sturm-liouville differential operators with deviating ...

TAMKANG JOURNAL OF MATHEMATICS Volume 48, Number 1, 61-71, March 2017 doi:10.5556/j.tkjm.48.2017.2264 -

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This paper is available online at http://journals.math.tku.edu.tw/index.php/TKJM/pages/view/onlinefirst

STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT S. A. BUTERIN, M. PIKULA AND V. A. YURKO

Abstract. Non-selfadjoint second-order differential operators with a constant delay are studied. We establish properties of the spectral characteristics and investigate the inverse problem of recovering operators from their spectra. For this inverse problem the uniqueness theorem is proved.

1. Introduction We study an inverse spectral problem for non-selfadjoint Sturm-Liouville differential operators on a finite interval with a constant delay and with complex-valued potentials. Inverse spectral problems consist in recovering operators from their spectral characteristics. The greatest success in the inverse spectral theory has been achieved for the classical Sturm– Liouville operator (see the monographs [1–5] and the references therein) and afterwards for higher-order differential operators and other classes of differential operators and systems [4]−[7]. The classical methods of inverse spectral theory (transformation operator method [1]−[4] and method of spectral mappings [3]−[6]), which allow obtaining global solutions of inverse problems for differential operators, are not applicable for differential operators with deviating argument as well as for other classes of nonlocal operators such as integrodifferential, integral and other operators. Therefore, the general inverse spectral theory for nonlocal operators has not yet been constructed and there are only isolated results in this direction not forming the general picture [8]−[21]. In the present paper we consider the boundary value problems L j = L j (q, h), j = 0, 1, of the form −y ′′ (x) + q(x)y(x − a) = λy(x), ′

y (0) − h y(0) = y

(j)

x ∈ (0, π),

(π) = 0,

(1) (2)

Received July 21, 2016, accepted September 2, 2016. 2010 Mathematics Subject Classification. 34A55 34K29 47E05. Key words and phrases. Differential operators, deviating argument, spectral properties, inverse spectral problems. Corresponding author: S. A. Buterin.

61

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S. A. BUTERIN, M. PIKULA AND V. A. YURKO

where λ is the spectral parameter, a ∈ (0, π), h is a complex number, q(x) is a complex-valued function, q(x) ∈ L(a, π), and q(x) = 0 for x ∈ [0, a]. The following inverse problem is studied: given the spectra of the problems L j , j = 0, 1, find the potential q(x) and the coefficient h. Differential equations with delay arise in various problems of mathematics as well as in applications (see the monographs [22]−[25] and the references therein). Some results on the spectral theory of differential operators with delay can be found in [10, 18, 24, 26] and other works. The presence of delay in a mathematical model produces serious qualitative changes in the study of spectral problems. Therefore, up to now there are no comprehensive results in the inverse problem theory for operators with delay. In the next section we study spectral properties of the boundary value problems (1)−(2), in particular, properties of the characteristic functions and the eigenvalues of L j . In Section 3 we consider the inverse spectral problem of recovering the potential q(x) and the coefficient h from the given two spectra of the boundary value problems L 0 and L 1 . We provide a uniqueness result for this inverse problem. More precisely, we prove that if the eigenvalues of L j (q, h) are the same as for the zero potential, then q can be only zero. 2. Characteristic functions and spectra Let N ∈ N be such that aN < π ≤ a(N +1), i.e. a ∈ [π/(N +1), π/N ). Let C (x, λ), S(x, λ) and ϕ(x, λ) be solutions of Eq. (1) under the initial conditions C (0, λ) = S ′(0, λ) = ϕ(0, λ) = 1,

S(0, λ) = C ′ (0, λ) = 0,

ϕ′ (0, λ) = h.

For each fixed x, and ν = 0, 1 the functions C (ν) (x, λ), S (ν) (x, λ) and ϕ(ν) (x, λ) are entire in λ of order 1/2, and ϕ(x, λ) = C (x, λ) + hS(x, λ).

