CC BY
10
y^K 004.93
ON THE NORMING CONSTANTS OF THE STURM-LIOU VILLE PROBLEM
*Harutyunyan T., [email protected] **Pahlevanyan A., [email protected]
*Faculty of Mathematics and Mechanics, Yerevan State University, 1 Alex Manoogian, 0025, Yerevan, Armenia * "Institute of Mathematics of National Academy of Sciences, 24/5 Baghramianave,
0019, Yerevan, Armenia
We derive new asymptotic formulae for the norming constants of Sturm-Liouville problem with summable potentials, which generalize and make more precise previously known formulae. Moreover, our formulae take into account the smooth dependence of norming constants on boundary conditions. We also find some new properties of the remainder terms of asymptotics.
Keywords: Sturm-Liouville problem, norming constants, asymptotics of the solutions, asymptotics of spectral data.
1. Introduction and Statement of the Results
Let L(q, a ,/?) denote the Sturm-Liouville problem
wher eq is a real-valued, summable function on [0, n] (we write q G n])- By L(q, a,(3) we also denote the self-adjoint operator, generated by the problem (1.1)—(1.3) in Hilbert space L2[0,n] (see [1; 2]). It is well-known that the spectra of L(q,a,(3) is discrete and consists of real, simple eigenvalues (see [1; 2; 3]), which we denote by ¡.i7,(q, a,(3), n = 0,1,2 emphasizing the dependence of on q, a and /?. For ^ the following asymptotic formula have been proven in [4]:
— y" + q(x)y = piy, x e (0, n),ц e €, y(0) cos a + y'(0) sin a = 0, a e (0,7r], у(л) cos/3 + у'(л) sin/? = 0,/? e [0,7Г),
(1.1) (1.2) (1.3)
7
1 г
lin{q,a,(3) = [n + ôn(a,p)]2 +- I q(t) dt + rn(q,a,p), (1.4)
n ) 0
where Sn is the solution of the equation (for n > 2)
1 cos a
ôn(a,p) = — arccos-
7Г
2
(n + <5n(a,/?)) sin2 a + cos2 a
1 cos /? ---arccos —
(1.5)
П +0п(а,р)У sin2/I + cos2/I
and rn(q,a,P) = o( 1), when n-> co, uniformly in a, /? e [0,7r] and q from any bounded subset of ¿^[0,n] (we will write q E 7r]). It
follows from (1.5) (for details see [4]), that
cot/? — cot a
8n(a,p) =---+ 0(l/n ), a,p (1.5a)
nn
1 cot/? , 1
= ? + / A + 0(l/n ) = - + 0(l/n),
2 n(n + \) 2 (1.5b)
p e (O.tt),
1 cot a „ 1 <*„(«, 0) = ---?-- + 0(l/n2) = - + 0(l/n),
2
a G (0,7г),
7r(n+i) 2 ' (1.5c)
5n (jl, 0) = 1. (1.5d)
Let y = (p{x,fi, a, q) and y = xpÇx.fÂ.p, q) be the solutions of (1.1) with initial values
<p{0, [i, a, q) = sin a, <p'(0, a,q) = — cos a, ip(n, il, p, q) = sin /?, xp'Çîi, [i, p,q) = — cos /?.
8
The eigenvalues ¡J.n of L{q,a,(i) are the solutions of the equation
<J>(X) = (p(jz, [i, a, q) cos /? + (p'(jz, a, q) sin/? = = — [ip(0, q) sin a + ip'(0,[i,(3, q) cos a] = 0.
It is easy to see that for arbitrary n = 0,1,2,..., (pn(x): = = (p(x,fin (q, a,(3), a, q) and i/>n(x): = i(j(x,fin(q, a,(3),(3, q) are
eigenfunctions, corresponding to the eigenvalue nn(q,a,(3). Thes quares of the L2-norm of these eigenfunctions:
7T 7T
an(q,a,/3)\= J\<pn(x)\2 dx, bn{q,a,f3)\= j\ipn(x)\2 dx o o
are called the norming constants.
