Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators
We solve the inverse spectral problem for a class of Sturm - Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.
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| Цитувати: | Inverse eigenvalue problems for nonlocal Sturm - Liouville operators / L.P. Nizhnik // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 41-47. — Бібліогр.: 4 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860042865318559744 |
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| author | Nizhnik, L.P. |
| author_facet | Nizhnik, L.P. |
| citation_txt | Inverse eigenvalue problems for nonlocal Sturm - Liouville operators / L.P. Nizhnik // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 41-47. — Бібліогр.: 4 назв. — англ. |
| collection | DSpace DC |
| description | We solve the inverse spectral problem for a class of Sturm - Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.
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| first_indexed | 2025-12-07T16:56:35Z |
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 1, pp. 41–47
INVERSE EIGENVALUE PROBLEMS FOR NONLOCAL
STURM–LIOUVILLE OPERATORS
L. P. NIZHNIK
To the memory of A. Ya. Povzner
Abstract. We solve the inverse spectral problem for a class of Sturm–Liouville
operators with singular nonlocal potentials and nonlocal boundary conditions.
1. Introduction
This paper is dedicated to the memory of A. Ya. Povzner whose outstanding works
have opened for me the area of spectral theory of Schrödinger operators. My personal
acquaintance with A. Ya. Povzner took place in 1961 when I was presenting my PhD
Thesis and he was an opponent. Talks with him in that time and the scientific discussions
were very instructive and left an unforgettable trace in all my scientific life.
A one-dimensional Schrödinger operator with nonlocal potential has the form
Ly ≡ − d2
dx2
y(x) +
∫
K(x, s)y(s) ds,
where K(x, s) = K̄(x, s) is a Hermitian-symmetric kernel. If K(x, s) = v(x)δ(s − x0) +
δ(x− x0)v̄(s), where δ is Dirac’s function, we have a Schrödinger operator with nonlocal
point potential, Ly ≡ −d2y(x)
dx2 +v(x)y(x0)+ δ(x−x0)(y, v)L2 [1]. When considering such
an operator, one can avoid using Dirac’s δ-function if the differential expression ly(x) =
d2y(x)
dx2 + v(x)y(x0), x �= x0, in the point x = x0 is supplemented with the boundary-value
conditions y(x0 − 0) = y(x0 + 0) = y(x0), y(x0 − 0)− y(x0 + 0) = (y, v)L2 [1]. Note that
such nonlocal operators appear not only in quantum mechanics but in other areas such
as the theory of diffusion processes, see the related references in [2].
In this paper, we study a one-dimensional Schrödinger operator with nonlocal point
potential on a bounded interval with periodic boundary-value conditions. Because of the
periodicity, we can limit the considerations to only the case where the point nonlocal
potential has its support at an endpoint of the interval. Then we have the following
nonlocal Sturm–Liouville eigenvalue problem:
(1) ℓ(y) := −y′′(x) + v(x)y(1) = λy(x), 0 ≤ x ≤ 1,
subject to the boundary-value condition
(2) y(0) = y(1), y′(1) − y′(0) + (y, v)L2 = 0.
Here v ∈ L2(0, 1) is the nonlocal “potential” and λ ∈ C is a spectral parameter. This
problem is close to the problem studied in [2] for equation (1) with the boundary-value
conditions y(0) = y′(1) + (y, v)L2 = 0. However, this unperturbed problem (1)–(2), for
v = 0, has a simple eigenvalue, λ = 0, and double eigenvalues λ = (2nπ)2, n ∈ N ,
2000 Mathematics Subject Classification. Primary 47A55, 47B25, 35J10.
Key words and phrases. Inverse spectral problems, Sturm–Liouville operators, nonlocal potentials,
nonlocal boundary conditions.
41
42 L. P. NIZHNIK
whereas the unperturbed problem considered in [2] has all simple eigenvalues λ = (nπ)2,
n ∈ N . This influences the way for studying problem (1)–(2) and solving the inverse
eigenvalue problem, that is, the problem of recovering the function v in equation (1) from
a known collection of all eigenvalues of problem (1)–(2).
