
1275
1290
(
2005
).
18.
X.-C.
Xu
and
N. P.
Bondarenko
, “
Local solvability and stability of the generalized inverse Robin-Regge problem with complex coefficients
,”
J. Inverse Ill-Posed Probl.
31
,
711
(
2023
).
19.
B. M.
Brown
,
V. S.
Samko
,
I. W.
Knowles
, and
M.
Marletta
, “
Inverse spectral problem for the Sturm–Liouville equation
,”
Inverse Probl.
19
,
235
252
(
2003
).
20.
Q.
Gao
,
X.
Cheng
, and
Z.
Huang
, “
Modified Numerov’s method for inverse Sturm–Liouville problems
,”
J. Comput. Appl. Math.
253
,
181
199
(
2013
).
21.
Q.
Gao
,
X.
Cheng
, and
Z.
Huang
, “
On a boundary value method for computing Sturm–Liouville potentials from two spectra
,”
Int. J. Comput. Math.
91
,
490
513
(
2014
).
22.
M.
Ignatiev
and
V.
Yurko
, “
Numerical methods for solving inverse Sturm–Liouville problems
,”
Results Math.
52
,
63
74
(
2008
).
23.
A.
Kammanee
and
C.
Böckmann
, “
Boundary value method for inverse Sturm–Liouville problems
,”
Appl. Math. Comput.
214
,
342
352
(
2009
).
24.
V. V.
Kravchenko
, “
On a method for solving the inverse Sturm–Liouville problem
,”
J. Inverse Ill-Posed Probl.
27
,
401
407
(
2019
).
25.
V. V.
Kravchenko
, “
Spectrum completion and inverse Sturm–Liouville problems
,”
Math. Methods Appl. Sci.
46
(
5
),
5821
5835
(
2023
).
26.
V. V.
Kravchenko
,
K. V.
Khmelnytskaya
, and
F. A.
Çetinkaya
, “
Recovery of inhomogeneity from output boundary data
,”
Mathematics
10
(
22
),
4349
(
2022
).
27.
V. V.
Kravchenko
,
E. L.
Shishkina
, and
S. M.
Torba
, “
A transmutation operator method for solving the inverse quantum scattering problem
,”
Inverse Probl.
36
,
125007
(
2020
).
28.
V. V.
Kravchenko
and
S. M.
Torba
, “
A direct method for solving inverse Sturm–Liouville problems
,”
Inverse Probl.
37
(
1
),
015015
(
2021
).
29.
V. V.
Kravchenko
and
S. M.
Torba
, “
A practical method for recovering Sturm–Liouville problems from the Weyl function
,”
Inverse Probl.
37
(
6
),
065011
(
2021
).
30.
A.
Neamaty
,
S.
Akbarpoor
, and
E.
Yilmaz
, “
Solving inverse Sturm–Liouville problem with separated boundary conditions by using two different input data
,”
Int. J. Comput. Math.
95
,
1992
2010
(
2018
).
31.
A.
Neamaty
,
S.
Akbarpoor
, and
E.
Yilmaz
, “
Solving symmetric inverse Sturm–Liouville problem using Chebyshev polynomials
,”
Mediterr. J. Math.
16
,
74
(
2019
).
32.
N.
Röhrl
, “
A least-squares functional for solving inverse Sturm–Liouville problems
,”
Inverse Probl.
21
,
2009
2017
(
2005
).
33.
W.
Rundell
and
P. E.
Sacks
, “
Reconstruction techniques for classical inverse Sturm–Liouville problems
,”
Math. Comput.
58
,
161
183
(
1992
).
34.
P. E.
Sacks
, “
An iterative method for the inverse Dirichlet problem
,”
Inverse Probl.
4
,
1055
1069
(
1988
).
35.
S. A.
Avdonin
,
K. V.
Khmelnytskaya
, and
V. V.
Kravchenko
, “
Recovery of a potential on a quantum star graph from Weyl’s matrix
,”
Inverse Probl. Imaging
18
,
311
(
2024
).