The main results of this paper are the following theorems: Theorem 1.1. For norming constants an and bn the following asymptotic formulae hold {when n -» oo):
л
<*Ti(q.a.p) = 2
2 xn(q,a,(3) 1 + —?-——^ + rn
л [n + 0(а,р))
sin2 a +
+
л
2 [n + 5n(a,pW
1 +
+ r„
л
bn(q,a,(3) = -
2 xn{q,a,(3) 1 + —-——— + pn
л [n + S(a,(3)]
2xn(q, a,(3) л[п + ô(a,(3)]
sin2 p +
(1.6)
2
cos a,
+
л
2 [п + 0п(а,Ю]2
2кп(д, a,p) _ л[п + 5(а,р)] Pn
cos2 /?,
where
71
1 Г
кп = Kn(q,a,(3) = —— I (л — t)q(t) sin2[n + ôn(a,(3)] t dt, (1.7)
2 J о
9
rn = rn(q,a,(3) = 0 (J^j and fn = fn(q,a,(3) = 0 (the same estimate is true for pn and prh), when n -» oo, uniformly in a,(3 G [0,7r] and qEBLUO.n],
Theorem 1.2. For both a, (3 G (0,7r) and a = n, /? = 0 cases function k, defined as the series
00
ZKn
——ir——^cos[n + 8n(a,p)]x n + on(a, B)
n=2
is absolutely continuous function on arbitrary segment [a, b] a (0,2n), i.e. k G /4C(0,2n).
The dependence of norming constants on a and p (as far as we know) hasn't been investigated before. The dependence of spectral data (by spectral data here we understand the set of eigenvalues and the set of norming constants) on a and p has been usually studied (see [13], [5-9]) in the following sense: the boundary conditions are separated into four cases:
1) sin cc 0, sinp 0, i.e.a,/? G (0,7r);
2) sin a = 0, sinp 0, i.e.a = n,p G (0,7r);
3) sin a * 0, sin p = 0, i.e.a G (0,7r), p = 0;
4) sin a = 0, sinp = 0, i.e.a = n, p = 0,
5)
and results are formulated separately for each case. For eigenvalues, formula (1.4) generalizes and unites four different formulae that were known before in four mentioned cases (see [4]).
So far, for norming constants the following is known.
In the case sin a 0 it is known that for smooth q
an(q,a,p)^n + o /J_y g)
sin2 a 2 Vn2/
10
For absolutely continuous q (we will write q E. AC[0,n]) the proof of (1.8) can be found in [2]). Let us note, that if q E AC[0,n],
then xn = 0 Q), and it is easy to see, that in this case (1.6) takes the
dF (x)
form (1.8). In [10], under the condition q(x) = ^ (almost everywhere), and sin or 0, where F is a function of bounded variation (we will write F G BV[0,n]), the author asserts that
an(q,a,P) n
-—-=n + an> (1 9)
sirrcr 2
where the sequence {a7i}^=o is characterized by the condition that the function f(x\. = Sn^o^ii cosnx has a bounded variation on [0, n], i.e. / G BV[0,n]. Our result is similar to this, but there are some differences, in particular, we assert that k G /4C(0, 2n). In [9], for q G Ljj[0, n], it was proved that
an{q,a,p) _n ^ Kn (110)
sin2 a 2 n'
where {Kn}£U G I2 (i.e. £n=oknl2 < oo), and Kn = xn + 0 (see (1.7)).
It is also important to note that norming constants an(q,a,P) are analytic functions on a and /?. It easily follows from formulae (3.1), (3.2) and (3.4) below and from the result in [4], which states that An(q,a,(J) (An(q,a,p) = [in(q,a,(3)) depend analytically on a and /?.
In the case sin a = 0, sin /? 0 it is known that for smooth q (for q G AC[0, n] the proof of (1.11) can be found in [2])
7Г
an(q,n,(3) =
2 (n + 1/2)2
l + O
Ш
(1.11)
11
Since 5n(n,(3) = ^ + 0 ^ (see (1.5b)), then it is easy to see that
(1.11) follows from (1.6). Besides, we see that (1.6) smoothly turns into (1.11) when a -» n (for q E AC[0, n]).
