Z. L. LEIBENZON

UNIQUENESS OF THE SOLUTION OF THE INVERSE PROBLEM FOR ORDINARY DIFFERENTIAL OPERATORS OF ORDER (n \ge 2) AND TRANSFORMATIONS OF SUCH OPERATORS

(Presented by Academician V. I. Smirnov, 11 IX 1961)

Let (L) be a linear differential operator of the form

[
Ly = y^{(n)} + \sum_{k=0}^{n-2} p_k(x)y^{(k)},
\tag{1}
]

where (p_k^{(k)}(x)) ((0 \le x \le 1)) are bounded; denote by (\Lambda) the class of operators of the form (1). For (n=2) the inverse problem of spectral analysis, i.e. the problem of determining an operator from its spectral data, was solved by I. M. Gel'fand, M. G. Krein, B. M. Levitan, and V. A. Marchenko ((^{1-3})). In these investigations an important role is played by Volterra transformations of operators (1). In the case of entire analytic coefficients (p_k(x)), Volterra transformations also exist for (n>2) and were applied by L. A. Sakhnovich ((^4)); for nonanalytic (p_k(x)), when (n>2), Volterra transformations, generally speaking, do not exist, as was established by V. I. Matsayev ((^5)). Here, for operators from (\Lambda) with (n \ge 2), another method is proposed which made it possible to prove uniqueness of the solution of the inverse problem, and transformations of operators (1), different from those constructed by M. K. Fage ((^6)), are built.

The question of transformation plays the fundamental role in the problem of reconstructing an operator from spectral data.

(1^\circ.) First we describe the spectral boundary-value problems considered for the operator (L) and prove uniqueness of the solution of the inverse problem. Preliminarily, for an arbitrary function (F(x)) denote by (\vec F(x)) the vector in the (n)-dimensional complex vector space (E^n) with components (F(x), F^1(x), \ldots, F^{(n-1)}(x)). We introduce boundary conditions by means of linear forms (C_\nu^a) in (E^n), where (\nu=1,\ldots,n) and (a=0) or (1), and for the vector (z=(z_0,\ldots,z_{n-1})\in E^n)

[
C_\nu^a z = z_{\nu-1} + \sum_{0\le \mu \le \nu-2} c_{\nu\mu}^a z_\mu
\qquad
(c_{\nu\mu}^a=\mathrm{const};\ 1\le \nu \le n;\ a=0,1).
\tag{2}
]

Formulation of the system of spectral boundary-value problems. For (1\le k\le n-1) we consider the spectral boundary-value problem

[
L\varphi=\lambda\varphi,\qquad
C_\nu^0\vec\varphi(0)=0\quad (k<\nu\le n);\qquad
C_{\nu'}^1\vec\varphi(1)=0\quad (n-k<\nu'\le n).
\tag{3}
]

The spectral boundary-value problem adjoint to (3) for the adjoint operator (\widetilde L) is specified by means of the linear forms (\widetilde C_\nu^a) in (E^n). By (\lambda_{kl}), (\varphi_{kl}(x)), and (f_{kl}(x)) ((l=1,2,\ldots)) we denote, respectively, the eigenvalues and the unnormalized eigenfunctions of problem (3) and of the adjoint problem,

[
|\lambda_{k1}| \le |\lambda_{k2}| \le \cdots,
]

[
\alpha_{kl}
=
\frac{1}{
C_k^0\vec\varphi_{kl}(0)\,\widetilde C_k^0 f_{kl}(0)
}
\int_0^1 \varphi_{kl}(t)\,\overline{f}_{kl}(t)\,dt.
\tag{4}
]

Definition 1. The totality of numbers ({\lambda_{kl},\alpha_{kl};\ 1\le k\le n-1;\ l=1,2,\ldots}) is called the ({C_\nu^a})-spectral characteristic of the operator (L). The functions (\varphi_{kl}(x)), (f_{kl}(x)) ((1\le k\le n-1;\ l=1,2,\ldots)) are called the ({C_\nu^a})-eigenfunctions of the operators (L) and (\widetilde L). Here ({C_\nu^a}) is the system consisting of the forms (C_1^0,\ldots,C_n^0,C_1^1,\ldots,C_n^1).

At the same time, for simplicity we assume* that:

(\alpha)) All eigenvalues (\lambda_{kl}) are simple and are distinct from one another for different (k).

For (1 \leq k \leq n-1), the complex numbers (\lambda_{kl}, \alpha_{kl}) ((l=1,2,\ldots)) determine the spectral measure (d\sigma_k(\lambda)) by the formula
[
\int F(\lambda)\,d\sigma_k(\lambda)=\sum_l \frac{1}{\alpha_{kl}}F(\lambda_{kl})
]
and the corresponding Parseval equality is satisfied.