For solving the inverse eigenvalue problem, as in the case of the usual potential, in
addition to eigenvalues it is also necessary to have some additional information [3]. The
same is true for the real nonlocal potential, — we need to have signs of its even Fourier
sine coefficients.
2. Direct spectral analysis
On the space L2(0, 1), problem (1)–(2) is naturally related to an operator Tv that has
domain
domT = {y ∈ W 2
2 (0, 1) | y(0) = y′(1) − y′(0) + (y, v)L2 = 0}
and is defined by Tvy = ℓ(y). The operator Tv is a closed symmetric operator, since
integration by part formula easily yields that the quadratic form (Tvy, y)L2 is real,
(3) (Tvy, y) = (y′, y′)L2 + 2Re [y(1)(v, y)L2 ].
For v = 0, the operator T0 is self-adjoint on the space L2(0, 1), with domain consisting of
functions periodic on (0, 1) and belonging to the Sobolev space W 2
2 (0, 1). Eigenfunctions
of the operator T0 are the functions 1, cos 2nπx, sin 2nπx, n ∈ N , and eigenvalues are
λ0 = 0, λn = (2nπ)2. The operators Tv and T0 can be considered as extensions of the
symmetric operator Tmin that has domain
(4) domTmin = {y ∈ W 2
2 (0, 1) | y(0) = y(1) = 0, y′(0) = y′(1), (y, v)L2 = 0}
and is defined by Tminy = Tvy = T0y = −y′′. Since dim(domTv/domTmin) ≤ 2, the
self-adjoint operator Tv is a rank r ≤ 2 perturbation of the self-adjoint operator T0. Since
the operator T0 has discrete spectrum, which consists of the numbers λn = (2nπ)2, the
operator Tv has discrete spectrum consisting of real numbers λn → +∞ for n → ∞.
Theorem 1. 1. All eigenvalues of problem (1), (2), distinct from (nπ)2, n ∈ N ,
are simple.
2. The number λ = (2nπ)2 is an eigenvalue of problem (1), (2) if and only if
(5) vn =
1
2
∫ 1
0
v(x) sin nπxdx =
{
0, if n even,
4nπ, if n odd.
3. The number λn = (2nπ)2 is a double eigenvalue of problem (1), (2) if and only
if, in addition to (5), we have
(6)
∑
k �=n
[1 + (−1)k+1]kπ[vk + v̄k] − 1
2 |vk|2
(kπ)2 − (nπ)2
= 0.
Problem (1), (2) has no eigenvalues with multiplicity exceeding 2.
4. The number λ = 0 is an eigenvalue of problem (1), (2) if and only if (6) holds
for n = 0.
Proof. The eigenvalues of problem (1), (2) coincide with the eigenvalues of the operator
Tv. Let y(x; z) be an eigenfunction of the operator Tv corresponding to an eigenvalue
λ = z2. If y(0; z) = y(1; z) = 0, then it follows from equation −y′′ = z2 that y = C sinπx
and z2 = (nπ)2, n ∈ N . Hence, for λ �= (nπ)2, eigenfunctions take nonzero equal values at
the endpoints of the interval (0, 1). This leads to simplicity of the eigenvalues λ �= (nπ)2,
n ∈ N . Indeed, if an eigenvalue were multiple, there would exist at least two linearly
independent eigenfunctions yi(x; z), i = 1, 2, corresponding to the same eigenvalue. But
then y(x; z) = y1(0; z)y2(x; z)−y2(0; z)y1(x; z) would be a nontrivial eigenfunction which
INVERSE EIGENVALUE PROBLEMS FOR NONLOCAL STURM–LIOUVILLE OPERATORS 43
vanishes at the endpoints of the interval (0, 1). This is impossible for λ = z2 �= (nπ)2.
Claim 1 of the theorem is proved.
Let now λ = (nπ)2, n ∈ N , be an eigenvalue of the operator Tv, that is, there exists
a nontrivial solution y(x; nπ) of problem (1), (2) for λ = (nπ)2. Since the nonlocal
potential satisfies v ∈ L2, this function can be represented by a Fourier sine series,
(7) v(x) =
∞
∑
k=1
vk sinkπxdx, vk = 2
∫ 1
0
v(x) sin kπxdx.