In the case sin a = 0,sin/? = 0 the following result can be found in [2] for q E AC[0,n]:
an(q>TC, 0) =
л
2 n2
1 + 0
(n2).
We think that it is more correct to write this result in the form (note that Sn(n,0) = 1)
an(q, л, 0) =
л
2 (n + l)2
1 + 0
(n2).
(1.12)
to keep the beginning of the enumeration of eigenvalues and norming constants starting from 0, but not from 1, as in [2].
Our proofs of the theorems are based on the detailed study of the dependence of eigenfunctions (p7l and ipn on parameters cr and p. We will present it in the sections 3 and 4. But first we need to prove some properties of the solutions of the equation (1.1).
2. Asymptotics of the solutions
Let q E Lj;[0,n], i.e. q is a complex-valued, summable function on [0,7r], and let us denote by yi(x,A), i = 1,2, 3,4, the solutions of the equation
-y" + q(x)y = Л2у, Satisfying the initial conditions
(2.1)
yi(O.A) = L yi(0.A) = 0,
y2(CU) = 0, y5( 0Д) = 1,
Уз(7Г,Я) = 1,
УзОД) = 0,
у4(л,Л) = 0,
У4 (ТС, Я) = 1.
(2.2)
12
Let us recall that by a solution of (2.1) (which is the same as (1.1)) we understand the function y, such that y, y' Gi4C[0,7r] and which satisfies (2.1) almost everywhere (see [1]).
The solutions yx and y2 (as well as the second pair y3 and y4) form a fundamental system of solutions of (1.1), i.e. any solution y of (1.1) can be represented in the form:
y(x) = y(0)y1(x,l) + y'(0)y2(x,X) = (2
= + y'(n)y4(x,A).
The existence and uniqueness of the solutions yL, i = 1,2,3,4 (under the condition q GL^[0,7t]) were investigated in [1], [11-14]. The following lemma in some sense extends the results of the mentioned papers related to asymptotics (when|A| -> oo) of the solutions y/, i = 1, 2, 3,4.
Lemma 2.1. Let q G Lf [0, n]. Then for the solutions yit i = = 1,2,3,4, the following representations hold {when |2| > 1):
1
y1(x,l) = cosAx + — a(x,A), (2.4)
2A
sin Ax 1
y2(Xjl)=_--—b(x,A), (2.5)
1
y3(x,l) = cos A(n — x) + — c(x,l), (2.6)
ZA
sin A(jz — x) 1 y4(x,l) =----d(x,A), (2 7)
where a, b, c, d are twice differentiable with respect to x and entire functions with respect to A, and have the form
Л Л
а(х,Я) = sin Ax J q(t)dt + J q(t) sinA(x — 21) dt + (2.8)
13
+R1(x, A,q),
X X
J J 1 CZV)
о 0
b(x. A) = cos Ax I q(t) dt - f q(t) cos A(x-2t)dt +
0
+ R2(x,A,q),
71 71
c{x, A) = sinA(n — x) J q(t)dt + J q(t) sin A(2t — n — x) dt + (2 10)
X X
+R3(x, A,q),
71 71
d(x, X) = cos A(n — x) | q(t)dt + J q(t) cos A(n + x — 2t) dt) +
J J \ ■ * *)
X X
+R4(x, A, q),
and Ru i = 1,2, 3,4, satisfy the estimates (when |A| > 1)
Rx{x,X,q\ R2(x,A,q) = 0 ( ——j, (2.12)
/g\ImA\(n -x)\
R3(x,A,q), R4(x,A,q) = 0 -—-j, (2.13)
uniformly with respect to q G BL\ [0,n].
Proof In [14] the authors have proved that y2(x,A) can be obtained as a sum of series
y2(x, A, q) = ^ Sk(x, A, q) ,
k=о
which converge to y2(x, A, q) uniformly on bounded subsets of the set [0, n] x (C x n], and where S0(x,A, q) = sin/b^
14
Г sin А(х — t)
Sk(x,A,q) = ! ---q(t)Sk_1(t,A,q)dt , к = 1,2,....
