Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions

We study the inverse problem for non-selfadjoint Sturm-Liouville operators on a finite interval with possibly multiple spectra. We prove the uniqueness theorem and obtain constructive procedures for solving the inverse problem along with the necessary and sufficient conditions of its solvability and also prove the stability of the solution.

MSC:34A55, 34B24, 47E05.

Inverse spectral problems consist of recovering operators from given spectral characteristics. Such problems play an important role in mathematics and have many applications in natural sciences and engineering (see, for example, monographs [1–7] and the references therein). We study the inverse problem for the Sturm-Liouville operator corresponding to the boundary value problem L=L(q(x),T) of the form

where q(x)\in L_1(0,T) is a complex-valued function. The results for the non-selfadjoint operator (1), (2) that we obtain in this paper are crucial in studying inverse problems for Sturm-Liouville operators on graphs with cycles. Here also lies the main reason of considering the case of Dirichlet boundary conditions (2) and arbitrary length T of the interval.

For the selfadjoint case, i.e., when q(x) is a real-valued function, the inverse problem of recovering L from its spectral characteristics was investigated fairly completely. As the most fundamental works in this direction we mention [8, 9], which gave rise to the so-called transformation operator method having become an important tool for studying inverse problems for selfadjoint Sturm-Liouville operators. The inverse problems for non-selfadjoint operators are more difficult for investigation. Some aspects of the inverse problem theory for non-selfadjoint Sturm-Liouville operators were studied in [10–14] and other papers.

In the present paper, we use the method of spectral mappings [7], which is effective for a wide class of differential and difference operators including non-selfadjoint ones. The method of spectral mappings is connected with the idea of the contour integration method and reduces the inverse problem to the so-called main equation of the inverse problem, which is a linear equation in the Banach space of bounded sequences. We prove the uniqueness theorem of the inverse problem, obtain algorithms for constructing its solution together with the necessary and sufficient conditions of its solvability. In general, by sufficiency one should require solvability of the main equation. Therefore, we also study those cases when the solvability of the main equation can be proved or easily checked, namely, selfadjoint case, the case of finite perturbations of the spectral data and the case of small perturbations. The study of the latter case allows us to prove also the stability of the inverse problem.

In the next section, we introduce the spectral data, study their properties and give the formulation of the inverse problem. In Section 3, we prove the uniqueness theorem. In Section 4 we derive the main equation and prove its solvability. Further, using the solution of the main equation, we provide an algorithm for solving the inverse problem. In Section 5, we obtain another algorithm, which we use in Section 6 for obtaining necessary and sufficient conditions of solvability of the inverse problem and for proving its stability.

Let {\{{\lambda }_n\}}_{n\ge 1} be the spectrum of the boundary value problem (1), (2). In the self-adjoint case, the potential q(x) is determined uniquely by specifying the classical discrete spectral data {\{{\lambda }_n,{\alpha }_n\}}_{n\ge 1}, where {\alpha }_n are weight numbers determined by the formula

while S(x,\lambda ) is a solution of equation (1) satisfying the initial conditions

In the non-selfadjoint case, there may be a finite number of multiple eigenvalues and, hence, for unique determination of the Sturm-Liouville operator, one should specify some additional information. In the present section, we introduce the so-called generalized weight numbers, as was done for the case of operator (1) with Robin boundary conditions (see [11, 12]) and study the properties of the generalized spectral data.

Let the function \psi (x,\lambda ) be a solution of equation (1) under the conditions

For every fixed x\in [0,T], the functions S(x,\lambda ), \psi (x,\lambda ) and their derivatives with respect to x are entire in λ. The eigenvalues {\lambda }_n, n\ge 1 of the problem L coincide with the zeros of its characteristic function

where 〈y,z〉:=y{z}^{\prime }-y^{\prime }z. It is known (see, e.g., [2]) that the spectrum {\{{\lambda }_n\}}_{n\ge 1} has the asymptotics

where

Denote by m_n the multiplicity of the eigenvalue {\lambda }_n ({\lambda }_n={\lambda }_{n+1}=\cdots ={\lambda }_{n+m_n-1}) and put \mathbb{S}=\{n:n-1\in \mathbb{N},{\lambda }_{n-1}\ne {\lambda }_n\}\cup \{1\}. Note that by virtue of (7) for sufficiently large n, we have m_n=1. Denote

Hence, for \nu \ge 1 we have

Moreover, for n\in \mathbb{S} formula (6) yields

Put

Thus, {\{S_n(x)\}}_{n\ge 1}, {\{{\psi }_n(x)\}}_{n\ge 1} are complete systems of eigen- and the associated functions of the boundary value problem L. Together with the eigenvalues {\lambda }_n we consider generalized weight numbers {\alpha }_n, n\ge 1, determined in the following way:

We note that the numbers {\alpha }_n for sufficiently large n coincide with the classical weight numbers (4) for the selfadjoint Sturm-Liouville operator.

Definition 1 The numbers {\{{\lambda }_n,{\alpha }_n\}}_{n\ge 1} are called the generalized spectral data of L.

Consider the following inverse problem.

