Instability intervals for Hill’s equation with symmetric single-well potential
UDC 517.9 We deduce some explicit estimates for the periodic and semiperiodic eigenvalues and the lengths of the instability intervals of Hill’s equation with symmetric single-well potentials by using an auxiliary eigenvalue problem. We also give bounds for the gaps of the Dirichlet and Neumann eig...
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| author | Başkaya, E. Coşkun, H. Kabataş, A. Башкая, Е. Кошкун, Х. Кабаташ, А. |
| author_facet | Başkaya, E. Coşkun, H. Kabataş, A. Башкая, Е. Кошкун, Х. Кабаташ, А. |
| author_sort | Başkaya, E. |
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| datestamp_date | 2019-12-05T08:56:42Z |
| description | UDC 517.9
We deduce some explicit estimates for the periodic and semiperiodic eigenvalues and the lengths of the instability intervals
of Hill’s equation with symmetric single-well potentials by using an auxiliary eigenvalue problem. We also give bounds
for the gaps of the Dirichlet and Neumann eigenvalues. |
| first_indexed | 2026-03-24T02:06:34Z |
| format | Article |
| fulltext |
UDC 517.9
H. Coşkun, E. Başkaya, A. Kabataş (Karadeniz Techn. Univ., Trabzon, Turkey)
INSTABILITY INTERVALS FOR HILL’S EQUATION
WITH SYMMETRIC SINGLE-WELL POTENTIAL
IНТЕРВАЛИ НЕСТАБIЛЬНОСТI ДЛЯ РIВНЯННЯ ХIЛЛА
З СИМЕТРИЧНИМ ОДНОЯМНИМ ПОТЕНЦIАЛОМ
We deduce some explicit estimates for the periodic and semiperiodic eigenvalues and the lengths of the instability intervals
of Hill’s equation with symmetric single-well potentials by using an auxiliary eigenvalue problem. We also give bounds
for the gaps of the Dirichlet and Neumann eigenvalues.
За допомогою допомiжної задачi на власнi значення отримано деякi явнi оцiнки для перiодичних i напiвперiодичних
власних значень та довжин iнтервалiв нестабiльностi для рiвняння Хiлла з симетричним одноямним потенцiалом.
Також наведено оцiнки для щiлин у множинах власних значень Дiрiхле та Ноймана.
1. Introduction. We consider the differential equation
y\prime \prime (t) + (\lambda - q(t)) y(t) = 0, (1.1)
where \lambda is a real parameter and q(t) is a real-valued, continuous and periodic function with period
a. Our interest is with two eigenvalue problems associated with (1.1) on [0, a] . The periodic problem
of (1.1) with the boundary conditions y(0) = y(a), y\prime (0) = y\prime (a). This problem has a countable
infinity of eigenvalues denoted by \{ \lambda n\} . We are also concerned with the semiperiodic problem of
(1.1) with the boundary conditions y(0) = - y(a), y\prime (0) = - y\prime (a) and the eigenvalues are denoted
by \{ \mu n\} . It is known [4] that the two sets of eigenvalues satisfy the relation
- \infty < \lambda 0 < \mu 0 \leq \mu 1 < \lambda 1 \leq \lambda 2 < \mu 2 \leq \mu 3 < . . . .