о
For Sk we have the estimate (when|A| > 1):
e\lmX\x ak(x}
\Sk(x,A,q)\< °k/, k = 0,1,2..........(2.14)
where <r0(x) = j*\q(t)\dt (see [14]). To prove (2.5), (2.9) and the estimate (2.12), we write S1 in the form
X
sin A(x — t) sin It -^--—— dt =
A A
[cosl(x — 21) — со sAx]q(t)dt =
51(x,A,q) = J
о
x
= 2Я2 /
2 A2
and note that
о
x x
cos Ax
+
J q(t)dt + 2J2 J cos— 2t) q(t) dt,
о 0
л.
S^(x, A, q) = J cos A(x — t)q (iOSfc.jXt, A, q)dt, к = 1, 2,
This implies that S'k £ /1C[0, ti]. By writing y2(x, A, q) — S0 -I- 44" "^k (*.A, <7), we obtain
sin Ax 1 y2(x,A,q) = —---— 6(x,A),
15
where — — b(x,A) = ^-1Sk(x,A,q) and therefore b(x,A) has the
form (2.9), where R2(x,A) = — 2A2 'Z<k=2^k(x>^)- Now, from the estimate (2.14), we obtain that
k-2 k-2
00
z-
e|lmA|x akx e|CmA|*(r2(x)
\A\k+1 k\
, |ImA|x_2
|яр
,|1тЯ|зс_2/
z
k-2
00
|l|fc_2/c!
<
"о (X) v °oOO
|Я|з ¿j |l|fc_2 (/c — 2)!
plImAIsc^^ g0(X) /т-2
■e w =
■z
|Я|3 ¿j |Я|пП!
71 = 0
gpM
|Я|
3
|Я| =
|Я|
This implies (2.12) for |A| > 1. Since y2 e 0,7r] and
50(x,l) = s'n/bc, then we obtain that —7-b (x,A) = y2 — 50 is also a
/1 2/i"
twice differentiable function (more precisely b' 6 AC[ 0,7r]). Assertions for yx, y3, y4 can be proven similarly.
3. The proof of the Theorem 1.1
According to (2.3), the solution (p{x,[i,a,q), which we will denote by <p(x, A2, a) for brevity, has the form
<f>(x, A2, a) = уг(х,А) sin a — y2(x,A) cos a,
(3.1)
and according to (2.4) and (2.5) we arrive at:
<p(x, A2, a) =
cos Ax + — а(х,Я) 2Я
sin a
sin Ax 1
Я 2 Я2
b(x,A)
cos a.
Taking the squares of both sides of the last equality, we obtain:
16
<р2 (х, Л2, а) = cos2 Лх sin2 а + —
л
а(х,Я) coslx +
а2(х, Я)
4Я
. 2
sin а
Ь(х,Я)со5Ях а(х,Я)5тЯх cos Ях sin Ях--—--1- -
2Я
а(х, Я)Ь(х, Я)
2Я
4Я2
sin2 Ях
х sin a cos а Н--—— cos2 а +
Я2
Ь2(х,Я) Ь(х,Я)5тЯх
4Я4
Я3
2
cosz а.
(3.2)
Recalling the formulae cos2 Ax = -(14- cos22x) and 2 1
sin Ax = - (1 — cos 2Ax), from (3.2), we obtain:
7Г r
1 J \<P
0
/ 7Г ' f
я| J
\o
7Г 2:
sin 2/br
4Я
■sin/ а +
7Г
if
4Я J
\
sin2 /br
Я2
■sin a cos а +
7Г 7T
•f — ( J b(x, A) cos Ях dx — J а(х,Я) 51пЯх dx ) sin a cos а +
(3.3)
о
)
sin a cos a f n sin 2Я7Г
H--ГТ7- a(x,A)b(x,A)dx + —-^cos2 a--cos^
LA3 J Z/iz 4/i-5 о
/ 7Г Я
— — I I Ь(х,Я) sin Ях dx — — I b2(x,X)dx cos2 a.
\ J 4Я J
\o о
7Г
17
We are going to receive the asymptotic formula (1.6) by the substitution A = An(q,a,(3) = yj[in{q,a,(3) in (3.3). To this aim, we estimate each term of the right-hand side of (3.3) for A = An. It can be easily deduced from (1.4) that for An = ^we have the following asymptotic formula:
[q]
An(q,a,P) = n + 8n(a,l3)+ + ln, (3.4)
2 (n + Sn(a,p))
where [q]: = R(t)dt, ln = ln(q, a,(3) = o Q) uniformly with respect to a,(3 G [0, n] and q 6 BL^[0,n] (see [15]).
