On inverse problem for singular Sturm-Liouville operator from two spectra
In the paper, an inverse problem with two given spectra for second order differential operator with singularity of type $\cfrac{2}{r} + \cfrac{l(l+1)}{r^2}$ (here, $l$ is a positive integer or zero) at zero point is studied. It is well known that two spectra $\{\lambda_n\}$ and $\{\mu_n\}$ uniquely...
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| author | Panakhov, E. S. Yilmazer, R. Панахов, Є. С. Йилмазер, Р. |
| author_facet | Panakhov, E. S. Yilmazer, R. Панахов, Є. С. Йилмазер, Р. |
| author_sort | Panakhov, E. S. |
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| description | In the paper, an inverse problem with two given spectra for second order differential operator with singularity
of type $\cfrac{2}{r} + \cfrac{l(l+1)}{r^2}$
(here, $l$ is a positive integer or zero) at zero point is studied. It is well known that
two spectra $\{\lambda_n\}$ and $\{\mu_n\}$ uniquely determine the potential function $q(r)$ in a singular Sturm-Liouville equation defined on interval $(0, \pi]$.
One of the aims of the paper is to prove the generalized degeneracy of the kernel $K(r, s)$. In particular, we obtain a new proof of Hochstadt's theorem concerning the structure of the difference $\widetilde{q}(r) - q(r)$. |
| first_indexed | 2026-03-24T02:42:35Z |
| format | Article |
| fulltext |
UDC 517.9
E. S. Panakhov, R. Yilmazer (Firat Univ., Elaziğ, Turkey)
ON INVERSE PROBLEM FOR SINGULAR
STURM – LIOUVILLE OPERATOR FROM TWO SPECTRA
PRO OBERNENU ZADAÇU DLQ SYNHULQRNOHO
OPERATORA ÍTURMA – LIUVILLQ VID DVOX SPEKTRIV
In the paper, an inverse problem with two given spectra for second order differential operator with singularity
of type
2
r
+
�(� + 1)
r2
(here, l is a positive integer or zero) at zero point is studied. It is well known that
two spectra {λn} and {µn} uniquely determine the potential function q(r) in a singular Sturm – Liouville
equation defined on interval (0, π].
One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular,
we obtain a new proof of Hochstadt’s theorem concerning the structure of the difference q̃(r) − q(r).
Vyvça[t\sq obernena zadaça z vykorystannqm dvox zadanyx spektriv dlq dyferencial\noho operatora
druhoho porqdku z synhulqrnistg typu
2
r
+
�(� + 1)
r2
(l — dodatne cile çyslo abo nul\) u nul\ovij
toçci. Vidomo, wo dva spektry {λn} ta {µn} vstanovlggt\ [dynym çynom funkcig potencialu q(r) u
synhulqrnomu rivnqnni Íturma – Liuvillq, vyznaçenomu na intervali (0, π].
Odni[g z cilej roboty [ dovedennq uzahal\neno] vyrodΩenosti qdra K(r, s). Zokrema, oderΩano
nove dovedennq teoremy Hoxßtadta wodo struktury riznyci q̃(r) − q(r).
Introduction. We will consider the equation
d2R
dr2
+
2
r
dR
dr
− �(� + 1)
r2
R +
(
E +
2
r
)
R = 0, 0 < r < ∞. (1)
In quantum mechanics, the study of the energy levels of a hydrogen atom leads to this
equation [1]. The substitution R = y/r reduces equation (1) to the form
d2y
dr2
+
{
E +
2
r
− �(� + 1)
r2
}
y = 0. (2)
Just as in the case of Bessel’s equation, one can show that, in a finite interval [0, b],
the spectrum is discrete.
As known [2, 3], for a solution of (2) which is bounded at zero, one has the following
asymptotic formula for λ → ∞ (E = λ) :
ϕ (r, λ) =
e
π
2
√
λ∣∣∣∣Γ
(
� + 1 +
i√
λ
)∣∣∣∣
1√
λ
cos
[√
λr +
1√
λ
ln
√
λr − (� + 1)
π
2
+ α
]
+ o (1) ,
(3)
where α = arg Γ
(
� + 1 +
i√
λ
)
.