We also denote the eigenvalues of (1.1) with the Dirichlet boundary conditions y(0) = y(a) = 0 by
\Lambda n and the Neumann boundary conditions y\prime (0) = y\prime (a) = 0 by \nu n. It is also known [4] that, for
n = 0, 1, 2, . . . ,
\mu 2n \leq \Lambda 2n \leq \mu 2n+1, \lambda 2n+1 \leq \Lambda 2n+1 \leq \lambda 2n+2, (1.2)
and
\mu 2n \leq \nu 2n+1 \leq \mu 2n+1, \lambda 2n+1 \leq \nu 2n+2 \leq \lambda 2n+2. (1.3)
The instability intervals of (1.1) are defined to be ( - \infty , \lambda 0) , (\mu 2n, \mu 2n+1), (\lambda 2n+1, \lambda 2n+2) and
called the zeroth, (2n+ 1)th and (2n+ 2)th instability interval, respectively. The length of the nth
instability interval of (1.1), whether it is absent or not, will be denoted by ln. We note that the
absence of an instability interval means that there is a value of \lambda for which all solutions of (1.1)
have either period a or semiperiod a. Instability intervals for Hill’s equation with various types of
restrictions on potential have been investigated by many authors over the years [1, 4, 9]. We refer
in particular to [7, 8] in which q(t) is a symmetric single-well potential. Some results about the
c\bigcirc H. COŞKUN, E. BAŞKAYA, A. KABATAŞ, 2019
858 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
INSTABILITY INTERVALS FOR HILL’S EQUATION WITH SYMMETRIC SINGLE-WELL POTENTIAL 859
first instability interval were obtained in [7] and the eigenvalue gap for Schrödinger operators on an
interval with Dirichlet and Neumann boundary conditions was considered in [8].
In this paper we obtain estimates about the instability intervals of (1.1) with q(t) being of a
symmetric single-well potential with mean value zero. By a symmetric single-well potential on
[0, a] , we mean a continuous function q(t) on [0, a] which is symmetric about t =
a
2
and non-
increasing on
\Bigl[
0,
a
2
\Bigr]
. Our analysis is based on the following theorem of Hochstadt, which involves
\Lambda n(\tau ) the eigenvalues of (1.1) considered on the interval [\tau , \tau + a] where 0 \leq \tau < a with Dirichlet
boundary conditions
y(\tau ) = y(\tau + a) = 0.
We refer to this problem as “auxiliary eigenvalue problem”. Here we note that this problem is
equivalent to the following problem [6]:
y\prime \prime (t) + (\lambda - q (t+ \tau )) y(t) = 0,
y(0) = y(a) = 0.
We note that q\prime (t) exists since a monotone function on an interval I is differentiable almost
everywhere on I [5].
We now state an asymptotic approximation previously obtained for the auxiliary eigenvalues
[1 – 3] which will be used to prove our results. It was shown in [3, p. 1275] (for N = 2) as n \rightarrow \infty :
\Lambda 1/2
n (\tau ) =
(n+ 1)\pi
a
+
a
4 (n+ 1)2 \pi 2
\times
\times
\left[ \mathrm{c}\mathrm{o}\mathrm{s}\biggl( 2 (n+ 1)\pi
a
\tau
\biggr) \tau +a\int
\tau
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt -
- \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2 (n+ 1)\pi
a
\tau
\biggr) \tau +a\int
\tau
q\prime (t) \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt
\right] -
- a2
8 (n+ 1)3 \pi 3
a\int
0
q2 (t) dt+ o
\bigl(
n - 3
\bigr)
. (1.4)
As an illustration of our results, we give the following theorem.
Theorem 1.1. Let q(t) be a symmetric single-well potential on [0, a] . Then, as n \rightarrow \infty ,
\Lambda 2n+1 - \Lambda 2n
\nu 2n+2 - \nu 2n+1
\geq (4n+ 3)\pi 2
a2
- 1
2 (n+ 1)\pi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
4 (n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
860 H. COŞKUN, E. BAŞKAYA, A. KABATAŞ
- 1
(2n+ 1)\pi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2 (2n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| + o
\bigl(
n - 2
\bigr)
and
\Lambda 2n+1 - \Lambda 2n
\nu 2n+2 - \nu 2n+1
\leq (4n+ 3)\pi 2
a2
+
1
2 (n+ 1)\pi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
4 (n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+
1
(2n+ 1)\pi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2 (2n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| + o
\bigl(
n - 2
\bigr)
.
The following theorem [4] which involves the auxiliary eigenvalues \Lambda n(\tau ) plays an important
role to obtain periodic and semiperiodic eigenvalues.
Theorem 1.2 [4]. The ranges of \Lambda 2n(\tau ) and \Lambda 2n+1(\tau ) as functions of \tau are [\mu 2n, \mu 2n+1] and
[\lambda 2n+1, \lambda 2n+2] , respectively.
