Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential
UDC 517.9 This paper is devoted to determine the asymptotic formulae for eigenfunctions of the Hill's equation with symmetric single well potential under periodic and semi-periodic boundary conditions.  The obtained results for eigenvalues by H. Coşkun and the othe...
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| author | Kabataş, A. Kabataş, A. |
| author_facet | Kabataş, A. Kabataş, A. |
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| description | UDC 517.9
This paper is devoted to determine the asymptotic formulae for eigenfunctions of the Hill's equation with symmetric single well potential under periodic and semi-periodic boundary conditions.  The obtained results for eigenvalues by H. Coşkun and the others (2019) are used.  With these estimates on the eigenfunctions, Green's functions related to the Hill's equation are obtained.  The method is based on the work of C. T. Fulton (1977) to derive Green's functions in an asymptotical manner.  We need the derivatives of the solutions in this method.  Therefore, the asymptotic approximations for the derivatives of the eigenfunctions are also calculated with different types of restrictions on the potential. |
| doi_str_mv | 10.37863/umzh.v74i2.6246 |
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DOI: 10.37863/umzh.v74i2.6246
UDC 517.9
A. Kabataş (Karadeniz Techn. Univ., Trabzon, Turkey)
EIGENFUNCTION AND GREEN’S FUNCTION ASYMPTOTICS
FOR HILL’S EQUATION WITH SYMMETRIC SINGLE WELL POTENTIAL
АСИМПТОТИКА ВЛАСНОЇ ФУНКЦIЇ ТА ФУНКЦIЇ ГРIНА
ДЛЯ РIВНЯННЯ ХIЛЛА IЗ СИМЕТРИЧНИМ ПОТЕНЦIАЛОМ
ОДНIЄЇ СВЕРДЛОВИНИ
This paper is devoted to determine the asymptotic formulae for eigenfunctions of the Hill’s equation with symmetric single
well potential under periodic and semi-periodic boundary conditions. The obtained results for eigenvalues by H. Coşkun
and the others (2019) are used. With these estimates on the eigenfunctions, Green’s functions related to the Hill’s equation
are obtained. The method is based on the work of C. T. Fulton (1977) to derive Green’s functions in an asymptotical manner.
We need the derivatives of the solutions in this method. Therefore, the asymptotic approximations for the derivatives of
the eigenfunctions are also calculated with different types of restrictions on the potential.
Статтю присвячено встановленню асимптотичних формул для власних функцiй рiвняння Хiлла iз симетричним
потенцiалом однiєї свердловини при перiодичних та напiвперiодичних граничних умовах. При цьому використано
результати для власних значень, отриманi в роботi H. Coşkun та iн. (2019). За допомогою вiдповiдних оцiнок для
власних функцiй отримано функцiї Грiна, пов’язанi з рiвнянням Хiлла. Метод базується на роботi Ч. Т. Фултона
(1977) щодо асимптотичного отримання функцiй Грiна. У цьому методi нам потрiбнi похiднi розв’язкiв, тому
обчислено також асимптотичнi наближення похiдних власних функцiй iз рiзними типами обмежень на потенцiал.
1. Introduction. The Hill’s equation is the second-order linear differential equation
y\prime \prime + q(x)y = 0, (1.1)
where q(x) is a real-valued and periodic function. This equation has numerous applications in
engineering and physics. Some of them contain the problems in mechanics, astronomy, circuits,
electric conductivity of metals, cyclotrons, quadrupole mass spectrometers, quantum optics of two-
level systems and accelerator physics.
Furthermore, the theory related to the Hill’s equation can be extended to every differential equa-
tion written in the general form
a0(x)y
\prime \prime + a1(x)y
\prime + a2(x)y = 0 (1.2)
such that the coefficients a0, a1 and a2 have enough regularity. This is due to the fact that, with
a suitable transformation, (1.2) can be reduced into one of the type of (1.1) (details can be seen in
[10, 11]).
As an example let us consider a mathematical (or inverted) pendulum studied in [9]. If we assume
that the oscillations of the pendulum are small and that the suspension point of the string vibrates
vertically with an acceleration a(t), then, as it is proved in [9], the movement would be modelled by
the equation (which follows the form (1.1))
\theta \prime \prime (t) - 1
l
(g + a(t))\theta (t) = 0,
c\bigcirc A. KABATAŞ, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2 191
192 A. KABATAŞ
where g denotes the gravity, l is the length of the string and \theta represents the angle between the string
and the perpendicular line to the base [4].
