Inverse Wave Spectral Problem with Discontinuous Wave Speed
The inverse problem for the Sturm{Liouville operator with complex pe- riodic potential and positive discontinuous coe±cients on the axis is studied. The main characteristics of the fundamental solutions and the spectrum of the operator are studied. The formulation of the inverse problem and a constr...
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irk-123456789-1066432016-10-02T03:02:54Z Inverse Wave Spectral Problem with Discontinuous Wave Speed Efendiev, R.F. Orudzhev, H.D. The inverse problem for the Sturm{Liouville operator with complex pe- riodic potential and positive discontinuous coe±cients on the axis is studied. The main characteristics of the fundamental solutions and the spectrum of the operator are studied. The formulation of the inverse problem and a constructive procedure for its solution are given. The uniqueness theorem of the inverse problem is proven. 2010 Article Inverse Wave Spectral Problem with Discontinuous Wave Speed / R.F. Efendiev, H.D. Orudzhev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 255-265. — Бібліогр.: 14 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106643 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The inverse problem for the Sturm{Liouville operator with complex pe- riodic potential and positive discontinuous coe±cients on the axis is studied. The main characteristics of the fundamental solutions and the spectrum of the operator are studied. The formulation of the inverse problem and a constructive procedure for its solution are given. The uniqueness theorem of the inverse problem is proven. |
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Article |
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Efendiev, R.F. Orudzhev, H.D. |
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Efendiev, R.F. Orudzhev, H.D. Inverse Wave Spectral Problem with Discontinuous Wave Speed Журнал математической физики, анализа, геометрии |
| author_facet |
Efendiev, R.F. Orudzhev, H.D. |
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Efendiev, R.F. |
| title |
Inverse Wave Spectral Problem with Discontinuous Wave Speed |
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Inverse Wave Spectral Problem with Discontinuous Wave Speed |
| title_full |
Inverse Wave Spectral Problem with Discontinuous Wave Speed |
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Inverse Wave Spectral Problem with Discontinuous Wave Speed |
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Inverse Wave Spectral Problem with Discontinuous Wave Speed |
| title_sort |
inverse wave spectral problem with discontinuous wave speed |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/106643 |
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Inverse Wave Spectral Problem with Discontinuous Wave Speed / R.F. Efendiev, H.D. Orudzhev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 3. — С. 255-265. — Бібліогр.: 14 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT efendievrf inversewavespectralproblemwithdiscontinuouswavespeed AT orudzhevhd inversewavespectralproblemwithdiscontinuouswavespeed |
| first_indexed |
2025-07-07T18:48:35Z |
| last_indexed |
2025-07-07T18:48:35Z |
| _version_ |
1837015097435750400 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, v. 6, No. 3, pp. 255–265
Inverse Wave Spectral Problem with Discontinuous
Wave Speed
R.F. Efendiev
Institute of Applied Mathematics, Baku State University
23 Z. Khalilov Str., Baku, AZ1148, Azerbaijan
E-mail:efendievrakib@bsu.az
H.D. Orudzhev
Qafqaz University
16 km Baku–Sumqayit Road, Baku, AZ0101, Azerbaijan
E-mail:hamzaga@yahoo.com
Received March 16, 2009
The inverse problem for the Sturm–Liouville operator with complex pe-
riodic potential and positive discontinuous coefficients on the axis is studied.
The main characteristics of the fundamental solutions and the spectrum of
the operator are studied. The formulation of the inverse problem and a
constructive procedure for its solution are given. The uniqueness theorem
of the inverse problem is proven.
Key words: spectral singularities, inverse spectral problem, continuous
spectrum.
Mathematics Subject Classification 2000: 34A36, 34L05, 47A10, 47A70.