By this theorem and the fact that \Lambda n(\tau ) is a continuous function of \tau , we observe that
\mathrm{m}\mathrm{a}\mathrm{x}
\tau
\Lambda 2n(\tau ) = \mu 2n+1, \mathrm{m}\mathrm{i}\mathrm{n}
\tau
\Lambda 2n(\tau ) = \mu 2n,
\mathrm{m}\mathrm{a}\mathrm{x}
\tau
\Lambda 2n+1(\tau ) = \lambda 2n+2, \mathrm{m}\mathrm{i}\mathrm{n}
\tau
\Lambda 2n+1(\tau ) = \lambda 2n+1.
(1.5)
2. Proof of the result. Before we prove the results, we first state the following lemma.
Lemma 2.1. If q(t) is a symmetric single-well potential, then
(i)
\int \tau +a
\tau
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt = 2
\int a/2
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt,
(ii)
\int \tau +a
\tau
q\prime (t) \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt = 0,
(iii)
\int a
0
q2(t)dt = aq2(a) + 2a
\int a/2
0
q(t)q\prime (t)dt - 4
\int a/2
0
tq(t)q\prime (t)dt.
Proof. (i) Since q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
is a periodic function with period a, we get
\tau +a\int
\tau
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt =
a\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt =
=
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt+
a\int
a/2
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt =
=
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt -
a\int
a/2
q\prime (a - t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt =
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
INSTABILITY INTERVALS FOR HILL’S EQUATION WITH SYMMETRIC SINGLE-WELL POTENTIAL 861
= 2
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt.
The last equality holds since q(t) is symmetric and q\prime (t) exists.
(ii) It can be proved similarly.
(iii) By using integration by parts, we have
a\int
0
q2(t)dt = tq2(t)
\bigm| \bigm| a
0
- 2
a\int
0
tq(t)q\prime (t) dt =
= aq2(a) - 2
\left\{
a/2\int
0
tq(t)q\prime (t)dt+
a\int
a/2
tq(t)q\prime (t) dt
\right\} =
= aq2(a) - 2
\left\{
a/2\int
0
tq(t)q\prime (t)dt -
a\int
a/2
tq (a - t) q\prime (a - t) dt
\right\} =
= aq2(a) - 2
\left\{
a/2\int
0
tq(t)q\prime (t)dt+
0\int
a/2
(a - t) q(t)q\prime (t)dt
\right\} =
= aq2(a) + 2a
a/2\int
0
q(t)q\prime (t) dt - 4
a/2\int
0
tq(t)q\prime (t)dt.
Theorem 2.1. The periodic and semiperiodic eigenvalues of (1.1) satisfy, as n \rightarrow \infty ,
\lambda
1/2
2n+1
\lambda
1/2
2n+2
=
2 (n+ 1)\pi
a
\mp a
8 (n+ 1)2 \pi 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
4(n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
- a2
64 (n+ 1)3 \pi 3
\times
\times
\left[ aq2(a) + 2a
a/2\int
0
q(t)q\prime (t)dt - 4
a/2\int
0
tq(t)q\prime (t)dt
\right] + o
\bigl(
n - 3
\bigr)
and
\mu
1/2
2n
\mu
1/2
2n+1
=
(2n+ 1)\pi
a
\mp a
2 (2n+ 1)2 \pi 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(2n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
862 H. COŞKUN, E. BAŞKAYA, A. KABATAŞ
- a2
8 (2n+ 1)3 \pi 3
\times
\times
\left[ aq2(a) + 2a
a/2\int
0
q(t)q\prime (t)dt - 4
a/2\int
0
tq(t)q\prime (t)dt
\right] + o
\bigl(
n - 3
\bigr)
.
Proof. From (1.4) and Lemma 2.1, we observe that
\Lambda 1/2
n (\tau ) =
(n+ 1)\pi
a
+
+
a
2 (n+ 1)2 \pi 2
\mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
2 (n+ 1)\pi
a
\tau
\biggr) a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt -
- a2
8 (n+ 1)3 \pi 3
\times
\times
\left[ aq2(a) + 2a
a/2\int
0
q(t)q\prime (t)dt - 4
a/2\int
0
tq(t)q\prime (t)dt
\right] + o
\bigl(
n - 3
\bigr)
.