Theorem 1. The ({C_\nu^a})-spectral characteristic of an operator (L) uniquely determines the operator (L\in\Lambda) and the forms (C_\nu^a) of the form (2).

We outline the proof. Suppose that ({C_\nu^a}) is the spectral characteristic of the operator (L), equal to ({D_\nu^a}), the spectral characteristic of the operator (L_1). For complex (\lambda), define the functions (\varphi_{r\lambda}(x)) and (\psi_{r\lambda}(x)) ((r=1,\ldots,n)) by the relations
[
L\varphi_{r\lambda}=\lambda\varphi_{r\lambda},\qquad
C_\nu^0\vec{\varphi}{r\lambda}(0)=0,\qquad
C_r^0\vec{\varphi}(0)=1,\qquad
C_\nu^1\varphi_{r\lambda}(1)=0;
\tag{5}
]
[
L_1\psi_{r\lambda}=\lambda\psi_{r\lambda},\qquad
D_\nu^0\vec{\psi}{r\lambda}(0)=0,\qquad
D_r^0\vec{\psi}(0)=1,\qquad
D_\nu^1\psi_{r\lambda}(1)=0.
\tag{6}
]
[
(r<\nu\leq n,\quad n-r+1<\nu'\leq n).
]

Define the ((n,n))-matrix (T_\lambda^c(x)) by the equality
[
T_\lambda^c(x)\vec{\varphi}{r\lambda}(x)=\vec{\psi}(x)\qquad (1\leq r\leq n).
\tag{7}
]

Computing the poles of the functions (\varphi_{r\lambda}(x)) and (\psi_{r\lambda}(x)) as analytic functions of (\lambda), we obtain that (T_\lambda^c(x)) is an entire analytic function of (\lambda). From the asymptotics of (\varphi_{r\lambda}(x)) and (\psi_{r\lambda}(x)) it follows that (T_\lambda^c(x)=o(|\lambda|)) as (\lambda\to\infty). Therefore (T_\lambda^c(x)) does not depend on (\lambda); hence from (5)—(7) it follows readily that (L=L_1) and (C_\nu^a=D_\nu^a) for all (\nu) and (a).

Definition 2. Let (L) be a self-adjoint operator and, for (\alpha=1,\ldots,n-1), let problem (3) with (k=\alpha) be adjoint to problem (3) with (k=n-\alpha). Then the system of forms ({C_\nu^a}) is called self-adjoint.

Theorem 2. An operator (L) and a system of forms ({C_\nu^a}) of the form (2) are self-adjoint if and only if
[
\lambda_{n-k,l}=\overline{\lambda}{k,l},\qquad
\alpha\qquad}=\overline{\alpha}_{k,l
(1\leq k\leq n-1;\; l=1,2,\ldots),
]
where ({\lambda_{kl},\alpha_{kl}}) is the ({C_\nu^a})-spectral characteristic of the operator (L).

For (n=2), Theorems 1 and 2 yield the known theorem that the spectral function of the Sturm–Liouville equation on a finite interval uniquely determines the equation and the boundary conditions.

(2^\circ). We now construct transformations of the operator (L) into other operators from (\Lambda). We construct these transformations with the aid of a system of numbers (\delta_i^A) and functions (\varphi_i(x)) and (f_i(x)) satisfying the relations
[
\sum_i|\delta_i^A-1|<\infty,\qquad
L\varphi_i=\lambda_i\varphi_i,\qquad
\widetilde{L}f_i=\overline{\lambda}_i f_i,\qquad
[\varphi_i,\tilde f_i]=0,
\tag{8}
]
where the index (i) runs through a countable set of values, (\lambda_i) are complex numbers, and (\varphi,\tilde f) is the Lagrange form of the operator (L). We shall also assume that the functions (\varphi_i(x)) and (\tilde f_i(x)) satisfy the following condition:

(\beta)) There exist nonnegative bounded functions (\tilde{\varphi}_i(x)) and (\tilde f_i(x)) such that the series
[
\sum_i\left(1+|\lambda_i|^{\frac{n-1}{n}}\right)\tilde{\varphi}_i(x)\tilde f_i(x)
]
converges uniformly, and

* This restriction and restriction (2), imposed on the forms (C_\nu^a), are inessential.

the inequalities hold:
[
|\varphi_i^{(k)}(x)|\leq (1+|\lambda_i|^{k/n})\widetilde{\varphi}_i(x),\qquad
|f_i^{(k)}(x)|\leq (1+|\lambda_i|^{k/n})\widetilde{f}_i(x)\quad (k=0,\ldots,n);
]
[
\left|\int_0^x \varphi_i(t)\overline{f_i(t)}\,dt\right|\leq \widetilde{\varphi}_i(x)\widetilde{f}_i(x),
]
[
|\varphi_i,f_j|\leq |\lambda_i-\lambda_j|\widetilde{\varphi}_i(x)\widetilde{f}_j(x)
\quad (i\ne j,\;0\leq x\leq 1).
]