By substituting (7) into equation (1) and solving it, we get
(8) y(x; nπ) = A cosnπx + B sin nπx − A
2nπ
vnx cosnπx − A
∑
k �=n
vk sin kπx
(kπ)2 − (nπ)2
.
The boundary-value condition y(0) = y(1) leads to the identity
(9) A[1 − (−1)n +
vn
2nπ
(−1)n] = 0.
Hence, if A �= 0, then (5) holds. If A = 0 and the solution y is nontrivial, then it has
the form y = B sin nπx and B �= 0. Substituting this solution into the boundary-value
condition y′(1) − y′(0) + (y, v) = 0 we are again led to (5). Hence, problem (1)–(2) has
a nontrivial solution for λ = (nπ)2 only if condition (5) is satisfied. Also, condition (5)
is sufficient for the function y = sin nπx to be an eigenfunction. Claim 2 is proved.
To prove Claim 3, note that λ = (nπ)2 is a double eigenvalue if and only if condition (5)
is satisfied and the function in (8), with A = 1 and B = 0, satisfies the boundary-value
condition y′(1) − y′(0) + (y, v) = 0. Substituting (8) into this boundary-value condition
leads to identity (6). Since problem (1)–(2) deals with a second order equation on
the interval (0, 1), it can not have more than two linearly independent solutions and,
consequently, there are no eigenvalues of multiplicity exceeding 2.
Consider Claim 4. It follows from (1) for λ = 0 that
(10) y(x; 0) = A + Bx − y(1)
∞
∑
k=1
vk sinkπx
(kπ)2
.
The boundary-value condition y(1) = y(0), since the solution y(x; 0) is nontrivial, leads
to the condition B = 0, A �= 0, y(1) = A. Substituting (10) into the boundary-value
condition (2) we get identity (6) with n = 0. �
To give an exact description of the distribution of eigenvalues of problem (1), (2), it is
convenient to show that the eigenvalues are connected with zeros of an analytic function,
which is a characteristic function of problem (1), (2).
To this end, consider a special solution of equation (1) with λ = z2, satisfying the
condition y(0) = y(1),
(11) ϕ(x; z) =
sin zx + sin z(1 − x)
z
− sin zx
z
∞
∑
k=1
vk sin kπx
(kπ)2 − z2
.
The function ϕ is an eigenfunction of problem (1), (2) if it satisfies the boundary-value
condition (2). This gives the characteristic equation χ(z) = 0, where the characteristic
function χ(z) is defined by χ(z) = ϕ′(1) − ϕ′(0) + (ϕ, v) and has the form
(12) χ(z) = 2(cos z − 1) +
sin z
z
∞
∑
k=1
(−1)kkπαk
(kπ)2 − z2
,
where
(13) (−1)kαk = [1 + (−1)k+1](vk + v̄k) − 1
2kπ
|vk|2.
44 L. P. NIZHNIK
Lemma 1. The characteristic function χ(z) of the form (12) is an entire function of z
and, for z = nπ, n ∈ N , takes the values
(14)
χ(2mπ) =
|v2m|2
(4mπ)2
,
χ((2m − 1)π) = −|2 − v2m−1
2(2m − 1)π)
|2.
Proof. The proof is carried out by direct computations using the explicit form (12) of
the characteristic function. �
Theorem 2. The number λ = z2 is an eigenvalue of problem (1), (2) if and only if z is
a zero of the characteristic function χ(z). The number λ = z2 �= 0 is a double eigenvalue
of the problem (1), (2) if and only if z is a double zero of the characteristic function.
All zeros z �= nπ, n ∈ N , of the characteristic function are simple. The characteristic
function does not have zeros of multiplicities greater than 2.
Proof. It follows that the squares z2 �= (nπ)2 of zeros of the characteristic function χ(z)
are eigenvalues from the fact that the special solution (11) is an eigenfunction and vice
versa. For z2 = (nπ)2 = λ, the number λ = z2 is an eigenvalue if and only if (5) is
satisfied which, by (14), is equivalent to χ(nπ) = 0. It is easy to check that condition (6)
is equivalent to χ̇(nπ) = 0 which, in its turn, together with χ(nπ) = 0, is equivalent
to that nπ is a double eigenvalue. Condition (6), for n = 0, is equivalent to χ(0) = 0.