If we minimize and maximize the last equation, we find
\mathrm{m}\mathrm{i}\mathrm{n}
\tau
\Lambda 1/2
n (\tau ) =
(n+ 1)\pi
a
- a
2 (n+ 1)2 \pi 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
- a2
8 (n+ 1)3 \pi 3
\times
\times
\left[ aq2(a) + 2a
a/2\int
0
q(t)q\prime (t)dt - 4
a/2\int
0
tq(t)q\prime (t)dt
\right] + o
\bigl(
n - 3
\bigr)
(2.1)
and
\mathrm{m}\mathrm{a}\mathrm{x}
\tau
\Lambda 1/2
n (\tau ) =
(n+ 1)\pi
a
+
a
2 (n+ 1)2 \pi 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2(n+ 1)\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
- a2
8 (n+ 1)3 \pi 3
\times
\times
\left[ aq2(a) + 2a
a/2\int
0
q(t)q\prime (t)dt - 4
a/2\int
0
tq(t)q\prime (t)dt
\right] + o
\bigl(
n - 3
\bigr)
. (2.2)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
INSTABILITY INTERVALS FOR HILL’S EQUATION WITH SYMMETRIC SINGLE-WELL POTENTIAL 863
Now, (1.5), (2.1) and (2.2) prove the theorem.
Corollary 2.1. ln satisfies, as n \rightarrow \infty ,
ln =
2
n\pi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a/2\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
2n\pi
a
t
\biggr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| + o
\bigl(
n - 2
\bigr)
.
Proof. Follows from Theorem 2.1.
Proof of Theorem 1.1. Theorem 2.1, (1.2) and (1.3) are used to prove the theorem.
Corollary 2.2. Let q(t) be a constant. Then, as n \rightarrow \infty ,
\Lambda 2n+1 - \Lambda 2n
\nu 2n+2 - \nu 2n+1
=
(4n+ 3)\pi 2
a2
+ o
\bigl(
n - 2
\bigr)
.
Proof. Follows from Theorem 1.1.
3. An example. To illustrate the foregoing results, we consider an eigenvalue problem
y\prime \prime (t) + (\lambda - q(t)) y(t) = 0, t \in [0, \pi ) ,
where q(t) =
1
4
\Bigl(
t - \pi
2
\Bigr) 4
+
1
2
\Bigl(
t - \pi
2
\Bigr) 2
and extended by periodicity. Since we assumed that q(t)
has mean value zero in our results, we take q(t) as follows:
q(t) =
1
4
\Bigl(
t - \pi
2
\Bigr) 4
+
1
2
\Bigl(
t - \pi
2
\Bigr) 2
- \pi 2
24
- \pi 4
160
.
In this case, by evaluating integral terms in Theorem 1.1, Theorem 2.1 and Corollary 2.1, we obtain,
as n \rightarrow \infty ,
\lambda
1/2
2n+1
\lambda
1/2
2n+2
= 2(n+ 1)\mp \pi 2 + 4
256 (n+ 1)3 \pi
-
- 1
64 (n+ 1)3
1
1290240
\pi 4
\bigl[
35\pi 4 + 384\pi 2 + 1792
\bigr]
+ o
\bigl(
n - 3
\bigr)
,
\mu
1/2
2n
\mu
1/2
2n+1
= 2n+ 1\mp \pi 2 + 4
32 (2n+ 1)3 \pi
-
- 1
8 (2n+ 1)3
1
1290240
\pi 4
\bigl[
35\pi 4 + 384\pi 2 + 1792
\bigr]
+ o
\bigl(
n - 3
\bigr)
,
\Lambda 2n+1 - \Lambda 2n
\nu 2n+2 - \nu 2n+1
\geq 4n+ 3 - \pi 2 + 4
64 (n+ 1)2
- \pi 2 + 4
16 (2n+ 1)2
+ o
\bigl(
n - 2
\bigr)
,
\Lambda 2n+1 - \Lambda 2n
\nu 2n+2 - \nu 2n+1
\leq 4n+ 3 +
\pi 2 + 4
64 (n+ 1)2
+
\pi 2 + 4
16 (2n+ 1)2
+ o
\bigl(
n - 2
\bigr)
,
and
ln+1 =
\pi 2 + 4
8 (n+ 1)2
+ o
\bigl(
n - 2
\bigr)
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
864 H. COŞKUN, E. BAŞKAYA, A. KABATAŞ
References
1. Coşkun H., Harris B. J. Estimates for the periodic and semi-periodic eigenvalues of Hill’s equations // Proc. Roy.
Soc. Edinburgh Sect. A. – 2000. – 130. – P. 991 – 998.