Introduction
We consider the differential equation
−y′′ (x) + q (x) y (x) = λ2ρ (x) y (x) (1)
in the space L2 (−∞, +∞) , where the prime denotes the derivative with respect
to the space coordinate. We assume that the potential q (x) is of the form
q(x) =
∞∑
n=1
qneinx, (2)
This work is supported by PICS-4962.
c© R.F. Efendiev and H.D. Orudzhev, 2010
R.F. Efendiev and H.D. Orudzhev
the condition
∑∞
n=1 |qn|2 = q < ∞ is satisfied, λ is a complex number, and
ρ (x) =
{
1
β2
for
for
x ≥ 0,
x < 0, β 6= 1, β > 0 .
(3)
In the frequency domain this equation describes the wave propagation in
an inhomogeneous medium, where q(x) is the restoring force and 1/ρ (x) is the
wave speed. The discontinuity in ρ (x) usually expresses an abrupt change in the
propagation medium of a wave, for example, the tension or the mass density of
string, or the refractive index in a wave propagation medium.
In regard to the problems with discontinuous coefficients, we remark that
Sabatier and his co-writers [1–4] studied the scattering for the impedance-potential
equation and the similar problems were intensively studied by many authors in
different statements [5, 6], but for the periodic complex potential they are con-
sidered for the first time.
Firstly potential (2) was considered by M.G. Gasymov [7]. Later, in 1990,
the results obtained in [7] were extended by L.A. Pastur, V.A. Tkachenko [8].
As a final remark we mention some works of V. Guillemin, A. Uribe [9] and
[10–12].
In the paper, our primary aim is to study the spectrum and to solve the
inverse problem for singular nonself-adjoint operator by transmitting the coeffi-
cient and normalizing the numbers corresponding to quasieigenfunctions of the
Sturm–Liouville operator with complex periodic potential and positive discon-
tinuous coefficients on the axis. As the coefficient allows a bounded analytic
continuation to the upper half-plane of the complex plane z = x + it, we can
analyze the problem (1)–(3) in detail.
The paper consists of three sections. In Section 1 we study the properties
of fundamental system of solutions of equation (1). The spectrum of problem
(1)–(3) is studied in Section 2. In Section 3 we give a formulation of the inverse
problem, prove the uniqueness theorem and provide a constructive procedure for
the solution of the inverse problem.
1. Representation of Fundamental Solutions
Here we study the solutions of the main equation
−y′′ (x) + q (x) y (x) = λ2ρ (x) y (x)
that will be needed later.
We can prove the existence of these solutions if the condition
∑∞
n=1 |qn|2 =
q < ∞ is fulfilled for the potential. This will be unique restriction on the potential
and later on we will consider it to be fulfilled.
256 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
Inverse Wave Spectral Problem with Discontinuous Wave Speed
Theorem 1. Let q(x) be of the form (2) and ρ (x) satisfy condition (3). Then
equation (1) has special solutions of the form
f+
1 (x, λ) = eiλx
(
1 +
∞∑
n=1
1
n + 2λ
∞∑
α=n
Vnαeiαx
)
for x ≥ 0, (4)
f+
2 (x, λ) = e−iλβx
(
1 +
∞∑
n=1
1
n− 2λβ
∞∑
α=n
Vnαeiαx
)
for x < 0, (5)
where the numbers Vnα are determined from the following recurrent relations:
α(α− n)Vnα +
α−1∑
s=n
qα−sVns = 0, 1 ≤ n < α, (6)
α
α∑
n=1
Vnα + qα = 0, (7)
and the series
∞∑
n=1
1
n
∞∑
α=n
α |Vnα| (8)
converge.
The proof of the theorem is similar to that of [8] and therefore we do not cite
it here.
R e m a r k 1. If λ 6= −n
2 , λ 6= n
2β and Imλ > 0, then f+
1 (x, λ) ∈ L2 (0, +∞),
f+
2 (x, λ) ∈ L2 (−∞, 0).