It follows from (1.5a)-(1.5d) that sm2n5n(a,(3) = 0 (^J and
sin 2nAn(q, a,(3) = 0 ^, cos 2nAn(q, a,(3) = 1 — 0 (3.5)
for all (cr,/?) G (0,7r] x [O.tt).
Thus, the second term
sin 2nAVt /1 -:-- sin2 a = О I — | sin'- a.
An
= o(-)sin2a. (3.6)
Important is the third term: y- fn a(x, An) coslnx dx. According
An ■'U
to (2.8) and (2.12) we have
a(x,An) =A(x,An) + 0 (3.7)
where
18
Л л
Л(*.А„) = f q(t)dt sinAnx + I ,(t)sln^(x - 2t) dt. (3 8)
After multiplying both sides by cosA„x, integrating over [0, n] and changing the order of integration we get
n n
f Sin2ln7T f _
I A(x,An) cosAnx dx =--- I q(t) cos Ant dt —
J An J
(3-9)
sin 2Ar,n
n 2Ann Г If
—- q(t) sin 2Ant dt — — (n — t)q(t) sin 2Ant dt.
4An J 2 J
0 0
Taking into account the formulae (3.5) and denoting (see (1.7))
кп - Kn(q,a,p): = — — f (n — t)q(t) sin21nt dt,
2 J
о
we can rewrite (3.9) in the form
7Г
j A(x,An) cos Anx dx = кп + О (3.10)
о
Since sin 2Ant = sin 2 (n + Sn + О Qj j t = sin 2(n + Sn)t +
Oln holds uniformly with respect to tE[0,n/, then хп=хп+01п, and therefore the third term of (3.3) has the form
7Г
1 Г кп / 1 \
— a(x, A„) cos Ar,x dx =-———— + О — .
An J 71 n + Sn(a,(3) \n /
19
Now, let us focus on the remained terms of the equality (3.3) for
i
A = An. The terms from the fourth to the eighth have the coefficient -y,
71
wherey > 2, and therefore they have the order 0 Concerning the last four terms of (3.3), we observe that both sm2™Xn cos2 a an(j "¿t/c* b2(x,An) dx cos2 a have the same order 0 (-jj) cos2 a. An important term is C b(x,An) sin Anx dx. According to (2.9) and (2.12)
71 U
we can write b(x, An) in the form
b(x,An)=B(x,An) + 0 (3.11)
where
X X
S(x.A„) = fq(t)dtcosAnX-fq(t)cosAn(x-2t)dL (3 12) 0 0
A simple computation yields:
X X
BiX,An) sin^x = f m dtsin2AnX - J q(t) sin 2Ant dt - A(X,An) cos Anx. 0 0
After integrating the latter equality from 0 to n, changing the order of integration and taking into consideration (3.9) we get:
Г cos2 AnTi Г
I B(x,An) sin Anx dx =---- I q(t) sinz Ant dt +
J An J
о о
7Г 7Г
sin 2 An7i Г If
H--n i q(t) sin 2Ant dt — — I (л — t)q(t) sin 2Ant dt
4An J 2 J
20
and the refore the eleventh term of the equality (3.3) for A = An has the form
If 1 f xr, / 1Л
—т I b(x,An) sin Anx dx = —I-——— + 0 — .
A3nj nJ n Al\n + 8n{a,(3) \n J) о x '
i I / I \ Let us remark that from (3.4) we have 7---— = 0 I—-I. Thus,
v ' A -I- V >} J .'
Av, 71 + 0*, \пл /
n
an(q,a,(3) = -
1 +
2 Kr,
+ 0
n[n + Sn(a,(3)] +0 ^ sin a cos a +
Ш
sin2 a +
(3.13)
+ ■
л
1 +■
2яг
2[n + 5n(a,^)]2L ;r[n + <5n(^)]
+ 0
Ш
2
cosz a.
