### Abstract

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.

##### Keywords:

non-selfadjoint Sturm-Liouville operators; inverse spectral problems; method of spectral mappings; generalized spectral data; generalized weight numbers### 1 Introduction

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

where
*T* of the interval.

For the selfadjoint case, *i.e.*, when
*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.

### 2 Generalized spectral data. Inverse problem

Let

while

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

For every fixed
*x* are entire in *λ*. The eigenvalues
*L* coincide with the zeros of its characteristic function

where
*e.g.*, [2]) that the spectrum

where

Denote by
*n*, we have

Hence, for

Moreover, for

Put

Thus,
*L*. Together with the eigenvalues

We note that the numbers
*n* coincide with the classical weight numbers (4) for the selfadjoint Sturm-Liouville
operator.

**Definition 1** The numbers
*L*.

Consider the following inverse problem.

**Inverse Problem 1** Given the generalized spectral data

Let the functions

The functions
*L*, respectively. According to (6), we have

The function
*e.g.*, [2])

Let
*e.g.*, [3]), one can prove the following asymptotics.

**Lemma 1** (i) *For*
*the following asymptotics holds*

*uniformly with respect to*

(ii) *Fix*
*Then for sufficiently large*

*where*

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

Fix

where
*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

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

Choose

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
*N* and sufficiently small fixed

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

**Theorem 2***The coefficients*
*and the generalized weight numbers*
*determine each other uniquely by the formula*

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

where

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

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

Consider the following inverse problems.

**Inverse Problem 2** Given the spectra

**Inverse Problem 3** Given the Weyl function

**Remark 1** According to (14), (15), (24), (25) and (28), inverse Problems 1-3 are equivalent.
The numbers

### 3 The uniqueness theorem

We agree that together with *L* we consider a boundary value problem
*γ* denotes an object related to *L*, then this symbol with tilde

**Theorem 3***If*
*then*
*i*.*e*.,
*a*.*e*. *on*
*Thus*, *the specification of the generalized spectral data*
*determines the potential uniquely*.

*Proof* By virtue of (7), we have

Using (13) and (34), we calculate

It follows from (12) and (35) that

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

uniformly with respect to

Thus, if
*x*, the functions
*λ*. Together with (37) this yields

### 4 Main equation. Solution of the inverse problem

Let the spectral data
*e.g.*, one can take

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

Denote

For

where
*S* with

By the same way as in [2], using (7), (16), (33) and Schwarz’s lemma [15], we get the such estimates as

The analogous estimates are also valid for

**Lemma 2***The following relation holds*

*where the series converges absolutely and uniformly with respect to*

*Proof* Let real numbers *a*, *b* be such that
*λ*-plane consider a closed contour

where

uniformly with respect to
*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

Differentiating this with respect to *λ*, the corresponding number of times and then taking

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

where

For each fixed
*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

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

Note that if

Consider also a block-matrix

where

Analogously, we introduce

Consider the Banach space *B* of bounded sequences
*B* to *B*, are linear bounded ones and

**Theorem 4***For each fixed*
*the vector*
*satisfies the equation*

*in the Banach space**B*, *where**I**is the identity operator*.

*Proof* We rewrite (44) in the form

Substituting here (48), and taking into account our notations, we arrive at

which is equivalent to (53). □

For each fixed
*main equation* of the inverse problem. Thus, the nonlinear inverse problem is reduced to the solution
of the linear equation. Let us prove the unique solvability of the main equation.

**Theorem 5***For each fixed*
*the operator*
*has a bounded inverse operator*, *namely*
*i*.*e*., *the main equation* (53) *is uniquely solvable*.

*Proof* Acting in the same way as in Lemma 2 and using (37) and (38), we obtain

where

uniformly with respect to
*λ*, *μ* on bounded sets. Calculating the integral by the residue theorem and passing to the
limit as

According to the definition of

Further, taking the definition of

which is equivalent to

Hence the operator

Using the solution of the main equation, one can construct the function

**Algorithm 1***Let the spectral data*
*be given*. *Then*

(i) *construct*
*by solving the linear systems* (28);

(ii) *choose*
*and calculate*
*and*

(iii) *find*
*by solving equation* (53);

(iv) *choose*
*e*.*g*.,
*and construct*
*by the formula*

**Remark 2** In the particular case, when

whose determinant does not vanish for any

In the next section for the case

### 5 Algorithm 2

Here and in the sequel, we assume that

Denote

It is obvious that

**Lemma 3***The series in* (57) *converges absolutely and uniformly on*
*and allows termwise differentiation*. *The function*
*is absolutely continuous*, *and*

*Proof* It is sufficient to prove for the case

It follows from (33), (41) and (55) that the series in (59) converges absolutely
and uniformly on

Furthermore, using the asymptotic formulae (7), (16) and (33), we calculate

Hence

**Lemma 4***The following relation holds*

*Proof* Differentiating (46) twice with respect to *x* and using (57) and (58), we get

Using (1) and (8), we replace here the second derivatives, and then replace

where

Using (1) and (8) for

Applying this relation, we get

which together with (57) and (61) gives (60). □

Thus, we obtain the following algorithm for solving the inverse problem.

**Algorithm 2***Let the spectral data*
*be given*. *Then*

(i) *construct*
*by solving the linear systems* (28);

(ii) *choose*
*so that*
*and calculate*
*and*

(iii) *find*
*by solving equation* (53), *and calculate*
*by* (48);

(iv) *calculate*
*by formulae* (56), (57) *and* (60).

### 6 Necessary and sufficient conditions

In the present section, we obtain necessary and sufficient conditions for the solvability of the inverse problem. In the general non-selfadjoint case, they must include the requirement of the solvability of the main equation. In Section 7, some important cases will be considered when the solvability of the main equation can be proved by sufficiency, namely, the selfadjoint case, the case of finite-dimensional perturbations of the spectral data and the case of small perturbations.

**Theorem 6***For complex numbers*
*to be the spectral data of a certain boundary value problem*
*with*
*it is necessary and sufficient that*

(i) *the relations* (7) *and* (19) *hold with*

(ii)
*for all*

(iii) (*Condition* S) *for each*
*the linear bounded operator*
*acting from**B**to**B*, *has a bounded inverse one*. *Here*
*is chosen so that*

*The boundary value problem*
*can be constructed by Algorithms* 1 *and* 2.

The necessity part of the theorem was proved above; here, we prove the sufficiency.
We note that sufficiency condition (ii) of the theorem allows to solve linear systems
(28) for finding

Let

*i.e.*,

Similarly to Lemma 1.6.7 in [2] using (51) and (53), one can prove the following assertion.

**Lemma 5***For*
*the following relations hold*

*where* Ω *is defined in* (55) *and*

We define the functions

Furthermore, we construct the functions

Analogously to Lemma 1.6.8 in [2] using (41) and (69), one can prove the following assertion.

**Lemma 6**

**Lemma 7***For*
*the following relations hold*

*Proof* (1) According to the estimates (42), the series in (46) is termwise differentiable
with respect to *x*, and hence

(2) In order to prove (71) and (72), we first assume that

Differentiating (63) twice, we obtain

It follows from (50), (66) and (67) that the series in (75) converges absolutely
and uniformly for

Solving the main equation (53), we infer

According to (50) and (66), the series in (77) converges absolutely and uniformly
for

where according to (50), (67), (68) and (76), the series converges absolutely and
uniformly for

On the other hand, it follows from the proof of Lemma 3 and from (74) that

Together with (48) this implies that

Using (44), (57) and (60), we get

Similarly, using (46) and (47), we calculate

For