Inverse Problem 1 Given the generalized spectral data {\{{\lambda }_n,{\alpha }_n\}}_{n\ge 1}, find q(x).

Let the functions C(x,\lambda ), \mathrm{\Phi }(x,\lambda ) be solutions of equation (1) under the conditions

The functions \mathrm{\Phi }(x,\lambda ) and M(\lambda ):={\mathrm{\Phi }}^{\prime }(0,\lambda ) are called the Weyl solution and the Weyl function for L, respectively. According to (6), we have

The function d(\lambda ) is the characteristic function of the boundary value problem for the equation (1) with the boundary conditions y^{\prime }(0)=y(T)=0. Let {\{{\mu }_n\}}_{n\ge 0} be its spectrum. Clearly, {\{{\lambda }_n\}}_{n\ge 1}\cap {\{{\mu }_n\}}_{n\ge 0}=\mathrm{\varnothing }. Thus, M(\lambda ) is a meromorphic function with poles in {\lambda }_n, n\ge 1, and zeros in {\mu }_n, n\ge 0. Moreover, (see, e.g., [2])

Let \lambda ={\rho }^2 and put \tau =Im\rho. Using the known method (see, e.g., [3]), one can prove the following asymptotics.

Lemma 1 (i) For |\rho |\to \mathrm{\infty }, the following asymptotics holds

uniformly with respect to x\in [0,T].

1. (ii)

   Fix \delta >0. Then for sufficiently large |\lambda |

   |\mathrm{\Delta }(\lambda )|\ge \frac{C_{\delta }}{|\rho |}exp(|\tau |T),\lambda \in G_{\delta },

   (18)

where G_{\delta }=\{\lambda ={\rho }^2:|\rho -\pi k/T|\ge \delta ,k\in \mathbb{Z}\}.

Using (7), (10), (11) and (16), one can calculate

Fix k\in \mathbb{S}. According to (14), the function M(\lambda ) has a representation

where M_{k+m_k-1}\ne 0, and the function M_k^0(\lambda ) is regular in a vicinity of {\lambda }_k. The sequence {\{M_n\}}_{n\ge 1} is called the Weyl sequence for L. By virtue of (14), (17) and (18), the following estimate holds

Moreover, according to (6), (14), (16) and (17), for each fixed \delta >0, we have

The maximum modulus principle together with (7), (20) and (21) give

Choose \omega >0 such that \mathrm{\Delta }(\pm i\omega )\ne 0 and put

According to (7) and (23), we get

and hence the series

converges absolutely and uniformly in λ on bounded sets.

Theorem 1 The following representation holds

Proof Consider the contour integral

where the contour {\mathrm{\Gamma }}_N:=\{\mu :|\mu |={(\pi (N+1/2)/T)}^2\}, N\in \mathbb{N} has the counterclockwise circuit. According to (7), we have {\mathrm{\Gamma }}_N\subset G_{\delta } for sufficiently large N and sufficiently small fixed \delta >0. By virtue of (21), we obtain the estimate

uniformly with respect to λ in bounded subsets of ℂ, and hence

On the other hand, using the residue theorem [15], we calculate

where

Further, we calculate

Substituting this into (27) and using (26), we obtain M(\lambda )=N(\lambda )+b. By virtue of (22), we get b=a and arrive at (25). □

Theorem 2 The coefficients M_n and the generalized weight numbers {\alpha }_n determine each other uniquely by the formula

Proof Using (10), (14) and (20), one can calculate

where {\mathrm{\Delta }}_{p,n}={\mathrm{\Delta }}^{(p)}({\lambda }_n)/(p!). Obviously, {\psi }_n(x)={\psi }_n^{\prime }(0)S_n(x), n\in \mathbb{S}. Moreover, by virtue of (8) and (10), induction gives

Further, since

we get

and (4), (5) and (6) yield

Hence,

and we calculate

Using (8) and (10) and integrating by parts, we obtain

Substituting (30) in (31) and taking (11) into account, we arrive at

Finally, substituting (32) in (29), we get

Since {\psi }_n^{\prime }(0)\ne 0, n\in \mathbb{S}, by induction we obtain (28). □

According to (19) and (28), we have the asymptotics

Consider the following inverse problems.

Inverse Problem 2 Given the spectra {\{{\lambda }_n\}}_{n\ge 1}, {\{{\mu }_n\}}_{n\ge 0}, construct the function q(x).

Inverse Problem 3 Given the Weyl function M(\lambda ), construct the function q(x).

Remark 1 According to (14), (15), (24), (25) and (28), inverse Problems 1-3 are equivalent. The numbers {\{{\lambda }_n,M_n\}}_{n\ge 1} can also be used as spectral data.

We agree that together with L we consider a boundary value problem \tilde{L}=L(\tilde{q}(x),\tilde{T}) of the same form but with another potential. If a certain symbol γ denotes an object related to L, then this symbol with tilde \tilde{\gamma } denotes the analogous object related to \tilde{L} and \hat{\gamma }:=\gamma -\tilde{\gamma }.