(3)

Let λ = ρ 2 and ρ = σ + i τ, i.e. σ = Re ρ, τ = I m ρ. Lemma 1. The following representations hold ϕ(x, λ) = ϕ0 (x, λ) + ϕ1 (x, λ) + . . . + ϕN (x, λ), S(x, λ) = S 0 (x, λ) + S 1 (x, λ) + . . . + S N (x, λ), (4) where

sin ρx sin ρx , S 0 (x, λ) = , x ≥ 0, ρ ρ Zx  sin ρ(x − t )  ϕk (x, λ) = q(t )ϕk−1 (t − a, λ) d t ,   ρ Zkxa sin ρ(x − t )   S k (x, λ) = q(t )S k−1(t − a, λ) d t ,  ρ ka

ϕ0 (x, λ) = cos ρx + h

(5)

STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT

63

for x ≥ k a, and ϕk (x, λ) = S k (x, λ) = 0 for x ≤ k a. Moreover, for |ρ| → ∞, ν = 0, 1, uniformly in x the following estimates hold: ϕ(ν) (x, λ) = O(ρ ν−k exp(|τ|(x − k a))), k

S k(ν) (x, λ) = O(ρ ν−k−1 exp(|τ|(x − k a))).

(6)

Proof. The functions ϕ(x, λ) and S(x, λ) are the solutions of the integral equations Zx  sinρx sin ρ(x − t )  ϕ(x, λ) = cos ρx + h + q(t )ϕ(t − a, λ) d t ,    ρ ρ 0 Zx  sin ρ(x − t ) sin ρx    + q(t )S(t − a, λ) d t . S(x, λ) = ρ ρ 0

(7)

Solving integral equations (7) by the method of successive approximations we arrive at (4), where the functions ϕk (x, λ) and S k (x, λ), k = 1, N , are defined by (5). Moreover, it follows from (5) that ϕ′k (x, λ) S k′ (x, λ)

=

Zx

=

Zx

ka

ka

  cos ρ(x − t )q(t )ϕk−1(t − a, λ) d t ,   

(8)

   cos ρ(x − t )q(t )S k−1(t − a, λ) d t , 

for x ≥ k a. Clearly, ϕ0(ν) (x, λ) = O(ρ ν exp(|τ|x)), S 0(ν) (x, λ) = O(ρ ν−1 exp(|τ|x)). Using (5) and (8), we arrive at (6) by induction.



Corollary 1. The following representation holds C (x, λ) = C 0 (x, λ) +C 1 (x, λ) + . . . +C N (x, λ), C 0 (x, λ) = cos ρx, x ≥ 0, Zx sin ρ(x − t ) q(t )C k−1 (t − a, λ) d t , C k (x, λ) = ρ ka

(9)

(10)

for x ≥ k a, and C k (x, λ) = 0 for x ≤ k a. Moreover, for |ρ| → ∞, ν = 0, 1, uniformly in x the following estimate holds: C k(ν) (x, λ) = O(ρ ν−k exp(|τ|(x − k a))).

(11)

Relations (9), (10) and (11) follow from (4), (5) and (6), respectively, with h = 0. Remark 1. The functions C 1 (x, λ), S 1 (x, λ) and ϕ1 (x, λ) depend linearly on the potential q. In

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S. A. BUTERIN, M. PIKULA AND V. A. YURKO

particular, taking (5) and (10) into account, we calculate for x ≥ a: Z Z  1 x sin ρ(x − a) x   q(t ) d t + q(t ) sinρ(x − 2t + a) d t , C 1 (x, λ) =   2ρ 2ρ a  a    Zx Zx   cos ρ(x − a) 1  ′   C 1 (x, λ) = q(t ) d t + q(t ) cos ρ(x − 2t + a) d t ,   2 2 a a Z Zx  cos ρ(x − a) x 1   S 1 (x, λ) = − q(t ) d t + q(t ) cos ρ(2t − x − a) d t ,   2 2  2ρ 2ρ  a a    Zx Zx    1 sin ρ(x − a)  ′  q(t ) d t + q(t ) sinρ(2t − x − a) d t . S 1 (x, λ) = 2ρ 2ρ a a

