On the determination of a differential equation from its spectral function. (Über die Bestimmung einer Differentialgleichung nach ihrer Spektralfunktion.) (Russian) Zbl 0044.09301

Gel’fand, I. M.; Levitan, B. M.

Izv. Akad. Nauk SSSR, Ser. Mat. 15, 309-360 (1951).

Let the function \(q(x)\) be continuous for \(0\le x <l\) and \(h\) be a real constant. Furthermore, let \(\varphi(x, \lambda)\) be that solution of the differential equation
\[ y'' + (\lambda - q(x)) y = 0 \tag{1} \]
which satisfies the initial conditions
\[ \varphi(0, \lambda) = 1,\quad \varphi'(0, \lambda) = h. \tag{2} \]
Then if boundary conditions of Weyl’s type are prescribed at \(x = l\), the relations (1) and (2) define a nondecreasing function \(\rho(\lambda)\) such that
\[ \int_0^l \vert f(x)\vert^2 \,dx = \int_{-\infty}^\infty \vert E(\lambda)\vert^2 \,d\rho(\lambda) \tag{3} \]
holds for every function \(f(x)\in L^2(0,l)\), the function \(E(\lambda)\) being defined by the relations
\[ E_n(\lambda) = \int_0^n \varphi(x,\lambda) f(x) \,dx\text{ and }\lim_{n\to\infty} \int_{-\infty}^\infty \vert E(\lambda - E_n(\lambda)\vert^2 \,d\rho(\lambda) = 0. \]
Conversely, it is known that \(\rho(\lambda)\), \(h\) and the Weyl’s boundary condition being given, determine the “potential” function \(q(x)\) uniquely. (This was proved under special assumptions by N. Levinson [Danske Vid. Selsk., Mat.-Fys. Medd. 25, No. 9, 29 S. (1949; Zbl 0032.20702)], under Weyl’s conditions by V. A. Marchenko [Dokl. Akad. Nauk SSSR, n. Ser. 74, 657–660 (1950; Zbl 0040.34303)], and also by the reviewer [11. Skand. Mat.-Kongr., Trondheim 1949, 276–287 (1952; Zbl 0048.06802)].)
The paper under review presents existence and construction theorems, which correspond to these uniqueness theorems. A principal result is the following \((l = \infty)\):
Let \(\rho(\lambda) = \frac2{\pi} \sqrt{\lambda} + \sigma(\lambda)\), \(\lambda\ge 0\); \(\rho(\lambda) = \sigma(\lambda)\), \(\lambda < 0\) be a given nondecreasing function such that
a) the integral \(\int_{-\infty}^0 \exp\left(\sqrt{\vert\lambda\vert}x\right) \,d\rho(\lambda)\) exists for all \(x > 0\),
b) the function \(a(x) = \int_1^\infty \lambda^{-1}\cos\sqrt\lambda x \,d\sigma(\lambda)\) has 4 continuous derivatives for \(0\le x < \infty\).
Then there exists a function \(q(x)\), continuous for \(0\le x < \infty\) and a real number \(h\) which together define a function \(\varphi(x, \lambda)\) through (1) and (2), such that (3) holds with the given function \(\rho(\lambda)\). Interesting existence theorems in the case of the inverse Sturm-Liouville problems are also given.
In the proofs a linear integral equation (4) is important. Assume that (1) and \(h\) are given, then there exists a Fredholm kernel \(K(x, t)\) such that
\[ \varphi(x, \lambda) = \cos\sqrt\lambda x + \int _0^x K(x,t) \cos\sqrt\lambda t \,dt \]
holds. This relation can be transformed into
\[ f(x,y) + \int _0^x f(y,t) K(x,t)\,dt + K(x,y) = 0 \tag{4} \]
where \(f(x,y) = \int_{-\infty}^\infty \cos \sqrt\lambda x \cos \sqrt\lambda y \,d\sigma(\lambda)\).
If now, conversely, \(\rho(\lambda)\) is known whereas \(q(x)\) and \(\varphi(x, \lambda)\) are not, (4) can be used for the determination of \(K(x,t)\) and hence \(\varphi(x, \lambda)\).

Reviewer: Göran Borg

MSC:

34L05

General spectral theory of ordinary differential operators

Keywords:

ordinary differential equations; spectral function

Citations:

Zbl 0032.20702; Zbl 0040.34303; Zbl 0048.06802