By analogy to [7], it is easy to see that equation (1) has fundamental solutions
f+
1 (x, λ),f−1 (x, λ) (f+
2 (x, λ),f−2 (x, λ)) for which
W
[
f+
1 (x, λ), f−1 (x, λ)
]
= 2iλ,
W
[
f+
2 (x, λ), f−2 (x, λ)
]
= −2iλβ,
(where W [f, g] = f ′g − fg′) is satisfied
Then each solution of equation (1) may be represented as a linear combination
of these solutions
f+
2 (x, λ) = C11 (λ) f+
1 (x, λ) + C12 (λ) f−1 (x, λ) for x ≥ 0, (9)
f+
1 (x, λ) = C22 (λ) f+
2 (x, λ) + C21 (λ) f−2 (x, λ) for x < 0, (10)
where
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 257
R.F. Efendiev and H.D. Orudzhev
f−1,2 (x, λ) = f+
1,2 (x,−λ) ,
C11 (λ) =
W [f+
2 (0, λ) , f−1 (0, λ)]
2iλ
, (11)
C12 (λ) =
W [f+
1 (0, λ) , f+
2 (0, λ)]
2iλ
,
C22 (λ) = − 1
β
C11 (−λ) , C21 (λ) =
1
β
C12 (λ) . (12)
According to physical sense of the solutions f±1 (x, λ),f±2 (x, λ) , it is natural to
say that 1
C12(λ) is a transmission coefficient to the left and C11(λ)
C12(λ) is a reflection
coefficient from the left for equation (1).
Let
f±n (x) = lim
λ→∓n
2
(n± 2λ)f±1 (x, λ) =
∞∑
α=n
Vnαeiαxe−i n
2
x. (13)
It follows from relation (6) that f±n (x) 6≡ 0 is valid for Vnn 6= 0. From this
we obtained that W [f±n (x) , f∓1
(
x,∓n
2
)
] = 0, and consequently the functions
f±n (x) , f∓1
(
x,∓n
2
)
are linear dependent.
Therefore,
f±n (x) = Vnnf∓1
(
x,∓n
2
)
. (14)
2. Spectrum of Operator L
Let L be an operator generated by the differential expression 1
ρ(x)
{
− d2
dx2 + q (x)
}
in the space L2 (−∞, +∞, ρ (x)).
Divide the plane λ into sectors
Sk = {kπ < arg λ < (k + 1)π}, k = 0, 1.
By means of general method for the kernel of the resolvent of operator
(
L− λ2I
)
we get
R11 (x, t, λ) =
1
C12(λ)
{
f+
1 (x, λ) f+
2 (t, λ) for t < x
f+
1 (t, λ) f+
2 (x, λ) for t > x
λ ∈ S0 (15)
and
258 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
Inverse Wave Spectral Problem with Discontinuous Wave Speed
R12 (x, t, λ) =
1
C12(−λ)
{
f−1 (x, λ) f−2 (t, λ) for t < x
f−1 (t, λ) f−2 (x, λ) for t > x
λ ∈ S1. (16)
R e m a r k 2. Rλ2 = (L − λ2I)−1 exists and is bounded for all λ2 out of
positive half-line and C12 (±λ) 6= 0.
Lemma 2. The coefficient C12 (λ) is an analytic function in the Imλ > 0
and has a finite number of zeros, moreover, if C12 (λn) = 0, then
d
dλ
C12 (λ)|λ=λn
= −i
+∞∫
−∞
ρ (x) f+
1 (x, λn) f+
2 (x, λn) dx.
The proof of Lemma 2 is similar to that of [14, p. 173] and therefore we do
not cite it here.
For the solutions f±1 (x, λ) and f±2 (x, λ) we can obtain the asymptotic equal-
ities:
f
±(j)
1 (0, λ) = ± (iλ)j + o(1) for |λ| → ∞, j = 0, 1,
f
±(j)
2 (0, λ) = ∓ (iλβ)j + o (1) for |λ| → ∞, j = 0, 1.
For simplicity we prove the first equality.