(12)

Denote ∆ j (λ) := ϕ( j ) (π, λ), j = 0, 1. The functions ∆ j (λ) are entire in λ of order 1/2. The zeros of ∆ j (λ) coincide with the eigenvalues of the boundary value problems L j (q, h) (counting with multiplicities). The function ∆ j (λ) is called the characteristic function for L j (q, h). Lemma 2. For |ρ| → ∞ the following asymptotical formulae are valid: Z ³1 ´ sinρπ sin ρ(π − a) π ∆0 (λ) = cos ρπ + h q(t ) d t + o exp(|τ|(π − a)) , + ρ 2ρ ρ a Zπ ³ ´ cos ρ(π − a) ∆1 (λ) = −ρ sin ρπ + h cosρπ + q(t ) d t + o exp(|τ|(π − a)) . 2 a

(13) (14)

Proof. Using (9), (11) and (12), we obtain Z Z 1 π sin ρ(π − a) π q(t ) d t + q(t ) sin ρ(π − 2t + a) d t C (π, λ) = cos ρπ + 2ρ 2ρ a a +O(ρ −2 exp(|τ|(π − 2a))). Since Zπ

q(t ) sin ρ(π − 2t + a) d t =

a

1 2

Z(π−a)

q((π + a − ξ)/2) sin ρξ d ξ = o(exp(|τ|(π − a))),

−(π−a)

it follows that C (π, λ) = cos ρπ +

sin ρ(π − a) 2ρ

Zπ

q(t ) d t + o(|ρ|−1 exp(|τ|(π − a))).

(15)

a

Analogously we calculate  Z cos ρ(π − a) π  q(t ) d t + o(exp(|τ|(π − a))),     2 a     Zπ  sin ρπ cos ρ(π − a) −2 q(t ) d t + o(|ρ| exp(|τ|(π − a))), − S(π, λ) =  ρ 2ρ 2 a     Zπ   sin ρ(π − a)  −1 ′ S (π, λ) = cos ρπ + q(t ) d t + o(|ρ| exp(|τ|(π − a))).   2ρ a

C ′ (π, λ) = −ρ sinρπ +

(16)

STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT

65

By virtue of (3), one has ∆ j (λ) = C ( j ) (π, λ) + hS ( j )(π, λ),

j = 0, 1.

(17)

Substituting (15)−(16) into (17), we arrive at (13)−(14).



Lemma 3. The boundary value problem L j has a countable set of eigenvalues {λn j }n≥0 , j = 0, 1 (counting with multiplicities) and for n → ∞ : ρ n0 ρ n1

Z ³1´ ³ p cos(n + 1/2)a π 1´ h := λn0 = n + + q(t ) d t + o , + 2 πn 2πn n a Z ³1´ p cos na π h := λn1 = n + . + q(t ) d t + o πn 2πn a n

(18) (19)

Proof. It follows from (14) that ∆1 (λ) = −ρ sinρπ + g (λ),

|g (λ)| ≤ C exp(|τ|π).

(20)

Here and below, the symbol ”C” denotes various positive constants in estimates. Fix δ > 0. Denote G δ := {ρ : |ρ − k| ≥ δ, k ∈ Z}. Since | sin ρπ| ≥ C exp(|τ|π) for ρ ∈ G δ , it follows from (20) that |ρ sin ρπ| > |g (λ)|,

|ρ| ≥ ρ ∗ ,

ρ ∈ G δ,

(21)

for sufficiently large ρ ∗ . Let Γn := {λ : |λ| = (n + 1/2)2 }. Using (20), (21) and Rouché’s theorem [27, p.125], we conclude that the number of zeros of ∆1 (λ) inside Γn is equal to n + 1. Thus, in the circle |λ| < (n + 1/2)2 there exist exactly n + 1 eigenvalues of the boundary value problem L 1 : λ01 , . . . , λn1 . Applying now Rouche’s theorem to the circle γn (δ) := {ρ : |ρ − n| ≤ δ}, we conclude that for sufficiently large n, in γn there is exactly one zero of ∆(ρ 2 ), namely ρ n1 = p λn1 . Since δ > 0 is arbitrary, it follows that ρ n1 = n + εn ,

(22)

εn = o(1), n → ∞.

