Three spectra inverse Sturm–Liouville problems with overlapping eigenvalues

Three spectra inverse Sturm–Liouville problems with overlapping eigenvalues

Shouzhong Fu

, Zhong Wang

and Guangsheng Wei

1College of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, PR China

2College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China

Received 28 January 2017, appeared 9 May 2017 Communicated by Miklós Horváth

Abstract. In the paper we show that the Dirichlet spectra of three Sturm–Liouville differential operators defined on the intervals [0, 1],[0,a] and[a, 1]for some a∈ (0, 1) fixed, together with the knowledge of the normalizing constants corresponding to the overlapped eigenvalues, uniquely determine the potentialq on[0, 1]. In this situation we also provide the algorithm for recovering the potential by using the given spectral data.

Keywords: eigenvalue, normalizing constant, inverse eigenvalue problem.

2010 Mathematics Subject Classification: 34A55, 34B24.

1 Introduction

In this paper we are concerned with the inverse problem for recovering the potential q on interval[0, 1]of a Sturm–Liouville equation

lu:=−u⁰⁰+q(x)u=λu (1.1)

using the three spectra σ(L),σ(L⁻)andσ(L⁺)corresponding to three Sturm–Liouville oper- atorsL,L⁻andL⁺. These operators are generated inL² spaces by the differential expressions ldefined on[0, 1],[0,a]and[a, 1], respectively, and the Dirichlet boundary conditions

u(0) =0=u(1), (1.2)

u(0) =0=u(a), (1.3)

u(a) =0=u(1). (1.4)

Here the potential q∈ L²[0, 1]is a real-valued function and a∈(0, 1)is fixed.

Recently there has been much interest in three spectra inverse problems (see [1–7,9,16]

and the references therein). This problem was first investigated by Pivovarchik [15] under condition thata=1/2 andσ(L)andσ(L^±)are all Dirichlet spectra. Further investigation has

BCorresponding author. Email: shzhfu@sina.com

been carried out by Gesztesy and Simon [9] under the more general situations of a ∈ (0, 1) and no-Dirichlet spectra. In particular, Gesztesy and Simon [9] proved uniqueness of the reconstructed qwhenever the three spectra do not overlap and suggested a counterexample to uniqueness otherwise. They also raised an interesting assertion: We believe the analysis of the situation whereσ(L⁻)∩σ(L⁺)has k-points will yield k-parameter sets of potentials q consistent with the given sets of eigenvalues (see [9, p. 91]).

One of our purposes of this paper is to answer affirmatively the above assertion. More specifically, we shall show that the uniqueness result remains valid by using the three spectra provided that the normalizing constants α_n corresponding to λ_n are employed as the addi- tional spectral data, whenλ_n ∈ σ(L⁻)∩σ(L⁺). This implies that whenσ(L⁻)∩σ(L⁺)hask- points, there then exists ak-parameter set of potentialsqcorresponding to the given three spec- tra. Our second purpose here is to reconstruct potentials in terms of the three spectra, together with the normalizing constants{α_n}_n∈Λ whereΛ ⊂_Nsatisfying σ(L⁺)∩σ(L⁻) = {λ_n}_n∈Λ. The method used here is similar to that used solving half-inverse problem [15] and three spectra inverse problem without the overlapping eigenvalues [14].

Section 2 includes the uniqueness theorem and its proof. Section 3 gives the algorithm for the reconstruction ofqin terms of given spectral data.

2 Uniqueness

In this section we shall prove our uniqueness result concerning with three spectra inverse Sturm–Liouville problems with overlapping eigenvalues.