2. Coşkun H. Some inverse results for Hill’ s еquation // J. Math. Anal. and Appl. – 2002. – 276. – P. 833 – 844.
3. Coşkun H. On the spectrum of a second order periodic differential equation // Rocky Mountain J. Math. – 2003. –
33. – P. 1261 – 1277.
4. Eastham M. S. P. The spectral theory of periodic differential equations. – Edinburgh; London: Scottish Acad. Press,
1973.
5. Haaser N. B., Sullivan J. A. Real analysis. – New York: Van Nostrand Reinhold Co., 1991.
6. Hochstadt H. On the determination of a Hill’s equation from its spectrum // Arch. Ration. Mech. and Anal. – 1965. –
19. – P. 353 – 362.
7. Huang M. J. The first instability interval for Hill equations with symmetric single well potentials // Proc. Amer. Math.
Soc. – 1997. – 125. – P. 775 – 778.
8. Huang M. J., Tsai T. M. The eigenvalue gap for one-dimensional Schrödinger operators with symmetric potentials //
Proc. Roy. Soc. Edinburgh Sect. A. – 2009. – 139. – P. 359 – 366.
9. Ntinos A. Lengths of instability intervals of second order periodic differential equations // Quart. J. Math. – 1976. –
27. – P. 387 – 394.
Received 20.07.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
|
| id | umjimathkievua-article-1481 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:34Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6f/07648113bd9a30073fa0cff01265496f.pdf |
| spelling | umjimathkievua-article-14812019-12-05T08:56:42Z Instability intervals for Hill’s equation with symmetric single-well potential Інтервали нестабiльностi для рiвняння Хiлла з симетричним одноямним потенцiалом Başkaya, E. Coşkun, H. Kabataş, A. Башкая, Е. Кошкун, Х. Кабаташ, А. UDC 517.9 We deduce some explicit estimates for the periodic and semiperiodic eigenvalues and the lengths of the instability intervals of Hill’s equation with symmetric single-well potentials by using an auxiliary eigenvalue problem. We also give bounds for the gaps of the Dirichlet and Neumann eigenvalues. УДК 517.9 За допомогою допомiжної задачi на власнi значення отримано деякi явнi оцiнки для перiодичних i напiвперiодичних власних значень та довжин iнтервалiв нестабiльностi для рiвняння Хiлла з симетричним одноямним потенцiалом. Також наведено оцiнки для щiлин у множинах власних значень Дiрiхле та Ноймана. Institute of Mathematics, NAS of Ukraine 2019-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1481 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 6 (2019); 858-864 Український математичний журнал; Том 71 № 6 (2019); 858-864 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1481/465 Copyright (c) 2019 Başkaya E.; Coşkun H.; Kabataş A. |
| spellingShingle | Başkaya, E. Coşkun, H. Kabataş, A. Башкая, Е. Кошкун, Х. Кабаташ, А. Instability intervals for Hill’s equation with symmetric single-well potential |
| title | Instability intervals for Hill’s equation with symmetric
single-well potential |
| title_alt | Інтервали нестабiльностi для рiвняння Хiлла
з симетричним одноямним потенцiалом |
| title_full | Instability intervals for Hill’s equation with symmetric
single-well potential |
| title_fullStr | Instability intervals for Hill’s equation with symmetric
single-well potential |
| title_full_unstemmed | Instability intervals for Hill’s equation with symmetric
single-well potential |
| title_short | Instability intervals for Hill’s equation with symmetric
single-well potential |
| title_sort | instability intervals for hill’s equation with symmetric
single-well potential |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1481 |
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