Since ∆1 (ρ 2n1 ) = 0, it follows from (14) and (22) that ρ n1 sin ρ n1 π = h cos ρ n1 π +

cos ρ n1 (π − a) 2

Zπ

q(t ) d t + o(1),

n → ∞,

a

and, consequently, cos na n sinεn π = h + 2 This yields εn =

cos na h + πn 2πn

Zπ

Zπ a

q(t ) d t + o(1),

n → ∞.

a

q(t ) d t + o

³1´ , n

n → ∞,

and we arrive at (19). Relation (18) can be obtained by similar arguments.



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S. A. BUTERIN, M. PIKULA AND V. A. YURKO

Lemma 4. The specification of the spectrum {λn j }n≥0 uniquely determines the characteristic function ∆ j (λ) by the formulae ∆0 (λ) =

∞ λ −λ Y n0 , (n + 1/2)2 n=0

∆1 (λ) = π(λ01 − λ)

∞ λ −λ Y n1 . n2 n=1

(23)

Proof. By Hadamard‘s factorization theorem [27, p.289], ∆1 (λ) is uniquely determined up to a multiplicative constant by its zeros: ∆1 (λ) = C

∞ ³ Y

1−

n=0

λ ´ λn1

(24)

(the case when ∆1 (0) = 0 requires minor modifications). Consider the function ˜ 1 (λ) = −ρ sin ρπ = −λπ ∆

∞ ³ Y

1−

n=1

Then

λ´ . n2

∞ ³ ∞ n2 Y λn1 − n 2 ´ ∆1 (λ) C (λ − λ01 ) Y 1+ 2 . = ˜ 1 (λ) λ01 πλ n=1 λn1 n=1 n −λ ∆

Taking (14) and (19) into account we calculate ∆1 (λ) = 1, ˜ 1 (λ) λ→−∞ ∆ lim

∞ ³ Y λn1 − n 2 ´ 1+ 2 = 1, λ→−∞ n=1 n −λ

lim

and hence C = πλ01

∞ λ Y n1 . 2 n n=1

Substituting this into (24) we arrive at (23) for ∆1 (λ). For the function ∆0 (λ), the arguments are similar.



Denote L(ρ) := ∆1 (λ) + i ρ∆0 (λ).

(25)

The function L(ρ) is entire in ρ of exponential type, and L(ρ) is the characteristic function for the Redge-type boundary value problem for Eq. (1) with the boundary conditions y ′ (0) − h y(0) = y ′ (π) + i ρ y(π) = 0. Clearly, L(ρ) = L 0 (ρ) + L 1 (ρ) + . . . + L N (ρ),

L k (ρ) = ϕ′k (π, ρ) + i ρϕk (π, ρ).

In particular, L 0 (ρ) = (−ρ sin ρπ + h cosρπ) + (i ρ cos ρπ + i h sinρπ), and consequently, L 0 (ρ) = (i ρ + h) exp(i ρπ).

(26)

STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT

67

Let us calculate L 1 (ρ). For this purpose we use (5), (10), (12) and (26): L 1 (ρ) = (C 1′ (π, λ) + hS 1′ (π, λ)) + i ρ(C 1 (π, λ) + hS 1(π, λ)) Z Z cos ρ(π − a) π 1 π = q(t ) d t + q(t ) cos ρ(π − 2t + a) d t 2 2 a a Zπ Z h sin ρ(π − a) h π + q(t ) d t + q(t ) sinρ(2t − π − a) d t 2ρ 2ρ a a Z Z i π i sin ρ(π − a) π q(t ) d t + q(t ) sinρ(π − 2t + a) d t + 2 2 a a Zπ Zπ i h cos ρ(π − a) ih − q(t ) d t + q(t ) cos ρ(2t − π − a) d t . 2ρ 2ρ a a Therefore, L 1 (ρ) =