Another equation that fit on the framework of Hill’s equation is Mathieu’s equation
y\prime \prime (x) + (a+ b \mathrm{c}\mathrm{o}\mathrm{s}x)y(x) = 0
(see [14]).
At the moment of studying oscillation phenomena of the solutions of (1.1), it is observed that
these are determined by the potential q(x). In particular, solutions of (1.1) do not oscillate when
q(x) < 0 but they do it infinite times for q(x) > 0 large enough. Moreover, the larger the potential
q(x) is, the faster the solutions of (1.1) oscillate [4].
The theory on Hill’s equation takes on a new significance when the equation (1.1) involves a real
parameter \lambda in the form
y\prime \prime + [\lambda - q(x)]y = 0, x \in [0, a]. (1.3)
If we consider (1.3) coupled with suitable boundary value conditions, we have a spectral problem.
We introduce here two eigenvalue problems associated with (1.3) and the interval [0, a], where \lambda
is regarded as the eigenvalue parameter. The periodic eigenvalue problem is defined with (1.3) and
boundary conditions y(0) = y(a), y\prime (0) = y\prime (a) and the semiperiodic eigenvalue problem is given
with (1.3) and boundary conditions y(0) = - y(a), y\prime (0) = - y\prime (a). The periodic and semiperiodic
eigenvalue problems are self-adjoint and they have a countable infinity of eigenvalues denoted by
\lambda n and \mu n, n = 0, 1, 2, . . . , respectively. It is known [11] that the eigenvalues of periodic problem
satisfy
\lambda 0 \leq \lambda 1 \leq \lambda 2 \leq . . . , \lambda n \rightarrow \infty as n \rightarrow \infty
and the eigenvalues of semiperiodic problem satisfy
\mu 0 \leq \mu 1 \leq \mu 2 \leq . . . , \mu n \rightarrow \infty as n \rightarrow \infty .
The periodic and semiperiodic problems with various types of restrictions on the potential have
been widely studied in the literature [1, 2, 6, 7, 11, 15, 17]. Important results about the eigenvalues
and instability intervals were obtained in [1, 6, 13, 16]. The properties of the Green’s functions and
some criteria for the maximum and antimaximum principles were investigated in [3 – 5]. In addition,
Coşkun in [8] has studied on the inverse problem.
Throughout this paper, the equation (1.3) under the periodic and semiperiodic boundary con-
ditions is considered when the potential q(x) is a real-valued, absolutely continuous and periodic
function with period a. Here, the first purpose is to derive asymptotic formulae for the eigenfunc-
tions of the periodic and semiperiodic problems with q(x) 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(x) on [0, a] which is symmetric about x =
a
2
and nonincreasing on
\Bigl[
0,
a
2
\Bigr]
.
It was shown in [6] that the periodic and semiperiodic eigenvalues of (1.3) having symmetric
single well potential are, as n \rightarrow \infty ,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
EIGENFUNCTION AND GREEN’S FUNCTION ASYMPTOTICS FOR HILL’S EQUATION . . . 193
\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(n - 3) (1.4)
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|
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| -
- a2
8(2n+ 1)3\pi 3
\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(n - 3). (1.5)
In this work, the eigenfunctions corresponding to \lambda n and \mu n given by (1.4) and (1.5) are investigated.
The following results obtained in [11] will be used to determine the eigenfunctions.