Hence, the eigenvalues, counting multiplicities, coincide with squares of zeros of χ(z),
counting multiplicities.
If the function χ(z) had a multiple root z0 �= nπ, this would imply that ∂
∂z
ϕ(x; z)|z=z0
were a generalized eigenfunction, which is impossible since the operator Tv is self-adjoint.
In the same way, we can prove that there are no zeros of χ(z) that have multiplicities
greater than 2 for z = nπ. �
Theorem 3. The increasingly ordered sequence λ1 ≤ λ2 ≤ . . . ≤ λn ≤ λn+1 ≤ . . . of all
eigenvalues of problem (1), (2), counting multiplicities, has the following properties:
1. the sequence weakly alternates with the sequence (nπ)2,
(15) λn ≤ (nπ)2 ≤ λn+1, n ∈ N ;
2. there is an asymptotic representation,
(16)
√
λ2n = 2nπ − β2n
n
,
√
λ2n+1 = 2nπ + β2n+1
n
,
where βj ≥ 0 and
∑∞
j=1 β2
j < +∞.
Proof. An upper estimate for λn is easily obtained from the variation minimax princi-
ple [4],
λn = supϕ1,...,ϕn−1
inf
||ψ||=1, ψ⊥ϕk,
ψ(0)=ψ(1)
(Tvψ, ψ)
≤ supϕ1,...,ϕn−1
inf
||ψ||=1, ψ⊥ϕk,
ψ(0)=ψ(1)=0
(ψ′, ψ′) = (nπ)2.
For large n, we have the strict inequality
(17) [(2n − 1)π]2 < λ2n ≤ λ2n+1 < [(2n + 1)π]2.
Indeed, by the Rouché theorem, the entire function χ(z) and the function 2(cos z − 1)
have the same number of zeros, counting multiplicities, in the strip −(2n+1)π < Re z <
(2n + 1)π for large n. Since, by Theorem 2, the eigenvalues λn of problem (1), (2)
are squares of zeros of the function χ(z), the function χ(z) is even, and double zeros
INVERSE EIGENVALUE PROBLEMS FOR NONLOCAL STURM–LIOUVILLE OPERATORS 45
of the function 2(cos z − 1) are the numbers z = 2nπ, n = 0,±1, ..., which shows that
inequalities (17) hold.
On the other hand, if
(18) v2m �= 0, v2m+1 �= 4(2m + 1)π,
then, by (14), χ(nπ) �= 0 for any n and, moreover, χ(nπ)(−1)n > 0. Hence, in every
interval (nπ, (n + 1)π), the characteristic function χ has at least one zero z0 and, con-
sequently, there is one eigenvalue λ = z2 in the interval In = ((nπ)2, (n + 1)2π2). The
assumption that at least one interval In contains more than one eigenvalue leads to a
contradiction with estimate (17). Hence, if conditions (18) are satisfied, inequality (15)
holds,
(19) λn < (nπ)2 < λn+1.
Since condition (18) can be satisfied by an arbitrary small change of the potential, passing
to the limit in (19) we get (15).
To prove the asymptotic representations (16) let us first find an asymptotic represen-
tation of zeros of the characteristic function χ(z) defined by (12), since
√
λn = zn.
Write the characteristic function in (12), χ(z), as
(20) χ(z) = 2(cos z − 1) − 1
z
∫ 1
0
sin ztα(t) dt,
where α(t) ∈ L2, and the numbers αk are coefficients in the Fourier sine expansion. If
z = 2mπ + εm, then (20) for m → ∞ gives
(21) −ε2
m +
|v2m|2
(4mπ)2
− εm
w2m
2mπ
+ o(ε2
m) = 0,
where w2m =
∫ 1
0
cos 2mπtα(t)t dt and
∑
m w2
2m < ∞.