Zπ h ´ exp(i ρ(π − a)) q(t ) d t 2 2i ρ a Zπ ³1 h ´ exp(i ρ(π + a)) + − q(t ) exp(−2i ρt ) d t . 2 2i ρ a

³1

+

Lemma 5. For τ ≥ 0, |ρ| → ∞, k ≥ 1, the following estimate holds ´ ³ 1 Zπ |q(t ) exp(−i ρ(2t − π − k a))| d t . L k (ρ) = O k−1 ρ ka

(27)

(28)

Proof. Using (26), (5) and (8), we calculate Zπ ³ ´ L k (ρ) = cos ρ(π − t ) + i sin ρ(π − t ) q(t )ϕk−1(t − a, λ) d t , ka

and consequently, L k (ρ) =

Zπ ka

exp(i ρ(π − t ))q(t )ϕk−1(t − a, λ) d t ,

k ≥ 1.

(29)

On the other hand, it follows from (6) that |ϕk−1 (t − a, λ)| ≤ C |ρ|1−k | exp(−i ρ(t − k a))|, τ ≥ 0. Substituting (30) into (29), we arrive at (28).

(30)



3. The inverse problem In this section we consider the following inverse problem: given two spectra {λn j }n≥0 , j = 0, 1, find q(x) and h. In order to formulate a uniqueness result for this inverse problem, ˜ of the same form ˜ h) we consider together with L j the boundary value problems L˜ j := L j (q, ˜ We agree that if a certain symbol β denotes an object related to L j , but with different q˜ and h. then β˜ will denote the analogous object related to L˜ j .

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S. A. BUTERIN, M. PIKULA AND V. A. YURKO

˜ Let {λ˜ n j }n≥0 , j = 0, 1, be the eigenvalues of the boundary value problems L˜ j with q(x) ≡ 0. ˜ ˜ Let ∆ j (λ) be the characteristic functions of L j , and ˜ ˜ 1 (λ) + i ρ ∆ ˜ 0 (λ). L(ρ) =∆

(31)

˜ ˜ Theorem 1. If λn j = λ˜ n j for all n ≥ 0, j = 0, 1, then q(x) = q(x) a.e. on (a, π) and h = h. Proof. 1) By virtue of Lemma 4, one has ˜ 0 (λ), ∆0 (λ) = ∆

˜ 1 (λ). ∆1 (λ) = ∆

Using (25) and (31) we get ˜ L(ρ) = L(ρ).

(32)

˜ Since q(x) ≡ 0, it follows from (18)−(19) and (29) that Zπ

q(t ) d t = 0,

˜ h = h,

a

˜ L(ρ) ≡ L 0 (ρ) = (i ρ + h) exp(i ρπ).

(33)

In particular, (33) and (27) yield L 1 (ρ) =

³1

2

−

h ´ exp(i ρ(π + a)) 2i ρ

Zπ

q(t ) exp(−2i ρt ) d t .

(34)

a

Denote L + (ρ) := L 2 (ρ)+. . .+L N (ρ) for N ≥ 2, and L + (ρ) ≡ 0 for N = 1. Using (26), (32) and (33), we obtain L 1 (ρ) ≡ −L + (ρ).

