Abstract Generated abstract
This note addresses the effective reconstruction of a Sturm-Liouville equation on a finite interval from two spectra corresponding to the same left boundary condition and two different right boundary conditions. Building on the Gel'fand-Levitan reconstruction from eigenvalues and norming constants, it derives an explicit formula for the norming constants in terms of canonical products formed from the two spectral sequences, and computes the first correction term in their asymptotic expansion. The paper also states necessary and sufficient conditions under which two alternating real sequences with prescribed asymptotics arise as such spectra, and relates the number of exact asymptotic terms to the differentiability of the potential.
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Reports of the Academy of Sciences of the USSR
- Volume 150, No. 3
MATHEMATICS
B. M. LEVITAN
ON THE DETERMINATION OF A STURM–LIOUVILLE DIFFERENTIAL EQUATION FROM TWO SPECTRA
(Presented by Academician A. A. Dorodnitsyn on 27 XII 1962)
1. Consider the differential equation
\[ y''+\{\lambda-q(x)\}y=0 \tag{1} \]
with boundary conditions
\[ y'(0)-hy(0)=0, \tag{2} \]
\[ y'(\pi)+Hy(\pi)=0. \tag{3} \]
Here \(q(x)\) is a real continuous function; \(h, H\) are real numbers.
Denote by \(\lambda_0,\lambda_1,\lambda_2,\ldots,\lambda_n,\ldots\) the eigenvalues of the problem \((1)+(2)+(3)\), and by \(\psi_0(x),\psi_1(x),\ldots,\psi_n(x),\ldots\) the corresponding eigenfunctions, normalized by the condition
\[ \psi_n(0)=1. \]
It is well known that if \(q(x)\) is a sufficiently many times differentiable function, then, beginning with sufficiently large \(n\), the asymptotic formulas
\[ \sqrt{\lambda_n}=n+\frac{a_0}{n}+\frac{a_1}{n^3}+\cdots, \]
\[ \alpha_n=\int_0^\pi \psi_n^2(x)\,dx=\frac{\pi}{2}+\frac{b_0}{n^2}+\frac{b_1}{n^4}+\cdots, \tag{4} \]
hold, where
\[ a_0=\frac{h+H+h_1}{\pi},\qquad h_1=\frac12\int_0^\pi q(t)\,dt. \]
Replace condition (3) by the condition
\[ y'(\pi)+H_1y(\pi)=0, \tag{3'} \]
where \(H_1\ne H\). Denote the eigenvalues of the problem \((1)+(2)+(3')\) by \(\mu_0,\mu_1,\mu_2,\ldots,\mu_n,\ldots\).
By a known result of Borg \((^1)\), the numbers \(\{\lambda_n\}\) and \(\{\mu_n\}\) (where \(n=0,1,2,\ldots\)) uniquely determine the function \(q(x)\), as well as the numbers \(h,H\), and \(H_1\).
The problem of effectively constructing the Sturm–Liouville equation from two spectra was studied by M. G. Krein, who obtained a number of fundamental results in this direction \((^{2,3})\). In the present note another solution of this problem is given.
2. As I. M. Gel'fand and the author showed (see \((^4)\), especially § 11), equation (1) can be effectively reconstructed from the numbers \(\{\lambda_n\}\) and \(\{\alpha_n\}\), given together with their asymptotic expansions. Therefore the problem of the effective construction of equation (1) can be reduced to the computation of the num-
of the numbers \(\alpha_n\) from the two spectra of equation (1). This can be done in the following way. Put*
\[ \Phi_1(\lambda)=\prod_{n=0}^{\infty}\left(1-\frac{\lambda}{\lambda_n}\right),\qquad \Phi_2(\lambda)=\prod_{n=0}^{\infty}\left(1-\frac{\lambda}{\mu_n}\right). \]
Since \(\lambda_n=O(n^2)\), \(\mu_n=O(n^2)\), these infinite products converge for all \(\lambda\) and, consequently, are entire analytic functions.