Identity (21) gives two solutions for εm, one of which is nonpositive, ε+
m = −β2m,
β2m ≥ 0, and the other one is nonnegative, ε−m = β2m+1, β2m+1 ≥ 0. Here
∑
j |βj |2 <
+∞. This gives representation (16). �
3. Isospectral nonlocal potentials
Let Λ(v) = {λj}∞j=1 be an ordered sequence of all eigenvalues, counting multiplic-
ities, of problem (1), (2) with a nonlocal potential v. Two potentials v1 and v2 are
called isospectral if Λ(v1) = Λ(v2), that is, eigenvalues and their multiplicities of prob-
lems (1), (2), coincide for the potentials v1 and v2. It follows from Theorem 2 that
potentials v1 and v2 are isospectral if and only if the corresponding characteristic func-
tions coincide. Using representations (12), (13) the for characteristic functions we obtain
a criterion for two potentials v1 and v2 to be isospectral in terms of their Fourier sine
coefficients v
(j)
k , j = 1, 2, k ∈ N ,
(22)
|v(1)
2m| = |v(2)
2m|,
∣
∣
∣
1 − v
(1)
2m+1
4(2m+1)π
∣
∣
∣
=
∣
∣
∣
1 − v
(2)
2m+1
4(2m+1)π
∣
∣
∣
.
On the space L2(0, 1) of real-valued functions, define nonlinear projections Πs that
depend on a sequence of real numbers s = {s1, s2, ..} by
(23) Πs
(
∞
∑
n=1
vn sin nπx
)
=
∞
∑
n=1
v(s)
n sin nπx,
∞
∑
n=1
v2
n < +∞,
46 L. P. NIZHNIK
where
(24) v(s)
n =
|vn|, if sn = 0,
vn, if |vn| ≤ |2sn − vn|, sn �= 0,
2sn − vn, if |2sn − vn| < |vn|.
Theorem 4. Two real nonlocal potentials v1, v2 ∈ L2(0, 1) are isospectral if and only if
the corresponding nonlinear projections coincide,
(25) Πsv1 = Πsv2.
Here s = {sn}∞n=1 consists of the numbers s2m = 0 and s2m−1 = 2(2m − 1)π.
Proof. The proof follows from the isospectral criterion (22) and definitions (23), (24) of
the nonlinear projection Πs. �
4. The inverse spectral analysis
The main tool for describing eigenvalues of problem (1), (2) is the characteristic func-
tion χ(z) explicitly given by (12). This function can be written using the ordered set of
all eigenvalues, Λ = {λj}.
Theorem 5. The characteristic function χ(z) can be expressed in terms of the ordered
sequence {λj}, counting multiplicities, of problem (1), (2) as
(26) χ(z) = (λ1 − z2)
∞
∏
k=2
λk − z2
4[k
2 ]2π2
,
where [α] denotes the entire part of α.
Proof. Since χ(z) is an even function of exponential type 1 and
lim
ξ→∞
χ(iξ)
2(cos iξ − 1)
= 1,
it can uniquely be represented in terms of its zeros zk =
√
λk as the product in (26). �
Consider now the problem of recovering the nonlocal potential for problem (1), (2) in
terms of the set Λ = {λj} of all eigenvalues of the problem. To solve this problem, we
can propose the following algorithm:
Step 1. Using the eigenvalues Λ = {λj}∞j=1 construct the characteristic function χ(z) by
formula (26).
Step 2. Calculate the values χ(nπ), n ∈ N .
Step 3. Use formula (14) to find vn that have the minimal modulus.
Step 4. Use the Fourier coefficients vn to construct the nonlocal potential
(27) v(x) =
∞
∑
n=1
vn sin nπx.
Example. Let, for problem (1), (2), we have λ1 = π2, λ2n = λ2n+1 = (2nπ)2, n ∈ N .
Then χ(z) = z2−π2
z2 2(cos z − 1). In this case, χ(2nπ) = 0, χ((2n − 1)π) = −4 + 4
(2n−1)2 ,
n ∈ N . It follows from formulas (14) that v2n = 0, v2n−1 = 4π 1
2n−1+
√
(2n−1)2−1
, n ∈ N .
