MATHEMATICS

M. G. Gasymov

DETERMINATION OF A STURM–LIOUVILLE EQUATION WITH A SINGULARITY FROM TWO SPECTRA

(Presented by Academician I. N. Vekua on 21 X 1964)

1. V. A. Marchenko proved \((^1)\) that two different spectra of one singular Sturm–Liouville equation uniquely determine this equation. Later the author and B. M. Levitan, in the paper \((^2)\), indicated an effective method for constructing a singular equation from two spectra. However, these constructions are conditional in character, since it is assumed in advance that two sequences of numbers \(\lambda_0 < \lambda_1 < \cdots\) and \(\mu_0 < \mu_1 < \cdots\) are eigenvalues of one and the same equation. Therefore it is of interest to find conditions on the numbers \(\{\lambda_n\}\) and \(\{\mu_n\}\) under which they are two different spectra of one equation of a definite type. In the case of a regular equation this problem has been completely solved in papers \((^{3-7})\), and in the case of a singular equation defined on a half-axis, partially in paper \((^7)\). Paper \((^8)\) contains a survey of these results.

In the present paper we give a complete solution of the problem posed above in the case when the Sturm–Liouville equation is defined on the finite interval \([0,\pi]\), but at the point \(\pi\) has a singularity of the type \(l(l+1)/(\pi-x)^2\), where \(l\) is a positive integer. We note that in this paper we first solve the ordinary inverse Sturm–Liouville problem for such equations. The ordinary inverse problem for an equation with a singularity of the indicated type at both endpoints of the interval \((0,\pi)\) was solved in papers \((^{10,11})\).

1. Consider the differential equation

\[ -y''+\{l(l+1)/(\pi-x^2+q(x)\}y=sy \tag{1} \]

and the boundary conditions

\[ y'(0)-hy(0)=0; \tag{2} \]

\[ y(\pi)=0. \tag{3} \]

Here it is assumed that \(q(x)\) is a real function summable with its square on the interval \([0,\pi]\); \(l\) is a positive integer; \(s\) is a complex parameter; \(h\) is a real number. We shall denote the eigenvalues of problem (1)—(2)—(3) by \(\lambda_0<\lambda_1<\cdots\), and the corresponding eigenfunctions normalized by the condition \(y(0)=1\), by \(\varphi_n(x)\). The numbers

\[ \alpha_n=\int_0^\pi \varphi_n^2(x)\,dx \tag{4} \]

are called the norming constants of problem (1)—(2)—(3).

Theorem 1. In order that the sequences of numbers \(\lambda_0<\lambda_1<\cdots\) and \(a_0,a_1,\ldots\) respectively be the eigenvalues and norming constants of a boundary-value problem of the type (1)—(2)—(3) with a function \(q(x)\in L_2(0,\pi)\), it is necessary and sufficient that the following conditions be fulfilled:

\(1^\circ\). The numbers \(a_0,a_1,\ldots\) are positive.

2°. For large \(n\) the asymptotic formulas hold

\[ \lambda_n=(n+l/2)^2+a+a_n; \tag{5} \]

\[ \alpha_n=\pi/2+b_n/n, \tag{6} \]

where \(a\) is a constant number, and the series

\[ \sum_{n=0}^{\infty} a_n^2,\qquad \sum_{n=0}^{\infty} b_n^2 \tag{7} \]

converge.

The necessity of the conditions of the theorem follows from the known asymptotics for \(\lambda_n\) and \(\alpha_n\). The sufficiency follows from the subsequent theorems.

Theorem 2. If the sequences of numbers \(\lambda_0,\lambda_1,\ldots\) and \(\alpha_0,\alpha_1,\ldots\) are respectively the eigenvalues and norming constants of the boundary-value problem of type (1)—(2)—(3) for \(l=0\) with function \(q_1(x)\in L_2(0,\pi)\), then the numbers \(\lambda_p,\lambda_{p+1},\ldots\) and \(\alpha_p,\alpha_{p+1},\ldots\) are respectively the eigenvalues and norming constants of the problem of type (1)—(2)—(3) with potential

\[ 2p(2p+1)/(\pi-x)^2+q(x), \tag{8} \]

where \(q(x)\in L_2(0,\pi)\), and conversely, if the numbers \(\lambda_p,\lambda_{p+1},\ldots\) and \(\alpha_p,\alpha_{p+1},\ldots\) are the eigenvalues and norming constants of the problem of type (1)—(2)—(3) for \(l=2p\) and \(q(x)\in L_2(0,\pi)\), then the numbers \(\lambda_0,\lambda_1,\ldots,\lambda_p,\ldots\) and \(\alpha_0,\alpha_1,\ldots,\alpha_p,\ldots\), where \(\lambda_0,\lambda_1,\ldots,\lambda_{p-1}\) are arbitrary distinct numbers different from \(\lambda_p,\lambda_{p+1},\ldots\), and \(\alpha_0,\alpha_1,\ldots,\alpha_{p-1}\) are arbitrary positive numbers, are the eigenvalues and norming constants of the boundary-value problem of type (1)—(2)—(3) for \(l=0\).