It can be shown that
\[ \alpha_k=\frac{\pi^2}{C_1C_2(H-H_1)}\,\Phi_1'(\lambda_k)\Phi_2(\lambda_k), \]
where
\[ C_1=\frac{1}{\lambda_0}\prod_{n=1}^{\infty}\frac{n^2}{\lambda_n},\qquad C_2=\frac{1}{\mu_0}\prod_{n=1}^{\infty}\frac{n^2}{\mu_n}. \]
Theorem 1. Let
\[ \sqrt{\lambda_n}=n+\frac{a_0}{n}+\frac{a_1}{n^3}+O\left(\frac{1}{n^4}\right), \tag{5} \]
\[ \sqrt{\mu_n}=n+\frac{a_0'}{n}+\frac{a_1'}{n^3}+O\left(\frac{1}{n^4}\right), \tag{6} \]
with
\[ a_0'\ne a_0. \tag{7} \]
Then
\[ \alpha_k=\frac{\pi}{2}+\frac{b_0}{k^2}+O\left(\frac{1}{k^3}\right), \]
where
\[ b_0=\frac{\pi}{2}\left[-(\lambda_0+\mu_0)+2(a_0+a_0')+\frac{a_1'-a_1}{a_0'-a_0}+ a_0+\frac{1}{6}\pi^2(a_0+a_0')^2-(s_\lambda+s_\mu)\right], \]
with
\[ s_\lambda=\sum_{n=1}^{\infty}(\lambda_n-n^2-2a_0),\qquad s_\mu=\sum_{n=1}^{\infty}(\mu_n-n^2-2a_0'). \]
The proof is based on studying the asymptotic behavior of the expressions \(\Phi_1'(\lambda_k)\) and \(\Phi_2(\lambda_k)\) for large \(k\). In principle, our method makes it possible to compute arbitrarily many terms of the asymptotic expansion (4), but even the determination of \(b_1\) is associated with substantial computational difficulties. Therefore we restricted ourselves to computing \(b_0\).
- Our method also makes it possible to indicate necessary and sufficient conditions for two sequences of real numbers \(\{\lambda_n\}\) and \(\{\mu_n\}\) to be two spectra of a Sturm–Liouville equation.
We have proved the following theorem:
Theorem 2. Let the numbers \(\{\lambda_n\}\) and \(\{\mu_n\}\) satisfy the following conditions: a) the sequences \(\{\lambda_n\}\) and \(\{\mu_n\}\) alternate; b) the asymptotic formulas (5) and (6) and condition (7) hold.
Then there exists an equation of the form (1) with a continuous function \(q(x)\) and real numbers \(h,H\), and \(H_1\) such that the sequence \(\{\lambda_n\}\) is the spectrum
* If \(\lambda_j=0\), then the factor \((1-\lambda/\lambda_j)\) should be replaced by \(-\lambda\).
of the problem (1) + (2) + (3), and the sequence \(\{\mu_n\}\) is the spectrum of the problem (1) + (2) + (3′), and
\[ a'_0 - a_0 = \frac{1}{\pi}(H_1 - H). \]
If in the asymptotic expansions for \(\sqrt{\lambda_n}\) and \(\sqrt{\mu_n}\) there are \(k\) exact terms (not counting the first), then the function \(q(x)\) is \((k-2)\) times continuously differentiable. In particular, in order that an infinite classical asymptotic expansion exist for \(\sqrt{\lambda_n}\) and \(\sqrt{\mu_n}\), it is necessary and sufficient that the function \(q(x)\) be infinitely differentiable.
Received
22 XII 1962
REFERENCES
- G. Borg, Acta Math., 78, No. 2, 1 (1945).
- M. G. Krein, Dokl. Akad. Nauk SSSR, 76, 21 (1951).
- M. G. Krein, Dokl. Akad. Nauk SSSR, 76, 345 (1951).
- I. M. Gelfand, B. M. Levitan, Izv. Akad. Nauk SSSR, Ser. Matem., 15, 309 (1951).