We define the linearly independent solutions \phi 1(x, \lambda ) and \phi 2(x, \lambda ) of (1.3) with the initial
conditions
\phi 1(0, \lambda ) = 1, \phi \prime
1(0, \lambda ) = 0 (1.6)
and
\phi 2(0, \lambda ) = 0, \phi \prime
2(0, \lambda ) = 1. (1.7)
Theorem 1.1 ([11], \S 4.3). Let \phi 1(x, \lambda ) and \phi 2(x, \lambda ) be the solutions of (1.3) satisfying (1.6)
and (1.7), respectively. Assume that q(x) is an absolutely continuous function. Then, as \lambda \rightarrow \infty ,
\phi 1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
\lambda - 1
2Q(x) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 1
\biggl\{
q(x) - q(0) - 1
2
Q2(x)
\biggr\}
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1), (1.8)
\phi 2(x, \lambda ) = \lambda - 1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
- 1
2
\lambda - 1Q(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 3
2
\biggl\{
q(x) + q(0) - 1
2
Q2(x)
\biggr\}
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 3
2 ), (1.9)
where
Q(x) =
x\int
0
q(t)dt. (1.10)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
194 A. KABATAŞ
Theorem 1.2 ([11], \S 4.3). Let \phi 1(x, \lambda ) and \phi 2(x, \lambda ) be the solutions of (1.3) satisfying (1.6)
and (1.7), respectively. Assume that q(x) is a piecewise continuous function. Then, as \lambda \rightarrow \infty ,
\phi 1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ \lambda - 1
2
x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )dt+
+\lambda - 1
x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)dt
t\int
0
\mathrm{s}\mathrm{i}\mathrm{n}\{ (t - u)
\surd
\lambda \} q(u) \mathrm{c}\mathrm{o}\mathrm{s}(u
\surd
\lambda )du+O(\lambda - 3
2 ), (1.11)
\phi 2(x, \lambda ) = \lambda - 1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ \lambda - 1
x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{s}\mathrm{i}\mathrm{n}(t
\surd
\lambda )dt+
+\lambda - 3
2
x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)dt
t\int
0
\mathrm{s}\mathrm{i}\mathrm{n}\{ (t - u)
\surd
\lambda \} q(u) \mathrm{s}\mathrm{i}\mathrm{n}(u
\surd
\lambda )du+O(\lambda - 2). (1.12)
The second goal of this paper is to determine the Green’s function asymptotics related to the Hill’s
equation with the estimates on the eigenfunctions. The method developed by Fulton [12] is followed.
In this method since the derivatives of the solutions are needed, the asymptotic approximations for
the derivatives of \phi 1(x, \lambda ) and \phi 2(x, \lambda ) are also calculated with different types of restrictions on the
potential q(x).
2. Approximations for the eigenfunctions. In this section, we obtain approximations for the
solutions \phi 1(x, \lambda ) and \phi 2(x, \lambda ) of (1.3) satisfying the initial conditions (1.6) and (1.7), respectively.
Before, we give the following lemma for q(x) being of a symmetric single well potential.
Lemma 2.1. If q(x) is a symmetric single well potential on [0, a], then
x\int
0
q(t)dt = xq(x) +
a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt. (2.1)
Proof. Using integration by parts gives
x\int
0
q(t)dt = tq(t)
\bigm| \bigm| \bigm| x
t=0
-
x\int
0
tq\prime (t)dt =
= xq(x) -
\left[ a/2\int
0
tq\prime (t)dt+
x\int
a/2
tq\prime (t)dt
\right] = xq(x) -
\left[ - a/2\int
0
tq\prime (a - t)dt+
x\int
a/2
tq\prime (t)dt
\right] =
= xq(x) -
a/2\int
a
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt = xq(x) +
a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
EIGENFUNCTION AND GREEN’S FUNCTION ASYMPTOTICS FOR HILL’S EQUATION . . . 195
Theorem 2.1. Let q(x) be a symmetric single well potential on [0, a]. Then, as \lambda \rightarrow \infty , for the
solutions of (1.3) with the initial conditions (1.6) and (1.7), respectively, we have
\phi 1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
\lambda - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{s}\mathrm{i}\mathrm{n} \bigl( x\surd \lambda
\bigr)
+
+
1
4
\lambda - 1
\left\{ q(x) - q(0) - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1),
(2.2)
\phi 2(x, \lambda ) = \lambda - 1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
- 1
2
\lambda - 1
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{c}\mathrm{o}\mathrm{s} \bigl( x\surd \lambda
\bigr)
+
+
1
4
\lambda - 3
2
\left\{ q(x) + q(0) - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 3
2 ).
(2.3)
Proof. If we use Theorem 1.1 and substitute (2.1) in (1.10), the proof is done.