Hence, the nonlocal potential v(x) has the following Fourier series:
v(x) = 4π
∞
∑
n=1
sin(2n − 1)πx
2n− 1 +
√
(2n − 1)2 − 1
.
INVERSE EIGENVALUE PROBLEMS FOR NONLOCAL STURM–LIOUVILLE OPERATORS 47
Since
∞
∑
n=1
sin(2n − 1)πx
2n− 1
=
π
4
, we have
v(x) =
π2
2
+ 2π
∞
∑
n=1
sin(2n − 1)πx
(2n − 1)[2n − 1 +
√
(2n − 1)2 − 1]2
.
As was remarked in the previous section, the nonlocal potential v is defined by its
eigenvalues Λ(v) non-uniquely. However, it is easy to give conditions such that the
nonlocal potential can be found in a unique way, as it was done in [2]. In particular,
the inverse problem for (1), (2) with nonlocal potential has a unique solution if we
use Λ = {λj}∞j=1 to find a nonlocal potential with a minimal norm and an additional
assignment of signs of all nonzero even Fourier coefficients.
For the inverse problem, it is important to have a description of the initial data of the
problem, that is the set Λ = {λj}∞j=1 of all eigenvalues of problem (1), (2) with nonlocal
potential v ∈ L2.
Theorem 6. Conditions 1 and 2 in Theorem 3 are necessary and sufficient for a sequence
Λ = {λj}∞j=1 to be an ordered sequence of all eigenvalues of problem (1), (2) with a
nonlocal potential v ∈ L2.
Proof. Let conditions 1 and 2 of Theorem 3 be satisfied. Then the function χ(z) con-
structed using (26) admits representation (19) and the value χ(nπ) can be written as
in (14) with some vn,
∑∞
n=1 |vn|2 < ∞. The nonlocal potential v(x) =
∑
vk sin kπx ∈ L2
gives rise to a characteristic function that coincides with χ(z) that is constructed using
formula (26). Hence, problem (1), (2) with the potential constructed as above has eigen-
values that coincide with Λ = {λj}∞j=1. �
References
1. S. Albeverio and L. P. Nizhnik, Schrödinger operators with nonlocal point interactions, J. Math.
Anal. Appl. 332 (2007), 884–895.
2. S. Albeverio, R. Hryniv and L. Nizhnik, Inverse spectral problems for nonlocal Sturm-Liouville
operators, Inverse Problems 23 (2007), 523-535.
3. V. A. Marchenko, Sturm-Liouville Operators and Their Applications, Naukova Dumka Publ.,
Kiev, 1977 (in Russian); Engl. transl.: Birkhäuser Verlag, Basel, 1986.
4. M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators,
Academic Press, New York, 1978.
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka,
Kyiv, 01601, Ukraine
E-mail address: nizhnik@imath.kiev.ua
Received 01/11/2008
|
| id | nasplib_isofts_kiev_ua-123456789-5701 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1029-3531 |
| language | English |
| last_indexed | 2025-12-07T16:56:35Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nizhnik, L.P. 2010-02-02T12:59:20Z 2010-02-02T12:59:20Z 2009 Inverse eigenvalue problems for nonlocal Sturm - Liouville operators / L.P. Nizhnik // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 41-47. — Бібліогр.: 4 назв. — англ. 1029-3531 https://nasplib.isofts.kiev.ua/handle/123456789/5701 We solve the inverse spectral problem for a class of Sturm - Liouville operators with singular nonlocal potentials and nonlocal boundary conditions. en Інститут математики НАН України Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators Article published earlier |
| spellingShingle | Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators Nizhnik, L.P. |
| title | Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators |
| title_full | Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators |
| title_fullStr | Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators |
| title_full_unstemmed | Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators |
| title_short | Inverse Eigenvalue Problems for Nonlocal Sturm-Liouville Operators |
| title_sort | inverse eigenvalue problems for nonlocal sturm-liouville operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/5701 |
| work_keys_str_mv | AT nizhniklp inverseeigenvalueproblemsfornonlocalsturmliouvilleoperators |