Since
f±1 (0, λ) = 1 +
∞∑
n=1
∞∑
α=n
Vnα
n± 2λ
,
then
∣∣f±1 (0, λ)
∣∣ ≤ 1 +
∞∑
n=1
∞∑
α=n
|Vnα|
|n + 2λ| ≤ 1 +
∞∑
n=1
∞∑
α=n
|Vnα|√
(n + 2Reλ)2 + 4Im2λ
≤ 1 +
1
|Imλ|
∞∑
n=1
∞∑
α=n
α |Vnα|
n
.
Therefore, as |λ| → ∞, we obtain f±1 (0, λ) = 1 + o (1).
Analogously, we can prove the rest of asymptotic equalities as |λ| → ∞ for
the solutions f±2 (x, λ).
Then for the coefficients C12(λ), C12(−λ), we get the following asymptotic
equalities:
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 259
R.F. Efendiev and H.D. Orudzhev
C12 (λ) =
1
2iλ
(iλβ + iλ) + o (1) =
β + 1
2
+ o (1) , (17)
C12 (−λ) =
β + 1
2
+ o (1) .
These asymptotic equalities and analytical properties of the coefficients C12(λ),
C12(−λ) make valid the following statement.
Lemma 3. The eigenvalues of the operator L are finite and coincide with the
square of zeros of the functions C12(λ), C12(−λ) from the sectors Sk, k = 0, 1,
respectively.
R e m a r k 3. Taking into account (17), we can obtain the useful relation
β = 2 lim
Imλ→∞
C12 (λ)− 1. (18)
Theorem 2. The continuous spectrum of the operator L fills out the semi
axis [0,∞) and on the continuous spectrum may be spectral singularities at the
points of the form
(
n
2
)2 and
(
n
2β
)2
, n ∈ N .
P r o o f. First, we will prove that the operator L has no positive eigenvalues.
We recall that equation (1) has fundamental solutions f+
1 (x, λ) , f−1 (x, λ) .
Then for the case λ2 > 0 the solution of equation (1) can be written in the
form
y(x, λ) = C1e
i|λ|x
(
1 +
∞∑
n=1
1
n+2|λ|
∞∑
α=n
Vnαeiαx
)
+C2e
−i|λ|x
(
1 +
∞∑
n=1
1
n−2|λ|
∞∑
α=n
Vnαeiαx
)
.
So, y (x, λ) /∈ L2 (−∞,+∞) since the principle parts of the solutions are
periodic.
Taking into account Remark 2, we will study the function
R(x, t, λ) =
{
R11(x, t, λ),
R12(x, t, λ),
λ ∈ S0
λ ∈ S1
in the neighborhood of poles λ0 from [0,∞). Then the number λ0 coincides
with one of the numbers n
2α , n, α ∈ N . From (15)–(16) it follows that the limit
lim
λ→λ0
(λ− λ0)R (x, t, λ) = R0 (x, t) exists and R0 (x, t) is a bounded function with
respect to all the variables. Let θ(x) be an arbitrary finite function. Then ϕ (x) =
+∞∫
−∞
R0 (x, t)θ (t) dt is a bounded solution of equation (1) for λ = λ0. Therefore,
260 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
Inverse Wave Spectral Problem with Discontinuous Wave Speed
ϕ (x) = C0f
+
1 (x, λ0). Comparison of the last relation with formulae (15)–(16)
shows that if λ0 6= n
2 , n ∈ N , then C0 = 0 and so the kernel of the resolvent
has removable singularity at the point λ0. Thus, there is a case λ0 = n
2 , n ∈ N ,
where the kernel of resolvent has the poles of first order. Since f+
1 (x, λ0) /∈
L2 (−∞, +∞) , then λ2
0 = (n
2 )2 is a spectral singularity of the operator L in the
sense of M.N. Naimark [13]. (Analogously, we can show that the resolvent has
the poles of first order at the points λ0 = n
2β , n ∈ N and ( n
2β )2 are spectral
singularities of the operator L.)