Theorem 3 If {\lambda }_n={\tilde{\lambda }}_n, {\alpha }_n={\tilde{\alpha }}_n, n\ge 1, then L=\tilde{L}, i.e., T=\tilde{T}, q(x)=\tilde{q}(x) a.e. on (0,T). Thus, the specification of the generalized spectral data {\{{\lambda }_n,{\alpha }_n\}}_{n\ge 1} determines the potential uniquely.

Proof By virtue of (7), we have T=\tilde{T}. According to Remark 1, it is sufficient to prove that if M(\lambda )=\tilde{M}(\lambda ), then L=\tilde{L}. Define the matrix P(x,\lambda )={[P_{jk}(x,\lambda )]}_{j,k=1,2} by the formula

Using (13) and (34), we calculate

It follows from (12) and (35) that

By virtue of (16)-(18), this yields

uniformly with respect to x\in [0,T]. On the other hand, according to (12) and (35), we get

Thus, if \hat{M}(\lambda )\equiv 0, then for each fixed x, the functions P_{11}(x,\lambda ) and P_{12}(x,\lambda ) are entire in λ. Together with (37) this yields P_{11}(x,\lambda )\equiv 1, P_{12}(x,\lambda )\equiv 0. Substituting into (36), we get S(x,\lambda )\equiv \tilde{S}(x,\lambda ) and consequently L=\tilde{L}. □

Let the spectral data {\{{\lambda }_n,{\alpha }_n\}}_{n\ge 1} of L=L(q(x),T) be given. We choose an arbitrary model boundary value problem \tilde{L}=L(\tilde{q}(x),T) (e.g., one can take \tilde{q}(x)\equiv 0). Introduce the numbers {\xi }_n, n\ge 1 by the formulae

According to (7) and (33), we have

Denote

For i,j=0,1, n\in {\mathbb{S}}_i put

where k\ge 1, \nu =\overline{0,m_{n,i}-1}. Analogously, we define \tilde{D}(x,\lambda ,\mu ), {\tilde{D}}_{\nu ,\eta }(x,\lambda ,\mu ), {\tilde{A}}_{n,i}(x,\lambda ) and {\tilde{P}}_{n,i;k,j}(x), n,k\ge 1, i,j=0,1, replacing S with \tilde{S} in the definitions above.

By the same way as in [2], using (7), (16), (33) and Schwarz’s lemma [15], we get the such estimates as \pm Re\rho \ge 0, n,k\ge 1, i,j=0,1, \nu =0,1

The analogous estimates are also valid for {\tilde{S}}_{n,i}(x), \tilde{D}(x,\lambda ,{\lambda }_{k,j}), {\tilde{P}}_{n,i;k,j}(x).

Lemma 2 The following relation holds

where the series converges absolutely and uniformly with respect to x\in [0,T].

Proof Let real numbers a, b be such that , b>max|Im{\lambda }_{n,i}|, n\ge 1, i=0,1. In the λ-plane consider a closed contour {\gamma }_N:=\partial {\mathrm{\Xi }}_N (with a counterclockwise circuit), where {\mathrm{\Xi }}_N=\{\lambda :a\le Re\lambda \le {(N+1/2)}^2{\pi }^2/T^2,|Im\lambda |\le b\}. By the standard method (see [2]), using (12), (35)-(37) and Cauchy’s integral formula [15], we obtain the representation

where

uniformly with respect to x\in [0,T] and λ on bounded sets. Calculating the integral in (45) by the residue theorem and using (20), we get

for sufficiently large N. Taking the limit in (45) as N\to \mathrm{\infty }, we obtain

Differentiating this with respect to λ, the corresponding number of times and then taking \lambda ={\lambda }_{n,i}, we arrive at (44). □

Analogously to (46), one can obtain the following relation

where

For each fixed x\in [0,T], the relation (44) can be considered as a system of linear equations with respect to S_{n,i}(x), n\ge 1, i=0,1. But the series in (44) converges only with brackets, i.e., the terms in them cannot be dissociated. Therefore, it is inconvenient to use (44) as a main equation of the inverse problem. Below, we will transfer (44) to a linear equation in the Banach space of bounded sequences (see (53)).

Let w be the set of indices u=(n,i), n\ge 1, i=0,1. For each fixed x\in [0,T], we define the vector

(where T is the sign for transposition) by the formula

Note that if {\varphi }_{n,0}, {\varphi }_{n,1} are given, then S_{n,0}, S_{n,1} can be found by the formula

Consider also a block-matrix

where

Analogously, we introduce {\tilde{\varphi }}_{n,i}(x), \tilde{\varphi }(x) and {\tilde{H}}_{n,i;k,j}(x), \tilde{H}(x) by the replacement of S_{n,i}(x), P_{n,i;k,j}(x) in the preceding definitions with {\tilde{S}}_{n,i}(x), {\tilde{P}}_{n,i;k,j}(x), respectively. Using (41) and (43), we get the estimates

Consider the Banach space B of bounded sequences a={[a_u]}_{u\in w}^T with the norm {\parallel a\parallel }_B={sup}_{u\in w}|a_u|. It follows from (49) and (50) that for each fixed x\in [0,T], the operators H(x) and \tilde{H}(x), acting from B to B, are linear bounded ones and