Theorem 3. If the sequences of numbers \(\lambda_0,\lambda_1,\ldots\) and \(\alpha_0,\alpha_1,\ldots\) are respectively the eigenvalues and norming constants of the boundary-value problem

\[ -y''+q_1(x)y=sy; \tag{9} \]

\[ y'(0)-hy(0)=0; \tag{10} \]

\[ y'(\pi)+Hy(\pi)=0 \tag{11} \]

without singularities, then the numbers \(\lambda_p,\lambda_{p+1},\ldots\) and \(\alpha_p,\alpha_{p+1},\ldots\) are the eigenvalues and norming constants of the problem of type (1)—(2)—(3) for \(l=2p-1\); conversely, if the numbers \(\lambda_p,\lambda_{p+1},\ldots\) and \(\alpha_p,\alpha_{p+1},\ldots\) are the eigenvalues and norming constants of the boundary-value problem (1)—(2)—(3) for \(l=2p-1\), then the numbers \(\lambda_0,\lambda_1,\ldots,\lambda_p,\ldots\) and \(\alpha_0,\alpha_1,\ldots,\alpha_p,\ldots\), where \(\lambda_0,\lambda_1,\ldots,\lambda_{p-1}\) are distinct arbitrary numbers different from the numbers \(\lambda_p,\lambda_{p+1},\ldots\), and \(\alpha_0,\alpha_1,\ldots,\alpha_{p-1}\) are arbitrary positive numbers, are the eigenvalues and norming constants of the boundary-value problem of type (9)—(10)—(11) without singularities.

The converse assertions of these theorems follow from the asymptotic formulas (5), (6) and from the results of [9] (see also [8]).

We now outline the proofs of the direct assertion, for example of Theorem 2. Let \(\lambda_p,\lambda_{p+1},\ldots\) and \(\alpha_p,\alpha_{p+1},\ldots\) be the eigenvalues and norming constants of the boundary-value problem (1)—(2)—(3) for \(l=2p\), and let \(\varphi_p(x),\varphi_{p+1}(x),\ldots\) be the eigenfunctions normalized by the condition \(\varphi_n(0)=1\). Then one can verify that the functions

\[ \psi_n(x)=\varphi_n(x)+ \frac{\varphi_p(x)}{\alpha_p+\displaystyle\int_0^x \varphi_p^2(t)\,dt} \int_0^x \varphi_p(y)\varphi_n(y)\,dy \tag{12} \]

\[ (n=p+1,\ p+2,\ldots) \]

are eigenfunctions of an operator of type (1)—(2)—(3) for \(l=2p+2\), and prove our assertions.

We note that the transformation (12) makes it possible to reduce the study of the inverse problem for equation (1) with a singularity at the point \(\pi\) to the study of the inverse problem for an equation without singularities.

3. In this section results are formulated that give a complete solution of the inverse problem from two spectra for equation (1). In the boundary condition (2) we replace the number \(h\) by the number \(h_1\). We obtain a new boundary-value problem, whose eigenvalues we denote by \(\mu_0, \mu_1, \ldots\). By the methods of papers \((^{6-8})\) and using the results of Sec. 2, one can prove the following theorem.

Theorem 4. In order that the sequences of numbers \(\lambda_0, \lambda_1, \ldots\) and \(\mu_0, \mu_1, \ldots\) be eigenvalues of one and the same equation of type (1) with an integer \(l\) and with potential \(q(x) \in L_2(0,\pi)\), the following conditions are necessary and sufficient:

\(1^\circ.\) The numbers \(\lambda_n\) and \(\mu_n\) interlace.

\(2^\circ.\) The asymptotic formulas
\[ \lambda_n=\left(n+\frac{l}{2}\right)^2+a+a_n,\qquad \mu_n=\left(n+\frac{l}{2}\right)^2+b+b_n, \]
hold, where \(a\ne b\), and the series \(\sum_{n=0}^{\infty} a_n^2\) and \(\sum_{n=0}^{\infty} b_n^2\) converge.

\(3^\circ.\) The difference
\[ \mu_n-\lambda_n=b-a+\frac{c_n}{n}, \]
where the series \(\sum_{n=1}^{\infty} c_n^2\) converges.

Furthermore, if \(q(x)\) has \(m\) summable derivatives, then the function
\[ F(x)=\sum_{n=0}^{\infty}\left\{ \frac{\mu_n-\lambda_n}{b-a}\cos \sqrt{\lambda_n}\,x -\cos\left(n+\frac{l}{2}\right)x \right\} \]
has \(m\) absolutely continuous derivatives on the interval \((0,2\pi)\), and, conversely, if \(F(x)\) has \(m\) absolutely continuous derivatives on the interval \([0,2\pi]\), then \(q(x)\) has \(m\) summable derivatives on the interval \([0,\pi]\).

Corollary 1. If the numbers \(\lambda_n\) and \(\mu_n\) interlace and
\[ \lambda_n=\left(n+\frac{l}{2}\right)^2+a+\frac{a_1}{n^2}+O(1/n^3), \]
\[ \mu_n=\left(n+\frac{l}{2}\right)^2+b+\frac{b_1}{n^2}+O(1/n^3), \]
where \(a\ne b\), \(a_1,b_1\) are constant numbers, then they are eigenvalues of one and the same equation of type (1) with different boundary conditions at zero and with an absolutely continuous function \(q(x)\).

Apparently, Theorem 4 and its corollary remain valid also in the case of nonintegral \(l\ge \tfrac12\). But we have not succeeded in proving this hypothesis. We have only succeeded in proving the following theorem.

Theorem 5. If the numbers \(\lambda_n\) and \(\mu_n\) interlace and
\[ \lambda_n=\left(n+\frac{l}{2}\right)^2+a+\frac{a_1}{n^2}+O(1/n^3), \]
\[ \mu_n=\left(n+\frac{l}{2}\right)^2+b+\frac{b_1}{n^2}+O(1/n^3), \]
where \(l\) is any positive number greater than \(1/2\), \(a\ne b\), and \(a_1,b_1\) are constant numbers, then the numbers \(\lambda_n\) and \(\mu_n\) are eigenvalues of one and the same equation of the form
\[ -y''+q(x)y=sy\qquad (0\le x<\pi), \]
where, generally speaking, the function \(q(x)\) has a singularity at the point \(\pi\).

Moscow Institute of Physics and Technology
Received
16 IX 1964

REFERENCES

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