Theorem 2.2. The eigenfunctions of the periodic problem having symmetric single well potential
satisfy, as n \rightarrow \infty ,
\phi 1(x, n) = \mathrm{c}\mathrm{o}\mathrm{s}
2(n+ 1)\pi x
a
+
+
a
4(n+ 1)\pi
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{s}\mathrm{i}\mathrm{n} 2(n+ 1)\pi x
a
+
+
a2
16(n+ 1)2\pi 2
\left\{ q(x) - q(0) - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \times
\times \mathrm{c}\mathrm{o}\mathrm{s}
2(n+ 1)\pi x
a
+ o(n - 2),
\phi 2(x, n) =
a
2(n+ 1)\pi
\mathrm{s}\mathrm{i}\mathrm{n}
2(n+ 1)\pi x
a
-
- a2
8(n+ 1)2\pi 2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{c}\mathrm{o}\mathrm{s} 2(n+ 1)\pi x
a
+
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
196 A. KABATAŞ
+
a3
32(n+ 1)3\pi 3
\left\{ q(x) + q(0) - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \times
\times \mathrm{s}\mathrm{i}\mathrm{n}
2(n+ 1)\pi x
a
+ o(n - 3).
Theorem 2.3. The eigenfunctions of the semiperiodic problem having symmetric single well
potential satisfy, as n \rightarrow \infty ,
\phi 1(x, n) = \mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi x
a
+
+
a
2(2n+ 1)\pi
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{s}\mathrm{i}\mathrm{n} (2n+ 1)\pi x
a
+
+
a2
4(2n+ 1)2\pi 2
\left\{ q(x) - q(0) - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \times
\times \mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi x
a
+ o(n - 2),
\phi 2(x, n) =
a
(2n+ 1)\pi
\mathrm{s}\mathrm{i}\mathrm{n}
(2n+ 1)\pi x
a
-
- a2
2(2n+ 1)2\pi 2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi x
a
+
+
a3
4(2n+ 1)3\pi 3
\left\{ q(x) + q(0) - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \times
\times \mathrm{s}\mathrm{i}\mathrm{n}
(2n+ 1)\pi x
a
+ o(n - 3).
To prove Theorems 2.2 and 2.3, the related eigenvalues given by (1.4) and (1.5) are substituted
in Theorem 2.1.
We have also some approximations for the derivatives of \phi 1(x, \lambda ) and \phi 2(x, \lambda ). We will use
them in calculation of the Green’s functions.
Lemma 2.2. If q(x) is a piecewise continuous function, then the derivative of (1.11) is, as
\lambda \rightarrow \infty ,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
EIGENFUNCTION AND GREEN’S FUNCTION ASYMPTOTICS FOR HILL’S EQUATION . . . 197
\phi \prime
1(x, \lambda ) = - \lambda
1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )dt+
+\lambda - 1
2
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)dt
t\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(t - u)
\surd
\lambda
\bigr\}
q(u) \mathrm{c}\mathrm{o}\mathrm{s}(u
\surd
\lambda )du+O(\lambda - 1). (2.4)
Proof. The usual variation of constants formula [10] (\S 2.5) gives
\phi 1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ \lambda - 1
2
x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)\phi 1(t, \lambda )dt.
If we arrange this formula, one can write
\phi 1(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
+\lambda - 1
2
\left\{ \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr) x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )q(t)\phi 1(t, \lambda )dt - \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr) x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}(t
\surd
\lambda )q(t)\phi 1(t, \lambda )dt
\right\} . (2.5)
It is obtained by differentiating (2.5) with respect to x and substituting \phi 1(t, \lambda ) from (1.11) in the
integral that
\phi \prime
1(x, \lambda ) = - \lambda
1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ \lambda - 1
2
\left\{ \lambda
1
2 \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr) x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )q(t)\phi 1(t, \lambda )dt+
+\lambda
1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr) x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}(t
\surd
\lambda )q(t)\phi 1(t, \lambda )dt
\right\} =
= - \lambda
1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)\phi 1(t, \lambda )dt =
= - \lambda
1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )dt+
+\lambda - 1
2
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)dt
t\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(t - u)
\surd
\lambda
\bigr\}
q(u) \mathrm{c}\mathrm{o}\mathrm{s}(u
\surd
\lambda )du+O(\lambda - 1).
In [11] (\S 4.3), it is determined similarly
\phi \prime
2(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ \lambda - 1
2
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{s}\mathrm{i}\mathrm{n}(t
\surd
\lambda )dt+
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
198 A. KABATAŞ
+\lambda - 1
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)dt
t\int
0
\mathrm{s}\mathrm{i}\mathrm{n}\{ (t - u)
\surd
\lambda \} q(u) \mathrm{s}\mathrm{i}\mathrm{n}(u
\surd
\lambda )du+O(\lambda - 3
2 ).