We consider two singular Sturm – Liouville problems
−y′′ +
[
�(� + 1)
r2
− 2
r
+ q(r)
]
y = λy, 0 < r ≤ π, (4)
y (0) = 0, (5)
y′(π) + Hy(π) = 0, (6)
c© E. S. PANAKHOV, R. YILMAZER, 2006
132 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
ON INVERSE PROBLEM FOR SINGULAR STURM – LIOUVILLE OPERATOR ... 133
−y′′ +
[
�(� + 1)
r2
− 2
r
+ q̃(r)
]
y = λy, 0 < r ≤ π, (7)
y (0) = 0,
y′(π) + H̃y(π) = 0, (8)
in which the functions q(r) and q̃(r) are assumed to be real-valued and square integrable.
H and H̃ are finite real numbers.
We denote the spectrum of the first problem by {λn}∞0 and the spectrum of the second
by
{
λ̃n
}∞
0
.
Next, we denote by ϕ (r, λ) the solution of (4) and we denote by ϕ̃ (r, λ) the solution
of (7) satisfying the initial condition (5).
It is well known that there exists a function K(r, s) such that
ϕ̃ (r, λ) = ϕ (r, λ) +
r∫
0
K(r, s)ϕ (s, λ) ds. (9)
The function K(r, s) satisfies the equation
∂2K
∂r2
−
[
2
r
− �(� + 1)
r2
+ q̃(r)
]
K =
∂2K
∂s2
−
[
2
s
− �(� + 1)
s2
+ q (s)
]
K (10)
and the conditions
K(r, r) =
1
2
r∫
0
[q̃ (t) − q (t)] dt, (11)
K (r, 0) = 0. (12)
After the transformations
z =
1
4
(r + s)2, w =
1
4
(r − s)2, K(r, s) = (z − w)−ν+ 1
2u(z, w),
we obtain the following problem (−ν +
1
2
= β) :
∂2u
∂z∂w
− β
z − w
∂u
∂z
+
β
z − w
∂u
∂w
=
(q̃ − q)u
4
√
zw
− u√
z (z − w)
∂u
∂z
+
β
z
u =
1
4
[
q̃
(√
z
)
− q
(√
z
)]
zν−1, u (z, z − δ) = 0.
This problem can be solved by using the Riemann method [4 – 6].
We put
cn =
π∫
0
ϕ2 (r, λn) dr, c̃n =
π∫
0
ϕ̃2
(
r, λ̃n
)
dr,
ρ(λ) =
∑
λn<λ
1
cn
, ρ̃(λ) =
∑
λ̃n<λ
1
c̃n
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
134 E. S. PANAKHOV, R. YILMAZER
The function ρ(λ) (ρ̃(λ)) is called the spectral function of problem (4) – (6) ((7), (8)).
Problem (4) – (6) will be regarded as an unperturbed problem, while (7), (8) will be con-
sidered to be a perturbation of (4) – (6).
It is a known [7] fact that the knowledge of two spectra for a given singular Sturm –
Liouville equation makes it possible to recover its spectral function, i.e., to find numbers
{cn}. More exactly, suppose that, in addition to the spectrum of problem (4) – (6), we
also know the spectrum {µn} of the problem
−y′′ +
[
�(� + 1)
r2
− 2
r
+ q(r)
]
y = λy
y(0) = 0, y′(π) + H1y(π) = 0, H1 �= H.
(13)
Knowing {λn} and {µn}, we can calculate the numbers {cn}. Similarly, for (7), if
besides
{
λ̃n
}
we also know the spectrum {µ̃n} determined by the boundary conditions
y (0) = 0, y′(π) + H̃1y(π) = 0, H̃1 �= H̃, (14)
it then follows that we can determine the numbers {c̃n}.
It is also shown that
√
λn =
[
n +
�
2
]
+
1
π
ln (n + �/2)
n + �/2
+ O
(
1
n2
)
,
‖ϕn‖2 =
π∫
0
ϕ2
n(r)dr =
π
2
+
π2
2
1
n + �/2
+ O
(
lnn
n2
)
.
Theorem 1. Consider the operator
Ly = −y′′ +
[
�(� + 1)
r2
− 2
r
+ q(r)
]
y, (15)
subject to boundary conditions
y (0) = 0, (16)
y′(π) + Hy(π) = 0, (17)
where q is square integrable on (0, π]. Let {λn} be the spectrum of L subject to (16)
and (17).