If sin a 0, then 0 sin a cos a can be included into the term
0 (J^j sin2 a, and if sin a = 0, then these terms are absent. Finally, we
can write (3.13) in the form (1.6). For bn everything can be done similarly. Theorem 1.1 is proved.
4. The proof of the Theorem 1.2
In the sequel the following notations will be used:
m
Л л
: = (n — t)q(t) and o(x) := J q(t)dt = j(n — t)q (t) dt.
(4.1)
Now, we have
21
= 1 Г
n + Sn(a,p) 2[п + Sn(a,(3)] J
о
7Г
sin 2(n + <5n)t dt
<j(n) sin 27r<5r)
2 cn + «0 J Si" 2(" + W ' = " 2(n + Sn) + (4 2)
0
71
+ J a(t) cos 2(n + Sn)t dt.
o
It was observed in (3.5) that sin27r<5n = 0 If we denote by <j(x) := ff^and cn: = = 0 then we can rewrite k(x) in
the form
k(x) = fciCt) + k2(x), (4.3)
where
^iOO = ^ cn cos[n + 8n(a,p)]x,
(4-4)
71 = 2
oo 2tt
= ^ J cos[n + Sn(a,fi)]t dt cos[n + Sn(a,/3)]x . (4.5)
71=2 o
Since cn = 0 (j^V), then the series in (4.4) converges absolutely
and uniformly on [0,27r], and kt G AC[0,2n]. Next, we consider two cases: Case I: Ifa,(3 G (0,7r), then by (1.5a) we have
22
cot В — cot а /1\ d /1\ /1\ 6п(а,(3) =---+ 0 [ — \ =- + 0\—) = 0[-\,
пп \п// п \п// \п/
. . (cot5-cot а)
where d =-.
Recalling the Maclaurin expansions of the functions sinx and cos x around the point x = 0, we obtain
cosnx — d ■ x sinnx
cos[n + 8n(a, B)]x =-+ en(x), (4.6)
n
where en(x), as all the other entries of (4.6), is a smooth function (en £ C°°) and
= (4.7)
uniformly on x e [0,271*]. Therefore k2 can be written in the form
k2(x) = /x(x) + Z2(x) + Z3(x),
where
00 2 71
(x) = — d ■ x ^ — J d(t) cos nt dt sin nx —
71=2 0
-"■rj
2 n
2
a(t) sin nt dt cosnx +d
71=2 о
oo 2 71
(4.8)
x]T-f td(t) sin nt dt sinnx +
71=2 о
271 oo 271
V f
+ ^ J en(t)d(t) dt cosnx -d - x en(t)d(t) dt sinnx,
71=2 о 71=2 о
23
2л
h(x) = ^ еп(х) j S{t) cosntdt —
71=2 о
со 2л оо 2л
—d-^ —-j tô(t)smnt dt +^en(x)J en(t)ô(t) sinnt dt ,
oo 2n
h(x) = ^ J â(t) cos nt dt cos nx. (4.10)
71=2 o
Since â G AC[0,2n], then Fourier coefficients are
2tt 2TT
j <f(t) cos nt dt = 0 , J ta(t) sin nt dt = 0 (4.11)
Also we note that
2n
f en(t)d(t) dt = 0 (4 12)
o
Therefore the trigonometric series in (4.8) converges absolutely and uniformly on [0,27t], and G AC(0,2n).
It follows from (4.7), (4.11) and (4.12) that the terms of the series in (4.9) have the order 0 (-A-), and therefore l2 G AC[Q, 2n\.
About ¿3(x) we can say the following: Since a G AC[0,2n], then the Fourier series of d
a0(<J) v-1 = —2--^ / (^nCô1) cosnx + bn(o) sinx),
71=1
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1 2n 1 2n
where an(a) = - f d(t) cosnt dt, bn(a)=-f d(t) sin nt dt, con-
7T U 7T U
verges to <x(x) in every point of [0, 2tt] and this series is a function from AC[0,271"].The same is true for <r*(x) = a(2n — x):
a0(a*) v-1
a(2n — x) = —---1- ^(an(<7*) cosnx + bn(a*) sinx).
a0O*)
+
n=l
But it is easy to see, that an(a*) = an(<j)and bn(a*) = —bn(Jf). So
oo
N . N4 ao(o0
— (¿"(x) + a(27r — x) J = —---1- ^ an(<j) cos nx,
71=1
i.e. this is "the even part" of Fourier series ofcf(x), and is absolutely continuous on [0,277"]. Thus, for the case a,(3 G (0,77") Theorem 1.2 is proved.