Lemma 2.3. Let q(x) be absolutely continuous. Then the derivatives of (1.8) and (1.9) are, as
\lambda \rightarrow \infty ,
\phi \prime
1(x, \lambda ) = - \lambda
1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
Q(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 1
2
\biggl\{
q(x) + q(0) +
1
2
Q2(x)
\biggr\}
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1
2 ), (2.6)
\phi \prime
2(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
\lambda - 1
2Q(x) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
-
- 1
4
\lambda - 1
\biggl\{
q(x) - q(0) +
1
2
Q2(x)
\biggr\}
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1). (2.7)
Proof. If q(x) is absolutely continuous, this implies that q\prime (x) exists p.p. and is integrable.
Under these conditions, let consider the second term on the right-hand side of (2.4). We have
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )dt =
=
1
2
x\int
0
\Bigl[
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ \mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - 2t)
\surd
\lambda
\bigr\} \Bigr]
q(t)dt =
=
1
2
Q(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - 2t)
\surd
\lambda
\bigr\}
q(t)dt =
=
1
2
Q(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
\left[ - 1
2
\lambda - 1
2 q(t) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - 2t)
\surd
\lambda
\bigr\} \bigm| \bigm| \bigm| x
t=0
+
+
1
2
\lambda - 1
2
x\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - 2t)
\surd
\lambda
\bigr\}
dt
\right] =
=
1
2
Q(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
4
\lambda - 1
2
\bigl[
q(x) + q(0)
\bigr]
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 1
2
x\int
0
q\prime (t) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - 2t)
\surd
\lambda
\bigr\}
dt.
The last integral on the right-hand side is o(1) as \lambda \rightarrow \infty by the Riemann – Lebesgue lemma. So,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
EIGENFUNCTION AND GREEN’S FUNCTION ASYMPTOTICS FOR HILL’S EQUATION . . . 199
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )dt =
=
1
2
Q(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
4
\lambda - 1
2
\bigl[
q(x) + q(0)
\bigr]
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1
2 ). (2.8)
Also, from [11] (\S 4.3)
x\int
0
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t) \mathrm{c}\mathrm{o}\mathrm{s}(t
\surd
\lambda )dt =
=
1
2
Q(x) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
1
4
\lambda - 1
2
\bigl[
q(x) - q(0)
\bigr]
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1
2 ). (2.9)
For the third term on the right-hand side of (2.4), together with (2.9) we find
\lambda - 1
2
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)dt
t\int
0
\mathrm{s}\mathrm{i}\mathrm{n}\{ (t - u)
\surd
\lambda \} q(u) \mathrm{c}\mathrm{o}\mathrm{s}(u
\surd
\lambda )du =
=
1
2
\lambda - 1
2
x\int
0
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl\{
(x - t)
\surd
\lambda
\bigr\}
q(t)Q(t) \mathrm{s}\mathrm{i}\mathrm{n}(t
\surd
\lambda )dt+O(\lambda - 1) =
=
1
4
\lambda - 1
2
x\int
0
\Bigl[
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
- \mathrm{s}\mathrm{i}\mathrm{n}
\bigl\{
(x - 2t)
\surd
\lambda
\bigr\} \Bigr]
q(t)Q(t)dt+O(\lambda - 1) =
=
1
4
\lambda - 1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr) \biggl[ Q2(t)
2
\biggr] \bigm| \bigm| \bigm| \bigm| x
t=0
+ o(\lambda - 1
2 ) =
=
1
8
\lambda - 1
2Q2(x) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1
2 ), (2.10)
again by using the Riemann – Lebesgue lemma. (2.8) and (2.10) prove (2.6). The proof of (2.7) is
similar.
Lemma 2.4. Consider the equation (1.3) having symmetric single well potential. As \lambda \rightarrow \infty ,
for the derivatives of its solutions \phi 1(x, \lambda ) and \phi 2(x, \lambda ) which satisfy (1.6) and (1.7), respectively,
we have
\phi \prime
1(x, \lambda ) = - \lambda
1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{c}\mathrm{o}\mathrm{s} \bigl( x\surd \lambda
\bigr)
+
+
1
4
\lambda - 1
2
\left\{ q(x) + q(0) +
1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1
2 ),
(2.11)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
200 A. KABATAŞ
\phi \prime
2(x, \lambda ) = \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
1
2
\lambda - 1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right] \mathrm{s}\mathrm{i}\mathrm{n} \bigl( x\surd \lambda
\bigr)
-
- 1
4
\lambda - 1
\left\{ q(x) - q(0) +
1
2
\left[ xq(x) + a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt
\right]
2\right\} \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1).