Denote byu−(x,λ)andu+(x,λ)the solutions of Eq. (1.1) with the initial conditions u−(0,λ) =u⁰₋(0,λ)−1=0, (2.1) u+(_1,λ) =u⁰₊(_1,λ)−1=_{0. (2.2)} It is known (see, e.g., [8] and [13, Lemma 3.4.2 and proof of Theorem 3.4.1]) that, for each x∈ [0, 1],u±(x,λ)are entire functions ofλand the eigenvalues{λ_n}^∞_n=1of the operator Lare precisely zeros of the function∆(λ)_{defined by}

∆(λ) =

u−(a,λ) u+(a,λ) u⁰₋(a,λ) u⁰₊(a,λ)

=u−(1,λ), (2.3)

where∆(λ)is called the characteristic function of L, which also has the following representa- tion:

∆(λ) =

∏

λ_k−λ k²π²

. (2.4)

Similarly, lettingσ(L⁺) ={µ⁺_n}^∞_n=1andσ(L⁻) ={µ⁻_n}^∞_n=1, then we have

∆⁺(λ):=u+(a,λ) = (1−a)

∏

µ⁺_n −λ k²π²/(1−a)²

, (2.5)

∆⁻(λ):=u−(a,λ) =a

∏

µ⁻_n −λ k²π²/a²

, (2.6)

where∆^±(λ)denote the characteristic functions of the operators L^±. Moreover, for anyλ_n ∈ σ(L), the normalizing constant,α_n, associated with theλ_nis defined by

αn =

Z ₁

0 u²₋(x,λn)dx. (2.7)

In virtue of the above preliminaries, we are now in a position to give the uniqueness result of this paper.

Theorem 2.1. Let the Sturm–Liouville differential operators L and L^±be defined by(1.1)–(1.4). Let σ(L⁺)∩σ(L⁻) ={λn}_n∈Λ (2.8) whereΛ(⊂ _N)is the associated index set. Then the potential q is uniquely determined almost every- where on[0, 1]by the three spectraσ(L),σ(L^±)and the normalizing constants{α_n}_n∈Λ.

Proof. Let us consider another operators ˜L, ˜L⁻ and ˜L⁺ of the same form (1.1)–(1.4) but with different coefficient ˜q(x) _on [_{0, 1}]. Denote by ˜∆(λ)_{and ˜∆}^±(λ) the characteristic functions of operators ˜L and ˜L^±. Then by the hypotheses of Theorem 2.1, these operators and L and L^± have common spectra σ(L)and σ(L^±)respectively; and normalizing constants α_n for n ∈ _Λ corresponding to the common eigenvalues λ_n∈ σ(L)∩σ(L^±)_.

From (2.3)–(2.6) we have u±(a,λ) = u˜±(a,λ)andu−(1,λ) =u˜−(1,λ). This together with (2.3) implies

u−(a,λ) u+(a,λ) u⁰₋(a,λ)−u˜⁰₋(a,λ) u⁰₊(a,λ)−u˜⁰₊(a,λ)

=_∆(λ)−_∆^˜(λ)≡0. (2.9) We first prove thatu⁰_±(a,µ^±_k ) =u˜⁰_±(a,µ^±_k )for allk∈_N. Ifµ⁻_k ∈/ σ(L)then by (2.5) and (2.6) we haveu+(a,µ⁻_k )6=0,u−(a,µ⁻_k ) =0 and thereforeu⁰₋(a,µ⁻_k ) =u˜⁰₋(a,µ⁻_k ). Similar argument yields u⁰₊(a,µ⁺_k) =u˜⁰₊(a,µ⁺_k )for all µ⁺_k ∈/σ(L).

On the other hand, if there existk₁, k₂ andk ∈ _Nsuch that µ⁻_k

1 = µ⁺_k

2 = λ_k ∈ σ(L), then u−(a,λ_k) =0=u+(a,λ_k). This together with (2.9) yields

˙

u−(a,λ_k) u˙+(a,λ_k) u⁰₋(a,λ_k)−u˜⁰₋(a,λ_k) u⁰₊(a,λ_k)−u˜⁰₊(a,λ_k)

=_{0. (2.10)}

Here ˙u±=∂u±/∂λ. Sinceλ_k is a simple eigenvalue of the operators LandL^±, it follows that

˙

u±(a,λ_k)6=0. If

u⁰_±(a,λ_k)−u˜⁰_±(a,λ_k)6=0, (2.11) then we have the following equality

˙

u−(a,λ_k) u˙+(a,λ_k) = ^u

0−(a,λ_k)−u˜⁰₋(a,λ_k)

u⁰₊(a,λ_k)−u˜⁰₊(a,λ_k)^{. (2.12)} Note that ˙∆(λ_k) =−α_kκ_k whereκ_k = u⁰₊(a,λ_k)/u⁰₋(a,λ_k)(see [8]). Since α_k =α˜_k and∆(λ) =