In order all numbers λ2 > 0 belong to the continuous spectra, it suffices
to show that the domain of value RL−λ2I of the operator
(
L− λ2I
)
is dense
in L2 (−∞,+∞), so that the orthogonal complement of the set RL−λ2I consists
of only zero element. However, the orthogonal complement of the set RL−λ2I
coincide with the space of the solutions of equation L∗f = λ2f . It is easy to see
that the operator L∗ is adjoint to the operator L.
Let ψ (x) ∈ L2 (−∞, +∞) , ψ (x) 6= 0, and
+∞∫
−∞
(
Lf − λ2f
)
ψ (x)dx = 0 (19)
be satisfied for any f (x) ∈ D (L).
From (19) it follows that ψ (x) ∈ D (L∗) , and ψ (x) is an eigenfunction of the
operator L∗ corresponding to eigenvalues λ. More exactly, ψ (x) is the solution
of the equation
−z′′ + q (x) z = λ2ρ(x)z (20)
belonging to L2 (−∞, +∞). We obtained that ψ (x) = 0, since the operator gen-
erated by the expression in the left-hand side of (20) is an operator of type L. This
contradiction shows that the domain of value RL−λ2I of the operator
(
L− λ2I
)
is everywhere dense in L2 (−∞, +∞). The theorem is proved.
Now, taking (9)–(10) and (14) into account, we calculate
lim
λ→n/2
(n− 2λ) R11 (x, t, λ) = lim
λ→n/2
(n− 2λ) 1
2iλ [f+
1 (x, λ) f+
1 (t, λ) W [f+
2 ,f−1 ]
W [f+
1 ,f+
2 ]
+f+
1 (x, λ) f−1 (t, λ)] = 1
in [Vnnf+
1
(
x, n
2
)
f+
1
(
t, n
2
)
+Vnnf+
1
(
x, n
2
)
f+
1
(
t, n
2
)
] = 2
inVnnf+
1
(
x, n
2
)
f+
1
(
t, n
2
)
.
(21)
Analogously, taking into account that f̃+
2 (x, λ) = f+
2 (x, λ) (n− 2λβ) has no
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 261
R.F. Efendiev and H.D. Orudzhev
poles at the points λ = n
2β , n ∈ N , we get
lim
λ→n/2β
(n− 2λβ)R11 (x, t, λ) = [− 1
in
W
[
f−2
(
0,n/2β
)
,f+
1
(
0,n/2β
)]
W
[
f+
1
(
0,n/2β
)
,f̃+
2
(
0,n/2β
)] f̃+
2
(
x, n/2β
)
+
W
[
f+
1
(
0,n/2β
)
,f̃+
2
(
0,n/2β
)]
in f−2
(
x, n/2β
)
]f̃+
2
(
t, n/2β
)
= F
(
x, t, n/2β
)
3. Eigenfunction Expansions
To define the natural spectral data of the operator L, it is necessary to obtain
the eigenfunction expansion for the same operator. For this we consider the
following lemma.
Lemma 4. Let ψ (x) be an arbitrary twice continuously differentiable function
belonging to L2 (−∞, +∞, ρ (x)) . Then
+∞∫
−∞
R (x, t, λ) ρ (t) ψ (t) dt = −ψ (x)
λ2
+
1
λ2
+∞∫
−∞
R (x, t, λ) g (t) dt,
where
g (t) = −ψ′′ (x) + q (x) ψ (x) ∈ L2 (−∞, +∞) .
Integrating the both hand sides along the circle |λ| = R and passing to the
limit as R →∞, we get
ψ (x) = − lim
R→∞
1
2πi
∮
|λ|=R
2λdλ
+∞∫
−∞
R (x, t, λ) ρ (t) ψ (t) dt.