(2.12)
Proof. By using (2.1) in (1.10) and substituting this in (2.6) and (2.7), we prove the lemma.
3. Approximations for Green’s functions. In this section, we aim to improve asymptotic
formulae for Green’s functions of the periodic and semiperiodic problems with symmetric single
well potential. The Green’s function G(x, \xi , \lambda ) is given by
G(x, \xi , \lambda ) =
\left\{
\phi 1(\xi , \lambda )\phi 2(x, \lambda )
w(\lambda )
, 0 \leq \xi \leq x \leq a,
\phi 1(x, \lambda )\phi 2(\xi , \lambda )
w(\lambda )
, 0 \leq x \leq \xi \leq a
(3.1)
(see [12]). Here, \phi 1(x, \lambda ) and \phi 2(x, \lambda ) are linearly independent solutions of (1.3) satisfying (1.6)
and (1.7), respectively. And, we define w(\lambda ) as follows:
w(\lambda ) := \phi 1(x, \lambda )\phi
\prime
2(x, \lambda ) - \phi \prime
1(x, \lambda )\phi 2(x, \lambda ). (3.2)
It is known as the Wronskian function of \phi 1(x, \lambda ) and \phi 2(x, \lambda ).
Theorem 3.1. Suppose that the equation (1.3) has the symmetric single well potential and its
independent solutions \phi 1(x, \lambda ) and \phi 2(x, \lambda ) satisfy the initial conditions (1.6) and (1.7), respectively.
Then the Green’s function of the problem is, as \lambda \rightarrow \infty ,
G(x, \xi , \lambda ) = \lambda - 1
2 \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
-
- 1
2
\lambda - 1
\Bigl[
A(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
- A(\xi ) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr) \Bigr]
+
+
1
4
\lambda - 3
2
\biggl\{ \biggl[
q(\xi ) + q(x) - 1
2
\bigl(
A2(\xi ) +A2(x)
\bigr) \biggr]
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
-
- A(\xi )A(x) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr) \biggr\}
+ o(\lambda - 3
2 ), 0 \leq \xi \leq x \leq a,
where
A(x) := xq(x) +
a\int
a/2
(a - t)q\prime (t)dt -
x\int
a/2
tq\prime (t)dt.
Similar result holds for 0 \leq x \leq \xi \leq a changing the role of \xi and x.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
EIGENFUNCTION AND GREEN’S FUNCTION ASYMPTOTICS FOR HILL’S EQUATION . . . 201
Proof. We begin to the proof by evaluating the Wronskian function w(\lambda ). For this, we substitute
(2.2), (2.3), (2.11) and (2.12) into (3.2). Hence,
w(\lambda ) = 1 - 1
4
\lambda - 1
\biggl[
q(x) - q(0) +
1
2
A2(x)
\biggr]
\mathrm{c}\mathrm{o}\mathrm{s}2
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 1
\biggl[
q(x) + q(0) - 1
2
A2(x)
\biggr]
\mathrm{s}\mathrm{i}\mathrm{n}2
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 1A2(x) +
1
4
\lambda - 1
\biggl[
q(x) - q(0) - 1
2
A2(x)
\biggr]
\mathrm{c}\mathrm{o}\mathrm{s}2
\bigl(
x
\surd
\lambda
\bigr)
-
- 1
4
\lambda - 1
\biggl[
q(x) + q(0) +
1
2
A2(x)
\biggr]
\mathrm{s}\mathrm{i}\mathrm{n}2
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 1) =
= 1 - 1
4
\lambda - 1A2(x) +
1
4
\lambda - 1A2(x) + o(\lambda - 1) = 1 + o(\lambda - 1).