(35)

2) First of all, we note that if q(x) = 0 a.e. on (2a, π), then q(x) = 0 a.e. on (a, π). Indeed, under this assumption we have L + (ρ) ≡ 0, and according to (35) we infer L 1 (ρ) ≡ 0. Using (34) we obtain

Zπ

q(t ) exp(−2i ρt ) d t ≡ 0,

a

and consequently, q(x) = 0 a.e. on (a, π). In particular, this finishes the proof for N = 1, since in this case we have 2a ≥ π and automatically L + (ρ) ≡ 0. 3) Let N ≥ 2. Fix ν = 0, 2N − 3. Let us prove that ³ ³ (ν + 1)a ´ νa ´ , π , t hen q(x) = 0 a.e. on π − ,π . i f q(x) = 0 a.e. on π − 2 2

Indeed, it follows from (28) that L 2 (ρ) = O

³ 1 Zπ−νa/2

ρ

2a

´ |q(t ) exp(−i ρ(2t − π − 2a))| d t ,

τ ≥ 0, |ρ| → ∞.

(36)

STURM-LIOUVILLE DIFFERENTIAL OPERATORS WITH DEVIATING ARGUMENT

69

Let π − νa/2 > 2a, otherwise we arrive at the situation in 2) and the proof is finished. Then in the integral we have 2a − π ≤ 2t − π − 2a ≤ π − (ν + 2)a, which yields L 2 (ρ) = O

³1

ρ

´ exp(−i ρ(π − (ν + 2)a)) ,

τ ≥ 0, |ρ| → ∞.

(37)

For k > 2, the functions L k (ρ) have less growth than in (37). This means that L + (ρ) = O

´ ³ 1 exp(−i ρ(π − (ν + 2)a)) , |ρ|

τ ≥ 0, |ρ| → ∞.

(38)

It follows from (34), (35) and (38) that Zπ−νa/2 ³1 ´ exp(i ρ(π + a)) q(t ) exp(−2i ρt ) d t = O exp(−i ρ(π − (ν + 2)a)) , τ ≥ 0, |ρ| → ∞, ρ a or, which is the same, exp(i ρ(2π − (ν + 1)a))

Zπ−νa/2

q(t ) exp(−2i ρt ) d t = O

a

Moreover, one has Zπ−(ν+1)a/2 a

³1´ , ρ

³ ´ q(t ) exp(−2i ρt ) d t = O exp(−i ρ(2π − (ν + 1)a) ,

τ ≥ 0, |ρ| → ∞.

τ ≥ 0, |ρ| → ∞.

(39)

(40)

Let us introduce the function F (ρ) := exp(i ρ(2π − (ν + 1)a))

Zπ−νa/2

q(t ) exp(−2i ρt ) d t .

π−(ν+1)a/2

The function F (ρ) is entire in ρ. Clearly, F (ρ) = O(1) for τ ≤ 0. On the other hand, it follows from (39) and (40) that F (ρ) = O(1) for τ ≥ 0. By Liouville’s theorem [27, p.77], F (ρ) ≡ C −const . Since F (ρ) = o(1) for real ρ, |ρ| → ∞, it follows that F (ρ) ≡ 0, i.e. Zπ−νa/2

q(t ) exp(−2i ρt ) d t ≡ 0.

π−(ν+1)a/2

This yields q(x) = 0 a.e. on the interval (π − (ν + 1)a/2, π − νa/2), i.e. (36) is proved. Applying proposition (36) successively for ν = 0, 1, . . . , 2N − 3, we obtain q(x) = 0 a.e. on the interval (π − (N − 1)a, π) ⊃ (2a, π). According to 2) we get q(x) = 0 a.e. on (a, π). Theorem 1 is proved. Acknowledgement This work was supported in part by Grants 1.1436.2014K, 1.1660.2017/PCh of the Ministry of Education and Science of RF and by Grants 15-01-04864, 16-01-00015 of Russian Foundation for Basic Research.

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[27] J. B. Conway, Functions of One Complex Variable, 2nd ed., vol.I, Springer-Verlag, New York, 1995. Department of Mathematics, Saratov University, Astrakhanskaya 83, Saratov 410012, Russia. E-mail: [email protected] Department of Mathematics, Saratov University, Astrakhanskaya 83, Saratov 410012, Russia. E-mail: [email protected] Department of Mathematics, Informatics and Physics, University East Sarajevo, Alekse Šantica 1, East Sarajevo, Bosnia and Herzegovina E-mail: [email protected]