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