If (17) is replaced by a new boundary condition
y′(π) + H1y(π) = 0, (18)
a new operator and a new spectrum, say {µn}, result.
Consider now a second operator
L̃y = −y′′ +
[
�(� + 1)
r2
− 2
r
+ q̃(r)
]
y, (19)
where q̃ is square integrable on (0, π]. Suppose that L̃ has the spectrum
{
λ̃n
}
with
λ̃n = λn for all n under the boundary conditions (16) and
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
ON INVERSE PROBLEM FOR SINGULAR STURM – LIOUVILLE OPERATOR ... 135
y′(π) + H̃y(π) = 0, (20)
L̃ with the boundary conditions (16) and
y′(π) + H̃1y(π) = 0 (21)
is assumed to have the spectrum {µ̃n} . We assume that H, H1 �= H, H̃ and H̃1 �= H̃ are
real numbers which are not infinite.
We shall denote by Λ0 the finite index set for which µ̃n �= µn and by Λ the infinite
index set for which µ̃n = µn. Under the above assumptions, it follows that the kernel
K(r, s) is degenerate in the extended sense:
K(r, s) =
∑
Λ0
cnφ̃n(r)ϕn (s) , (22)
where ϕn, φ̃n are suitable solutions of (4) and (7).
Proof. It follows from (9) that
ϕ̃′ (r, λ) = ϕ′ (r, λ) + K(r, r)ϕ (r, λ) +
r∫
0
∂K
∂r
ϕ (s, λ) ds (23)
and
ϕ̃′ (r, λ) + H̃ϕ̃ (r, λ) =
= ϕ′ (r, λ) + H̃ϕ (r, λ) + K(r, r)ϕ (r, λ) +
r∫
0
(
∂K
∂r
+ H̃K
)
ϕ (s, λ) ds.
Substituting r = π, λ = λn into the last equation and using boundary conditions (17),
(20), we obtain
(
H̃ −H
)
ϕ (π, λn) + K (π, π)ϕ (π, λn) +
+
π∫
0
(
∂K
∂r
+ H̃K
)
r=π
ϕ (s, λn) ds = 0. (24)
As n → ∞ and ϕ (π, λn) → o (1) , the integral on the right-hand side tends to zero.
Therefore, from (24) we get
K (π, π) = H − H̃, (25)
π∫
0
(
∂K
∂r
+ H̃K
)
r=π
ϕ (s, λn) ds = 0, n = 0, 1, . . . . (26)
Since the system of functions ϕ (s, λn) is complete, it follows from the last equation that
(
∂K
∂r
+ H̃K
)
r=π
= 0, 0 < s ≤ π. (27)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
136 E. S. PANAKHOV, R. YILMAZER
We now use the condition imposed on the second-mentioned spectrum. Using (9) again,
we obtain
ϕ̃′ (r, λ) + H̃1ϕ̃ (r, λ) = ϕ′ (r, λ) + H̃1ϕ (r, λ) + K(r, r)ϕ (r, λ) +
+
r∫
0
(
∂K
∂r
+ H̃1K
)
ϕ (s, λ) ds. (28)
Putting r = π and λ = µn (n ∈ Λ) and using (18), (21), we obtain
π∫
0
(
∂K
∂r
+ H̃1K
)
r=π
ϕ (s, µn) ds +
(
H̃1 −H1
)
ϕ (π, µn) +
+K (π, π)ϕ (π, µn) = 0.
In the last equation, as n → ∞, the left-hand side tends to zero and ϕ (π, µn) → o (1) .
Therefore,
K (π, π) = H1 − H̃1, (29)
π∫
0
(
∂K
∂r
+ H̃1K
)
r=π
ϕ (s, µn) ds = 0, n ∈ Λ. (30)
Comparing (25) and (29), we obtain H− H̃ = H1− H̃1. For n ∈ Λ0, we obtain from
(28) (for r = π and λ = µn)
π∫
0
(
∂K
∂r
+ H̃1K
)
r=π
ϕ (s, µn) ds = ϕ̃′ (π, µn) + H̃1ϕ̃ (π, µn) . (31)
It follows from (30) and (31) that
(
∂K
∂r
+ H̃1K
)
r=π
=
∑
Λ0
ϕ̃′ (π, µn) + H̃1ϕ̃ (π, µn)
‖ϕ (s, µn)‖2 ϕ (s, µn) , 0 < s ≤ π. (32)
We derive from (27) and (32) the following equations:
K (π, s) =
1
H̃1 − H̃
∑
Λ0
ϕ̃′ (π, µn) + H̃1ϕ̃ (π, µn)
‖ϕ (s, µn)‖2 ϕ (s, µn) , (33)
∂K(r, s)
∂r
∣∣∣∣
r=π
= − H̃
H̃1 − H̃
∑
Λ0
ϕ̃′ (π, µn) + H̃1ϕ̃ (π, µn)
‖ϕ (s, µn)‖2 ϕ (s, µn) , (34)
0 < s ≤ π.