∆˜(λ)for allλ∈C, it follows thatκ_k =κ˜_k. This combined with (2.11) gives u⁰₋(a,λ_k)

u⁰₊(a,λ_k) = ^u˜

0−(a,λ_k)

˜

u⁰₊(a,λ_k) = ^u

0−(a,λ_k)−u˜⁰₋(a,λ_k)

u⁰₊(a,λ_k)−u˜⁰₊(a,λ_k) = ^u˙−(a,λ_k)

˙

u+(a,λ_k)^{. (2.13)} Moreover, because each zero of the characteristic function∆(λ)is simple, we have

∆˙(λ_k) =

˙

u−(a,λ_k) u˙+(a,λ_k) u⁰₋(a,λ_k) u⁰₊(a,λ_k)

6=0, (2.14)

which contradicts (2.13). Therefore,u⁰_±(a,λ_k) =u˜⁰_±(a,λ_k)_{for all}k ∈_Λ.

By the discussion above, we see that u⁰_±(a,µ^±_k )−u˜⁰_±(a,µ^±_k ) = 0 for all k ∈ _{N. Let us} consider the functionF(λ)defined as

F−(λ) = ^u

0−(a,λ)−u˜⁰₋(a,λ)

u−(a,λ) ^{. (2.15)}

It is clear to check thatF(λ)is an entire function and it has the following representation F−(λ) = ^u

0−(a,λ) u−(a,λ)−^u˜

0−(a,λ)

˜

u−(a,λ) =: ˜m−(a,λ)−m−(a,λ). (2.16) Herem−(a,λ)is called the Weylm-function associated with the Sturm–Liouville operatorL⁻. It is known that m−(a,λ) = i√

λ+o(₁) _as |λ| → _∞ in any sector ε < _Arg(λ) < π−ε for ε >0, where√

λis the square root branch with Im(√

λ)≥0, which implies that m−(a,iy)−

˜

m−(a,iy) =o(1)asy →+∞. It together with (2.16) yieldsF−(λ) =0 and thereforeu⁰₋(a,λ) =

˜

u⁰₋(a,λ)_{for all}λ∈_C.

The above argument implies that m±(a,λ) ≡ m˜±(a,λ). By the uniqueness result of Marchenko [13], we have either q = q˜ on [0,a] by using m−(a,λ) ≡ m˜−(a,λ) or q = q˜ on [a, 1]_{by using}m+(a,λ)≡m˜+(a,λ). The proof is complete.

3 Reconstruction

By the uniqueness theorem above, we know that the solution of the inverse problem discussed by us, if exists, is unique. In this section we provide the way of recovering the potential q by using three Dirichlet spectra σ(L), σ(L^±) and (if necessary) the normalizing constants {α_n}_n∈Λ corresponding to the overlapped eigenvalues λ_n ∈ σ(L), where Λ = {n : λ_n ∈ σ(L⁻)∩σ(L⁺)}. The method used here is similar to that used solving half-inverse problem [15] and three spectra inverse problem [14] without the overlapping eigenvalues.

Without loss of generality, we always assume the eigenvalues{λ_k}^∞_k=1,{µ^±_k}^∞_k=1are all pos- itive. Denote bys² =λandγ² =µ. Under these notations, we shall use increasing sequences {s_k}_k∈Z\{0} s_−k =−s_k,(s_k)² =λ_k fork ∈_N and{γ^±_k }_k∈Z\{0} γ^±_−k =−γ^±_k , (γ^±_k )²= µ^±_k for k∈_N, and{α_n}_n∈Λ to recoveringq.