36.
S. A.
Avdonin
and
V. V.
Kravchenko
, “
Method for solving inverse spectral problems on quantum star graphs
,”
J. Inverse Ill-Posed Probl.
31
,
31
42
(
2023
).
37.
N. P.
Bondarenko
, “
Inverse Sturm–Liouville problem with analytical functions in the boundary condition
,”
Open Math.
18
,
512
528
(
2020
).
38.
N. P.
Bondarenko
, “
Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition
,”
Math. Methods Appl. Sci.
43
,
7009
7021
(
2020
).
39.
F.
Gesztesy
,
R.
del Rio
, and
B.
Simon
, “
Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions
,”
Int. Math. Res. Not.
15
,
751
758
(
1997
).
40.
N. J.
Guliyev
, “
On two-spectra inverse problems
,”
Proc. Am. Math. Soc.
148
,
4491
4502
(
2020
).
41.
A. M.
Savchuk
and
A. A.
Shkalikov
, “
Inverse problem for Sturm–Liouville operators with distribution potentials: Reconstruction from two spectra
,”
Russ. J. Math. Phys.
12
,
507
514
(
2005
).
42.
D.
Colton
and
P.
Monk
, “
The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium
,”
Q. J. Mech. Appl. Math.
41
,
97
125
(
1988
).
43.
D.
Colton
,
Y. J.
Leung
, and
S. X.
Meng
, “
Distribution of complex transmission eigenvalues for spherically stratified media
,”
Inverse Probl.
31
,
035006
(
2015
).
44.
C. F.
Yang
and
S.
Buterin
, “
Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues
,”
J. Differ. Equations
260
,
4871
4887
(
2016
).
45.
Y. P.
Wang
,
W.
Zhao
, and
C. T.
Shieh
, “
Reconstruction for a class of the inverse transmission eigenvalue problem
,”
Math. Methods Appl. Sci.
42
,
6660
6671
(
2019
).
46.
V. V.
Kravchenko
and
S. M.
Torba
, “
Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schrödinger equation
,”
J. Math. Phys.
58
,
122107
(
2017
).
47.
V. V.
Kravchenko
,
L. J.
Navarro
, and
S. M.
Torba
, “
Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions
,”
Appl. Math. Comput.
314
,
173
192
(
2017
).
48.
M.
Abramovitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions
(
Dover
,
New York
,
1972
).
49.
E. L.
Shishkina
and
S. M.
Sitnik
,
Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics
(
Elsevier
,
Amsterdam
,
2020
).
50.
A.
Baricz
,
D.
Jankov
, and
T. K.
Pogány
, “
Neumann series of Bessel functions
,”
Integr. Transforms Spec. Funct.
23
(
7
),
529
538
(
2012
).
51.
A.
Baricz
,
D.
Jankov
, and
T. K.
Pogány
,
Series of Bessel and Kummer-type Functions
,
Lecture Notes in Mathematics
(
Springer
,
Cham
,
2017
), p.
2207
.
52.
G. N.
Watson
,
A Treatise on the Theory of Bessel Functions
, 2nd ed. (
Cambridge University Press
,
Cambridge, UK
,
1996
), p.
vi+804
.
53.
J. E.
Wilkins
, “
Neumann series of Bessel functions
,”
Trans. Am. Math. Soc.
64
,
359
385
(
1948
).
54.
P.
Hartman
,
Ordinary Differential Equations
(
SIAM
,
Philadelphia
,
2002
).
55.
V. V.
Kravchenko
and
V. A.
Vicente-Benitez
, “
Closed form solution and transmutation operators for Schrödinger equations with finitely many δ-interactions
,” arXiv: 2302.13218 (
2023
).
56.
V.
Ledoux
,
M. V.
Daele
, and
G. V.
Berghe
, “
MATSLISE: A MATLAB package for the numerical solution of Sturm–Liouville and Schrödinger equations
,”
ACM Trans. Math. Softw.
31
,
532
554
(
2005
).
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