The function K(r, s) satisfies (10). Therefore, it follows from the initial conditions
(33) and (34) that, in the triangle I (see Figure), we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
ON INVERSE PROBLEM FOR SINGULAR STURM – LIOUVILLE OPERATOR ... 137
K(r, s) =
1
H̃1 − H̃
∑
Λ0
ϕ̃′ (π, µn) + H̃1ϕ̃ (π, µn)
‖ϕ (s, µn)‖2 ×
×
[
c̃ (r, µn) − H̃s̃ (r, µn)
]
ϕ (s, µn) , (35)
where c̃ (r, λ) and s̃ (r, λ) are solutions of (7) satisfying the initial conditions
c̃ (π, λ) = s̃′ (π, λ) = 1, c̃′ (π, λ) = s̃ (π, λ) = 0.
The function K(r, s) and the sum (35) satisfy (12); therefore, they coincide in the tri-
angle II; consequently, they coincide in the triangle III as solutions of (10) satisfy the same
initial conditions on the line r = π/2, etc., i.e., K(r, s) is expressed by (35) throughout
the triangle 0 < s ≤ r ≤ π (see [8 – 10]).
Hence, we obtain Hochstadt’s result in a somewhat more general formulation.
Theorem 2. If the spectra {λn} and
{
λ̃n
}
coincide and {µn} and {µ̃n} differ in a
finite number of their terms, i.e., µ̃n = µn for n ∈ Λ, then
q̃(r) − q(r) =
∑
Λ0
c̃n
d
dr
(
φ̃n, ϕn
)
,
where ϕn , φ̃n are suitable solutions of (4) and (7).
Proof. We obtain from (11) the equation
q̃(r) − q(r) = 2
dK(r, r)
dr
.
Differentiating (35) and putting s = r, we obtain
q̃(r) − q(r) =
2
H̃1 − H̃
∑
Λ0
ϕ̃′ (π, µn) + H̃1ϕ̃ (π, µn)
‖ϕ (s, µn)‖2 ×
× d
dr
{[
c̃ (r, µn) − H̃s̃ (r, µn)
]
ϕ (π, µn)
}
.
Consequently,
q̃(r) − q(r) =
∑
Λ0
c̃n
d
dr
(
φ̃nϕn
)
,
where c̃ (r, µn) − H̃s̃ (r, µn) = φ̃n, ϕ (r, µn) = ϕn (r, µn) , and
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
138 E. S. PANAKHOV, R. YILMAZER
ĉn =
2
[
ϕ̃′ (π, µn) + H̃1ϕ̃ (π, µn)
]
(
H̃1 − H̃
)
‖ϕ (s, µn)‖2
.
This completes the proof of Theorem 2. We note that similar problems are investigated in
[11 – 14].
1. Blohincev D. I. Foundations of quantum mechanics. – Moscow: Gostekhteorizdat, 1949. (Engl. transl.:
Dordrecht: Reidel, 1964).
2. Fok V. A. Beginnings of quantum mechanics (in Russian). – Leningrad: Izdat. Leningrad. Univ., 1932.
3. Levitan B. M., Sargsyan I. S. Introduction to spectral theory. – Moscow: Nauka, 1970.
4. Courant R., Hilbert D. Methods of mathematical physics. – New York, 1953.
5. Coz M., Rochus P. Translation kernels for velocity dependent interaction // J. Math. Phys. – 1977. – 18,
# 11. – P. 2232 – 2240.
6. Volk V. Y. On inverse formulas for a differential equation with a singularity at // Uspekhi Mat. Nauk. –
1953. – 8 (56). – S. 141 – 151.