Based on condition that the potentialqexists corresponding to our spectral data, according to [13, p. 32] we have

u−(x,λ) = ^sin(sx)

s +

Z _x

0 K₁(x,t)^sin(st)

s dt, (3.1)

u+(x,λ) = ^{sin s}(1−x)

s +

Z ₁

x K2(x,t)^{sin s}(1−t)

s dt. (3.2)

Using (3.1) and (3.2), we obtain

u⁰₋(a,λ) =_cos(sa) +K₁sin(sa)

s + ^ψ1(s)

s , (3.3)

u⁰₊(a,λ) =cos s(1−a)−K₂sin s(1−a)

s +^ψ2(s)

s , (3.4)

whereψ₁(0) =0,ψ2(0) =0, K₁= ¹

2 Z _a

0 q(t)dt, K2= ¹ 2

Z ₁

a q(t)dt,

ψ₁(s) =

Z _a

0 K⁰_1,x(a,t)sin(st)dt, ψ₂(s) =

Z ₁

a K_2,x⁰ (a,t)sin s(1−t)dt. (3.5) Moreover, theK₁(x,t)andK2(x,t)possess partial derivatives of the first order, and belonging to L²(0,a) and L²(a, 1), respectively, as a function of each of its variables when the other variable is fixed. From [10, Theorem 8] we see that two functional sequences

sin γ⁻_k t ^∞_k=1 and

sin γ_k⁺(t−1) ^∞

k=1 are Riesz bases of L²[0,a] and L²[a, 1], respectively. This together with the fact thatK_1,x⁰ (a,t)∈ L²(0,a)andK_2,x⁰ (a,t)∈ L²(a, 1)implies

{ψ₁(γ⁻_k )}_k∈Z\{0}, {ψ₂(γ⁺_k )}_k∈Z\{0} ∈l₂ (3.6) (see [13, Lemma 1.4.3]). On the other hand, letting L^a denote the class of entire functions of exponential type ≤ a, which belong to L²(−_∞,∞) for real λ, then we have ψ₁ ∈ L^a and ψ2∈ L^(1−a).

Now we are in a position to constructq. We begin by introducing the following lemma.

Lemma 3.1. We have

u⁰₋(a,µ⁻_k ) =











− ^∆(µ⁻_k )

u+(a,µ⁻_k ) ^ifµ

−

k ∈/σ(L)_,

− ^αn∆˙(µ⁻_k )

∆˙(µ⁻_k )u˙−(a,µ⁻_k ) +α_nu˙+(a,µ⁻_k ) ^ifµ

−

k =λ_n∈ σ(L),

(3.7)

and

u⁰₊(a,µ⁺_k ) =















∆(µ⁺_k )

u−(a,µ⁺_k ) ^ifµ

+

k ∈/σ(L),

∆˙(µ⁺_k )²

∆˙(µ⁺_k )u˙−(a,µ⁺_k ) +α_nu˙+(a,µ⁺_k) ^ifµ

+

k =λ_n∈ σ(L).

(3.8)

Proof. By (2.3), if λ = µ⁻_k ∈/ σ(L) then we have u+(a,µ⁻_k ) 6= 0 and therefore u⁰₋(a,µ⁻_k) =

−4(µ⁻_k)/u+(a,µ⁻_k); moreover, if µ⁻_k = λn ∈ σ(L) then we haveµ⁻_k ∈ σ(L⁺). This together with (2.3) implies

∆˙(µ⁻_k ) =

˙ u− u˙+

u⁰₋ u⁰₊

(a,µ⁻_k ). (3.9)

Note that ˙∆(µ⁻_k ) = −α_nκ_n and κ_n = u⁰₊(a,µ⁻_k )/u⁰₋(a,µ⁻_k ), which together with (3.9) yields (3.7). Similar argument implies that (3.8) remains true. The proof is complete.

The above lemma helps us to solve the values of ψ₁(γ_k⁻) and ψ₂(γ_k⁺) for k ∈ _Z\{0} in terms of (3.3) and (3.4). Thus, fork ∈_Nwe have

ψ₁(γ⁻_k ) =γ⁻_k u⁰₋(a,µ⁻_k)−cos(aγ⁻_k )−K₁sin(aγ⁻_k ) γ_k⁻

!

, (3.10)

ψ₁(γ₋⁻_k) =γ⁻_−k u⁰₋(a,µ⁻_k )−cos(aγ⁻_−k)−K₁sin(aγ⁻_−k) γ⁻_−k

!