Case II: If a = 77",/? = 0, then Sn(ir, 0) = 1, and the function k2(-)
27T
takes the form k2(x) -Zn=3f0 cosnt dt cosnx and again it is "the even part" of Fourier series (without the zeroth, the first and the second terms) of an absolutely continuous function. Theorem 1.2 is proved.
Acknowledgment. T.N. Harntyunyan M'as supported by State Committee of Science MES in frame of the research project No. 15T-1A392.
References
1. Naimark, M.A. Linear Differential Equations, Nauka, Moscow, (in Russian), 1969.
2. Levitan, B.M. and Sargsyan, I.S. Introduction to Spectral Theory, Nauka, Moscow, (in Russian), 1970.
3. Marchenko, V.A. The Sturm -Li ouville Operators and their Applications, NaukovaDumka, Kiev, (in Russian), 1977.
4. Harutyunyan, T.N. "The Dependence of the Eigenvalues of the Sturm-Liouville Problem on Boundary Conditions". Matematicki Vesnik, 60, no. 4, (2008): 285-294.
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5. Isaacson, E.L. and Trubowitz, E. "The inverse Sturm-Liouville problem. I." Comm. Pure Appl. Math., 36, no. 6, (1983): 767-783.
6. Isaacson, E.L., McKean, H.P. and Trubowitz, E. "The inverse Sturm-Liouville problem. llrComm. Pure Appl. Math., 37, no. 1, (1984): 1-11.
7. Dahlberg, B.E. and Trubowitz, E. "The inverse Sturm-Liouville problem. III. Comm. Pure Appl. Math., 37, no. 2, (1984): 255-267.
8. Poschel, J. and Trubowitz, E. Inverse spectral theory, Academic Press, Inc., Boston, MA, 1987.
9. Yurko, V.A. Introduction to the theory of inverse spectral problems, Fizmatlit, Moscow, (in Russian), 2007.
10. Zhikov, V.V. "On inverse Sturm-Liouville problems on a finite segment." Izv. Akad.Nauk SSSR, ser. Math., 31, no. 5, (in Russian), (1967): 965-976.
11. Marchenko, V.A. "Some questions of the theory of one-dimensional linear differential operators of the second order." Trudy Moskov. Mat. Obsh., 1, (in Russian), (1952): 327-420.
12. Chudov, L.A. "The inverse Sturm-Liouville problem. Mat. Sbornik, 25(67), no. 3, (in Russian), (1949): 451-456.
13. Atkinson, F.V. Discrete and continuous boundary problems, Academic Press, New York-London, 1964.
14. Harutyunyan, T.N. and Hovsepyan, M.S. "On the solutions of the Sturm-Liouvilleuation. Mathematics in Higher School, 1, no. 3, (in Russian), (2005): 59-74.
15. Harutyunyan, T.N. "Asymptotics of the eigenvalues of Sturm-Liouville problem. Journal of Contemporary Mathematical Analysis, 51, no. 4, (2016): 173-181.
О НОРМИРОВОЧНЫХ ПОСТОЯННЫХЗАДАЧИ ШТУРМА-ЛИУВИЛЛЯ Арутюнян Т., Пахлеванян А.
Для нормировочных постоянных задачи Штурма-Лиувилля с суммируемым потенциалом доказывается новая асимптотическая формула, которая обобщает и уточняет ранее известные формулы. Кроме того, наши формулы учитывают гладкую зависимость нормировочных постоянных от краевых условий. Мы также находим некоторые новые свойства остаточных членов асимптотики.
Ключевые слова: задача Штурма-Лиувилля, нормировочные постоянные, асимптотика решений, асимптотика спектральных данных.
Дата поступления 14.08.2016.
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