7. Amirov R. Kh., Gülyaz S. Proc. Eigth Int. Colloq. Different. Equat. (Plovdiv, Bulgaria, 18 – 23 August
1998). – P. 17 – 24.
8. Hald O. H. Discontinuous inverse eigenvalue problems // Communs Pure and Appl. Math. – 1984. – 37.
– P. 539 – 577.
9. Levitan B. M. On the determination of the Sturm – Liouville operator from one and two spectra // Izv. AN
USSR. Ser. Mat. – 1978. – 42, # 1. – P. 185 – 199.
10. Panakhov E. S. The definition of differential operator with peculiarity in zero on two spectrum // J. Spectral
Theory Operators. – 1987. – 8. – P. 177 – 188.
11. Carlson R. Inverse spectral theory for some singular Sturm – Liouville problems // J. Different. Equat. –
1993. – 106. – P. 121 – 140.
12. Hochstadt H. The inverse Sturm – Liouville problem // Communs Pure and Appl. Math. – 1973. – 26. –
P. 715 – 729.
13. Krall A. M. Boundary values for an eigenvalue problem with a singular potential // J. Different. Equat. –
1982. – 45. – P. 128 – 138.
14. Rundell W., Sacks P. E. Reconstruction of a radially symmetric potential from two spectral sequences // J.
Math. Anal. and Appl. – 2001. – 264. – P. 354 – 381.
Received 16.05.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
|
| id | umjimathkievua-article-3440 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:35Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/98/b94ba0e510769125e5896cdc6f8bfa98.pdf |
| spelling | umjimathkievua-article-34402020-03-18T19:54:30Z On inverse problem for singular Sturm-Liouville operator from two spectra Про обернену задачу для сингулярного оператора Штурма-Ліувілля від двох спектрів Panakhov, E. S. Yilmazer, R. Панахов, Є. С. Йилмазер, Р. In the paper, an inverse problem with two given spectra for second order differential operator with singularity of type $\cfrac{2}{r} + \cfrac{l(l+1)}{r^2}$ (here, $l$ is a positive integer or zero) at zero point is studied. It is well known that two spectra $\{\lambda_n\}$ and $\{\mu_n\}$ uniquely determine the potential function $q(r)$ in a singular Sturm-Liouville equation defined on interval $(0, \pi]$. One of the aims of the paper is to prove the generalized degeneracy of the kernel $K(r, s)$. In particular, we obtain a new proof of Hochstadt's theorem concerning the structure of the difference $\widetilde{q}(r) - q(r)$. Вивчається обернена задача з використанням двох заданих спектрів для диференціального оператора другого порядку з сингулярністю типу $\cfrac{2}{r} + \cfrac{l(l+1)}{r^2}$ ($l$ — додатне ціле число або нуль) у нульовій точці. Відомо, що два спектри $\{\lambda_n\}$ та $\{\mu_n\}$ встановлюють єдиним чином функцію потенціалу $q(r)$ у сингулярному рівнянні Штурма-Ліувілля, визначеному на інтервалі $(0, \pi]$. Однією з цілей роботи є доведення узагальненої виродженості ядра $K(r, s)$. Зокрема, одержано нове доведення теореми Гохштадта щодо структури різниці $\widetilde{q}(r) - q(r)$.. Institute of Mathematics, NAS of Ukraine 2006-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3440 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 1 (2006); 132–138 Український математичний журнал; Том 58 № 1 (2006); 132–138 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3440/3619 https://umj.imath.kiev.ua/index.php/umj/article/view/3440/3620 Copyright (c) 2006 Panakhov E. S.; Yilmazer R. |
| spellingShingle | Panakhov, E. S. Yilmazer, R. Панахов, Є. С. Йилмазер, Р. On inverse problem for singular Sturm-Liouville operator from two spectra |
| title | On inverse problem for singular Sturm-Liouville operator from two spectra |
| title_alt | Про обернену задачу для сингулярного оператора Штурма-Ліувілля від двох спектрів |
| title_full | On inverse problem for singular Sturm-Liouville operator from two spectra |
| title_fullStr | On inverse problem for singular Sturm-Liouville operator from two spectra |
| title_full_unstemmed | On inverse problem for singular Sturm-Liouville operator from two spectra |
| title_short | On inverse problem for singular Sturm-Liouville operator from two spectra |
| title_sort | on inverse problem for singular sturm-liouville operator from two spectra |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3440 |
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