From that, we can write
1
w(\lambda )
=
1
1 + o(\lambda - 1)
= 1 + o(\lambda - 1). (3.3)
Finally, by using (2.2), (2.3), (3.3) in (3.1), we find
\phi 1(\xi , \lambda )\phi 2(x, \lambda )
w(\lambda )
=
\biggl\{
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
+
1
2
\lambda - 1
2A(\xi ) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
\xi
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 1
\biggl[
q(\xi ) - q(0) - 1
2
A2(\xi )
\biggr]
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
+ o(\lambda - 1)
\biggr\}
\times
\times
\biggl\{
\lambda - 1
2 \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
- 1
2
\lambda - 1A(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 3
2
\biggl[
q(x) + q(0) - 1
2
A2(x)
\biggr]
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 3
2 )
\biggr\} \bigl\{
1 + o(\lambda - 1)
\bigr\}
=
=
\Biggl\{
\lambda - 1
2 \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
- 1
2
\lambda - 1A(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 3
2
\biggl[
q(x) + q(0) - 1
2
A2(x)
\biggr]
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
2
\lambda - 1A(\xi ) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
- 1
4
\lambda - 3
2A(\xi )A(x) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
+
+
1
4
\lambda - 3
2
\biggl[
q(\xi ) - q(0) - 1
2
A2(\xi )
\biggr]
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
+ o(\lambda - 3
2 )
\Biggr\} \bigl\{
1 + o(\lambda - 1)
\bigr\}
=
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
202 A. KABATAŞ
= \lambda - 1
2 \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
- 1
2
\lambda - 1
\Bigl[
A(x) \mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr)
- A(\xi ) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr) \Bigr]
+
+
1
4
\lambda - 3
2
\Biggl\{ \biggl[
q(\xi ) + q(x) - 1
2
\bigl(
A2(\xi ) +A2(x)
\bigr) \biggr]
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
x
\surd
\lambda
\bigr)
-
- A(\xi )A(x) \mathrm{s}\mathrm{i}\mathrm{n}
\bigl(
\xi
\surd
\lambda
\bigr)
\mathrm{c}\mathrm{o}\mathrm{s}
\bigl(
x
\surd
\lambda
\bigr) \Biggr\}
+ o(\lambda - 3
2 ).
Theorem 3.2. Green’s functions of the periodic problem with symmetric single well potential
satisfy, as n \rightarrow \infty ,
G(x, \xi , n) =
a
2(n+ 1)\pi
\mathrm{c}\mathrm{o}\mathrm{s}
2(n+ 1)\pi \xi
a
\mathrm{s}\mathrm{i}\mathrm{n}
2(n+ 1)\pi x
a
- a2
8(n+ 1)2\pi 2
\times
\times
\biggl[
A(x) \mathrm{c}\mathrm{o}\mathrm{s}
2(n+ 1)\pi \xi
a
\mathrm{c}\mathrm{o}\mathrm{s}
2(n+ 1)\pi x
a
- A(\xi ) \mathrm{s}\mathrm{i}\mathrm{n}
2(n+ 1)\pi \xi
a
\mathrm{s}\mathrm{i}\mathrm{n}
2(n+ 1)\pi x
a
\biggr]
+
+
a3
32(n+ 1)3\pi 3
\biggl\{ \biggl[
q(\xi ) + q(x) - 1
2
\bigl(
A2(\xi ) +A2(x)
\bigr) \biggr]
\times
\times \mathrm{c}\mathrm{o}\mathrm{s}
2(n+ 1)\pi \xi
a
\mathrm{s}\mathrm{i}\mathrm{n}
2(n+ 1)\pi x
a
- A(\xi )A(x) \mathrm{s}\mathrm{i}\mathrm{n}
2(n+ 1)\pi \xi
a
\mathrm{c}\mathrm{o}\mathrm{s}
2(n+ 1)\pi x
a
\biggr\}
+ o(n - 3)
for 0 \leq \xi \leq x \leq a. Similar result holds for 0 \leq x \leq \xi \leq a changing the role of \xi and x.