Preliminarily, with the aid of (\varphi_i, f_i), and (\delta_i^A), we form the infinite matrix
[
A(x)=|A_{ij}(x)|,
]
[
A_{ij}(x)=\frac{\varphi_i,f_j}{\lambda_i-\lambda_j}\quad \text{for } i\ne j,\qquad
A_{ii}(x)=\delta_i^A+\int_0^x \varphi_i(t)\overline{f_i(t)}\,dt.
\tag{9}
]

We shall call (A(x)) the ({\delta_i^A,\varphi_i,f_i})-matrix; ({\delta_i^A,\varphi_i,f_i}) denotes the totality of the numbers (\delta_i^A) and the functions (\varphi_i(x)) and (f_i(x)), corresponding to all possible (i). Using (7), one can prove that the infinite determinant (\det A(x)) converges and depends continuously on (x). Assuming that
[
\det A(x)\ne 0\qquad (0\leq x\leq 1),
\tag{10}
]
we construct the matrix
[
[A(x)]^{-1}=B(x)=|B_{ij}(x)|
\tag{11}
]
and the functions
[
\psi_i(x)=\sum_j B_{ij}(x)\varphi_j(x);\qquad
g_i(x)=-\sum_j \overline{B_{ji}(x)}f_j(x).
\tag{12}
]

Lemma 1. If conditions (8), (\beta)), and (10) are satisfied, then there exists an operator (L_1\in\Lambda) such that, for all (i),
[
L_1\psi_i=\lambda_i\psi_i,\qquad \widetilde{L}_1 g_i=\overline{\lambda_i}g_i,\qquad [\psi_i,g_i]=0,
]
where (\psi_i(x)) and (g_i(x)) are determined by formulas (12), (11), (9); (\widetilde{L}_1) is the operator adjoint to (L_1); ([\psi,g]_1(x)) is the Lagrange form of the operator (L_1).

Definition 3. The ({\delta_i^A,\varphi_i,f_i})-transformation of the operator (L) is the operator (L_1) indicated in Lemma 1.

Theorem 3. Under the conditions of Lemma 1, if
[
\delta_i^B=\overline{B_{ii}(0)},
]
then (B(x)) is the ({\delta_i^B,\psi_i,g_i})-matrix, and (L) is the ({\delta_i^B,\psi_i,g_i})-transformation of the operator (L_1). There exists a triangular ((n,n))-matrix (T(x)) such that, if (\lambda\ne \lambda_i) for all (i), then the formula
[
\vec{\psi}{\lambda}(x)=T(x)\vec{\varphi}(x)
-\sum_i \frac{\varphi_{\lambda},f_i}{\lambda-\lambda_i}\vec{\psi}i(x)
\tag{13}
]
establishes a one-to-one correspondence between the functions (\varphi\lambda(x)) such that (L\varphi_\lambda=\lambda\varphi_\lambda), and the functions (\psi_\lambda(x)) such that (L_1\psi_\lambda=\lambda\psi_\lambda).

Formula (13) can be written with the aid of ((n,n))-matrices (T_\lambda(x)) and (Q_i(x)):
[
\vec{\psi}{\lambda}(x)=T}(x)\vec{\varphi{\lambda}(x),\qquad
T;}(x)=T(x)-\sum_i \frac{Q_i(x)}{\lambda-\lambda_i
\tag{14}
]
[
Q_i(x)\vec{\varphi}(x)=\varphi,f_i\vec{\psi}_i(x)
\tag{15}
]
for an arbitrary (\varphi(x)).

(3^\circ). Definition 4. The ({\varphi_{kl},f_{kl}})-transformation is the ({\delta_i^A,\varphi_i,f_i})-transformation of the operator (L) such that the index (i=(k,l)),

(\lambda_i=\lambda_{kl}), (\psi_i=\varphi_{kl}), (f_i=f_{kl}), and (\delta_i^A=1), where (k=1,\ldots,n-1); (l=1,2,\ldots); (\varphi_{kl}(x)) and (f_{kl}(x)) are the ({C_\nu^a})-eigenfunctions of the operators (L) and (\widetilde L), normalized arbitrarily; (\lambda_{kl}) are the corresponding eigenvalues; ({\varphi_{kl}, f_{kl}}) denotes the totality of the functions (\varphi_{kl}(x)) and (f_{kl}(x)), corresponding to all possible (k) and (l).