The function
∫ +∞
−∞ R (x, t, λ) ρ (t) ψ (t) dt is analytical inside the contour with re-
spect to λ excepting the points λ = λn, n = 1, 2, . . . , λ = n
2 , λ = n
2β , n =
1, 2, . . . . Denote by Γ+
0
(
Γ−0
)
the contour formed by segments [0, 1
2β − δ], [ 1
2β +
δ, 1
2 − δ], . . . [ n
2β + δ, n
2 − δ] and semicircles of radius δ with the centers at points
n
2 , n
2β n = 1, 2, . . . , located in the upper (lower) half plane. Then
ψ (x) = − 1
2iπ
+∞∫
−∞
2λρ (t) ψ (t) [
∫
Γ+
0
R11 (x, t, λ) dλ−
∫
Γ−0
R12 (x, t, λ) dλ]dt
262 Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3
Inverse Wave Spectral Problem with Discontinuous Wave Speed
= − 1
2iπ
+∞∫
−∞
2λρ (t) ψ (t)
∫
Γ−0
[R11 (x, t, λ)−R12 (x, t, λ)]dλdt
+Res
λ=λn
R11 (x, t, λ)+Res
λ= n
2β
R11 (x, t, λ) +Res
λ=n
2
R11 (x, t, λ) .
Calculate separately every term.
R11 (x, t, λ)−R12 (x, t, λ) =
f+
1 (x, λ) f+
1 (t, λ)
2iλC12 (λ) C22 (λ)
The residues of the resolvent R11 (x, t, λ) in λ1,λ2, . . . λl denote by G11 (λn, x, t).
Thus G11 (λn, x, t) will be equal to
G11 (λn, x, t) = lim
λ→λn
(λ− λn) R11 (x, t, λ) .
Then for every function ψ (x) belonging to L2 (−∞, +∞, ρ (x)) we get the eigen-
function expansion in the form
ψ (x) = − 1
2iπ
+∞∫
−∞
ρ (t) ψ (t)
[ ∮
Γ−0
f+
1 (x,λ)f+
1 (t,λ)
2iλC12(λ)C22(λ)dλ
+G11 (λn, x, t) + 2
inVnnf+
1
(
x, n
2
)
f+
1
(
t, n
2
)
+ F
(
x, t, n/2β
)]
dt
(22)
4. Solution of the Inverse Problem
Let us study the inverse problem for the problem (1–3). From the repre-
sentation (15)–(16) it also follows that for each x and t from (−∞,+∞) the
kernel R(x, t, λ) admits a meromorphic continuation from the sector S = {λ :
0 < argλ < π} and may have poles at the points
(
n
2
)2 and
(
n
2β
)2
, n ∈ N , out-
side of S. These poles of the resolvent are called quasi-stationary states of the
operator L.
Thus the quasistationary states of the operator L are the numbers
(
n
2
)2 and(
n
2β
)2
, n ∈ N . In spectral expansion (22) the numbers Vnn, n ∈ N , play a part of
the normalizing numbers corresponding to quasieigenfunction of the operator L.
So, it makes natural the formulation of the inverse problem about reconstruction
of the potential of the equation (1) and the number β.
Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3 263
R.F. Efendiev and H.D. Orudzhev
Inverse Problem
Given the spectral data { C12 (λ) , Vnn}, construct β and the potential q (x).
Using the results obtained above, we arrive at the following procedure for the
solution of the inverse problem:
1. Taking into account (14), we get
Vn,α+n = Vnn
α∑
m=1
Vmα
m + n
,
from which all the numbers Vnα, α = 1, 2, . . . , n = 1, 2, . . . , n < α, are defined.
2. From recurrent formula (6)–(8), find all numbers qn.
3. The number β is defined by the formula
β = 2 lim
Imλ→∞
C12 (λ)− 1.
So, the inverse problem has a unique solution and the numbers β and qn are
defined constructively by spectral data.
Theorem 3. The specification of spectral data uniquely determines β and the
potential q (x).
Acknowledgements. The paper was finished during the time R.F. Efendiev
visited Universite de Nantes as an invited professor. The author wishes to thank
Prof. A. Nachaoui and Prof. A. Boulkhemair from Nantes University for dis-
cussing the paper as well as the administration of University for invitation and
Laboratoire de Mathematiques Jean Leray for kind and warm hospitality.
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