Theorem 3.3. Green’s functions of the semiperiodic problem with symmetric single well poten-
tial satisfy, as n \rightarrow \infty ,
G(x, \xi , n) =
a
(2n+ 1)\pi
\mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi \xi
a
\mathrm{s}\mathrm{i}\mathrm{n}
(2n+ 1)\pi x
a
-
- a2
2(2n+ 1)2\pi 2
\biggl[
A(x) \mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi \xi
a
\mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi x
a
-
- A(\xi ) \mathrm{s}\mathrm{i}\mathrm{n}
(2n+ 1)\pi \xi
a
\mathrm{s}\mathrm{i}\mathrm{n}
(2n+ 1)\pi x
a
\biggr]
+
+
a3
4(2n+ 1)3\pi 3
\Biggl\{ \biggl[
q(\xi ) + q(x) - 1
2
\bigl(
A2(\xi ) +A2(x)
\bigr) \biggr]
\mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi \xi
a
\mathrm{s}\mathrm{i}\mathrm{n}
(2n+ 1)\pi x
a
-
- A(\xi )A(x) \mathrm{s}\mathrm{i}\mathrm{n}
(2n+ 1)\pi \xi
a
\mathrm{c}\mathrm{o}\mathrm{s}
(2n+ 1)\pi x
a
\Biggr\}
+ o(n - 3)
for 0 \leq \xi \leq x \leq a. Similar result holds for 0 \leq x \leq \xi \leq a changing the role of \xi and x.
To prove Theorems 3.2 and 3.3, the related eigenvalues given by (1.4) and (1.5) are used together
with Theorem 3.1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
EIGENFUNCTION AND GREEN’S FUNCTION ASYMPTOTICS FOR HILL’S EQUATION . . . 203
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Received 28.07.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
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| id | umjimathkievua-article-6246 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:41Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c8/22f60bf7abb6f33d56ef938922ebb6c8.pdf |
| spelling | umjimathkievua-article-62462025-03-31T08:45:58Z Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential Kabataş, A. Kabataş, A. Hill's equation, symmetric single well potential, periodic and semi-periodic eigenfunctions, Green's functions, asymptotics UDC 517.9 This paper is devoted to determine the asymptotic formulae for eigenfunctions of the Hill's equation with symmetric single well potential under periodic and semi-periodic boundary conditions.&nbsp;&nbsp;The obtained results for eigenvalues by H. Coşkun and the others (2019) are used.&nbsp;&nbsp;With these estimates on the eigenfunctions, Green's functions related to the Hill's equation are obtained.&nbsp;&nbsp;The method is based on the work of C. T. Fulton (1977) to derive Green's functions in an asymptotical manner.&nbsp;&nbsp;We need the derivatives of the solutions in this method.&nbsp;&nbsp;Therefore, the asymptotic approximations for the derivatives of the eigenfunctions are also calculated with different types of restrictions on the potential. УДК 517.9Aсимптотика власної функцiї та функцiї Грiна для рiвняння Хiлла iз симетричним потенцiалом з однiєю ямою Статтю присвячено встановленню асимптотичних формул для власних функцій рівняння Хілла із симетричним потенціалом однієї свердловини при періодичних та напівперіодичних граничних умовах.&nbsp;&nbsp;При цьому використано результати для власних значень, отримані в роботі H. Coşkun та ін. (2019).&nbsp;&nbsp;За допомогою відповідних оцінок для власних функцій отримано функції Гріна, пов’язані з рівнянням Хілла.&nbsp;&nbsp;Метод базується на роботі Ч. Т. Фултона (1977) щодо асимптотичного отримання функцій Гріна.&nbsp;&nbsp;У цьому методі нам потрібні похідні розв’язків,&nbsp;&nbsp;тому обчислено також асимптотичні наближення похідних власних функцій із різними типами обмежень на потенціал. Institute of Mathematics, NAS of Ukraine 2022-02-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6246 10.37863/umzh.v74i2.6246 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 2 (2022); 191 - 203 Український математичний журнал; Том 74 № 2 (2022); 191 - 203 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6246/9191 Copyright (c) 2022 Ayşe Kabataş |
| spellingShingle | Kabataş, A. Kabataş, A. Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential |
| title | Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential |
| title_alt | Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential |
| title_full | Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential |
| title_fullStr | Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential |
| title_full_unstemmed | Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential |
| title_short | Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential |
| title_sort | eigenfunction and green’s function asymptotics for hill’s equation with symmetric single well potential |
| topic_facet | Hill's equation symmetric single well potential periodic and semi-periodic eigenfunctions Green's functions asymptotics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6246 |
| work_keys_str_mv | AT kabatasa eigenfunctionandgreensfunctionasymptoticsforhillsequationwithsymmetricsinglewellpotential AT kabatasa eigenfunctionandgreensfunctionasymptoticsforhillsequationwithsymmetricsinglewellpotential |