In the case of ({\varphi_{kl}, f_{kl}})-transformations, the matrices (A(x)), (T_\lambda(x)), and (T(x)), introduced in §2 for arbitrary ({\delta_i^A,\varphi_i,f_i})-transformations, will be denoted respectively by (A^c(x)), (T_\lambda^c(x)), and (T^c(x)); for (i=(k,l)) we put (\psi_i(x)=\psi_{kl}(x)), (g_i(x)=g_{kl}(x)).

Lemma 2. If

[
\sum_{k=1}^{n-1}\sum_{l=1}^{\infty} e^{n-1}
\max_{0\le \alpha,\beta\le n-1}
\left| e^{-\alpha-\beta}\varphi_{kl}^{(\alpha)}(0) f_{kl}^{(\beta)}(0)\right|<\infty
\tag{16}
]

and (\det A^c(x)\ne0) ((0\le x\le1)), then there exists an operator (L_1)—a ({\varphi_{kl}, f_{kl}})-transformation of the operator (L). The matrix (T_\lambda^c(x)) satisfies relations of the form (7), (5), and (6), where

[
D_\nu^a=C_\nu^a[T^c(a)]^{-1}.
\tag{17}
]

On the basis of Lemma 2, for a fixed system of forms ({C_\nu^a}), by changing the normalization of the ({C_\nu^a})-eigenfunctions (\varphi_{kl}) and (f_{kl}) ((1\le k\le n-1;\ 1\le l<\infty)), we obtain an infinite set of ({\varphi_{kl}, f_{kl}})-transformations of the operator (L). The asymptotics of (\varphi_{kl}(x)), (f_{kl}(x)), (\lambda_{kl}), and (\alpha_{kl}) is determined by the methods of ({}^{(8)}). In particular, we obtain that

[
\alpha_{kl}=\varepsilon_k e^{-(n-2)}
\left[
1+\sum_{m=1}^{n} h_{km}(L,{C_\nu^a})e^{-m}+e^{-n}o_{kl}
\right],
\qquad
\sum_{kl}|o_{kl}|^2<\infty,
]

where (\varepsilon_k) are constants, and (h_{km}(L,{C_\nu^a})) are expressed in finite form in terms of (L) and ({C_\nu^a}).

Theorem 4. Let (L,L_1\in\Lambda); let ({\lambda_{kl},\alpha_{kl}}) and ({\mu_{kl},\beta_{kl}}) be respectively the ({C_\nu^a})-spectral characteristic of the operator (L) and the ({D_\nu^a})-spectral characteristic of the operator (L_1), and suppose that condition a) is satisfied for the eigenvalues (\lambda_{kl}) and (\mu_{kl}). In order that the operator (L_1) be a ({\varphi_{kl}, f_{kl}})-transformation of the operator (L) for some normalization of the ({C_\nu^a})-eigenfunctions (\varphi_{kl}(x)) and (f_{kl}(x)) ((1\le k\le n-1;\ 1\le l<\infty)), and that (17) hold, it is necessary and sufficient that the relations

[
\lambda_{kl}=\mu_{kl},\qquad
h_{km}(L,{C_\nu^a})=h_{km}(L_1,{D_\nu^a})
]

[
(1\le k\le n-1;\quad 1\le l<\infty;\quad 1\le m\le n)
]

be satisfied.

There is the equality

[
\beta_{kl}=
\frac{\alpha_{kl}}
{1+\displaystyle\int_0^1 \varphi_{kl}(t)\overline{f}_{kl}(t)\,dt}
\qquad
(1\le k\le n-1,\quad 1\le l<\infty).
]

The ({D_\nu^a})-eigenfunctions of the operators (L_1) and (\widetilde L_1) are the functions (\psi_{kl}(x)) and (g_{kl}(x)).

Received
22 VIII 1961

References Cited

({}^{1}) I. M. Gel'fand, B. M. Levitan, Izv. Akad. Nauk SSSR, Ser. Mat., 15, 309 (1951).
({}^{2}) M. G. Krein, Dokl. Akad. Nauk SSSR, 76, No. 1, 21 (1951).
({}^{3}) V. A. Marchenko, Tr. Mosk. Mat. Obshch., 1, 327 (1952); 2, 3 (1953).
({}^{4}) L. A. Sakhnovich, Mat. Sb., 46, No. 1 (1958).
({}^{5}) V. I. Macaev, Dokl. Akad. Nauk SSSR, 130, No. 3, 499 (1960).
({}^{6}) M. K. Fage, Tr. Mosk. Mat. Obshch., 7, 227 (1958).
({}^{7}) Helge von Koch, Acta Math., 24, 89 (1900).
({}^{8}) M. A. Naimark, Linear Differential Operators, Moscow, 1954.