;

ψ₂(γ⁺_k ) =γ⁺_k u⁰₊(a,µ⁺_k) +cos (1−a)γ⁺_k

+K₂sin (1−a)γ_k⁺ γ_k⁺

!

, (3.11)

ψ₂(γ₋⁺_k) =γ⁺_−k





u⁰₊(a,µ⁺_k ) +cos

(1−a)γ⁺_−k

+K₂ sin

(1−a)γ₋⁺_k

γ₋⁺_k





,

where γ⁻_−k = −γ⁻_k, γ⁺_−k = −γ_k⁺. Moreover, it is easy to check from (3.1) and (3.2) that both functionssu−(a,λ) andsu+(a,λ) are of sine-type (see, e.g., [11]), that is, there exist positive numbersm,M andp such that

me^|Ims|a ≤ |su−(a,λ)| ≤ Me^|Ims|a, me^|Ims|(1−a) ≤ |su+(a,λ)| ≤ Me^|Ims|(1−a) for|Ims|> p. We use Theorem A in [11] and find

ψ₁(s) =su−(a,s²)

∑

ψ₁(γ⁻_k )

dsu−(a,s²) ds |_s=γ−

k (s−γ⁻_k )^{, (3.12)} ψ₂(s) =su+(a,s²)

∑

ψ₂(γ⁺_k )

dsu+(a,s²) ds |_s=γ⁺

k (s−γ⁺_k )^{, (3.13)} where ψ₁(0) = ψ₂(0) = 0. The series on the right-hand sides of (3.12) and (3.13) converge uniformly on any compact subdomain ofC and in L₂(−_∞,∞) for realλ to functions which belong toL^a _andL^1−a. Then we find the characteristic functionsu⁰₋(a,λ)_andu⁰₊(a,λ)_{of the} Dirichlet-Neumann Sturm–Liouville problems on the intervals [0,a] and [a, 1], respectively.

Denote by{ν⁻_k }^∞_−∞,k6=0and{ν_k⁺}^∞_−∞,k6=0the sets of zeros of the functionsu⁰₋(a,λ)andu⁰₊(a,λ). We are able to construct the part of the potentialq∈ L₂(_0,a)by the two spectra{γ⁻_k }^∞_−∞,k6=0 and{ν_k⁻}^∞_−∞,k6=0 and the part of the potential q∈ L2(a, 1)by the two spectra{γ⁺_k }^∞_−∞,k6=0 and {ν_k⁺}^∞_−∞,k6=0 via the procedure of [12,13].

The method of reconstruction of potentials is described by the following algorithm.

Algorithm 3.2. Suppose that the three spectraσ(L) = {λ_n}^∞_n=1, σ(L⁻) = {µ⁻_n}^∞_n=1 andσ(L⁺) = {µ⁺_n}^∞_n=1 corresponding to Sturm–Liouville operators L, L⁻ and L⁺ defined by (1.1)–(1.4) and the normalizing constants{αn}_n∈ΛwhereΛ= {n:λn∈ σ(L⁻)∩σ(L⁺)}are given. Then the algorithm of constructing q from the given data follows.

(1) Construct∆(λ)and u±(a,λ)by(2.3)–(2.6)in Section 2.

(2) Findψ₁(s)andψ₂(s)by(3.12)–(3.13)and K₁and K₂by the asymptotics of{µ^±_n}^∞_n=1:

µ⁺_n =^nπ a

2

+ ^K1

a +o(1), µ⁺_n = nπ

1−a 2

+ ^K2

1−a+o(1); and then find u⁰₋(a,λ)and u⁰₊(a,λ)by(3.3)and(3.4).

(3) Solve the zeros,{ν_k⁻}^∞_−∞,k6=0and{ν_k⁺}^∞_−∞,k6=0, of functions u⁰₋(a,λ)and u⁰₊(a,λ).

(4) Using the algorithm of Gelfand and Levitan (see [12,13]) to construct the (unique) potential q on [0,a]and[a, 1]by{γ^±_k }_k∈Z\{0}and

ν_k^± _k∈Z\{0}.

Acknowledgements

The authors would like to thank the referees for their insightful suggestions and comments which improved and strengthened the presentation of this manuscript.

The research was supported in part by the NNSF of China (No. 11571212, 11171295).

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