Inverse Spectral Theory: The essence of Gel'fand-Levitan theory...

The Gelfand-Levitin Theorem establishes a relationship between a function's Fourier transform and the spectral density function of a self-adjoint operator. Specifically, it states that for a self-adjoint operator with a known spectral density function, the Fourier transform of the spectral function can be reconstructed from the kernel of the operator's resolvent. This theorem is particularly useful in quantum mechanics and signal processing for reconstructing potential functions or other operator characteristics from observed data.

Let us explain the essence of Gel'fand–Levitan theory in more detail. Let \( \psi(x, k) \) be as in equations (3.17) and (3.18). Then \( \psi(x, k) \) is an even and entire function of \( k \) in \( \mathbb{C} \) satisfying

$$ \psi(x, k) = \frac{\sin kx}{k} + o\left(\frac{e^{| \text{Im} k | x}}{|k|}\right) \text{ as } |k| \rightarrow \infty. $$

Here we recall the Paley–Wiener theorem. An entire function \( F(z) \) is said to be of exponential type \( \sigma \) if for any \( \epsilon > 0 \), there exists \( C_{\epsilon} > 0 \) such that

$$ |F(z)| \leq C_{\epsilon} e^{(\sigma + \epsilon)|z|}, \quad \forall z \in \mathbb{C}. $$

By virtue of Paley–Wiener theorem and the expression above, \( \psi(x, k) \) has the following representation

$$ \psi(x, k) = \frac{\sin kx}{k} + \int_{0}^{\infty} K(x, y) \frac{\sin ky}{k} \, dy. $$

Inserting this expression into equation (3.17), then \( K \) is shown to satisfy the equation

$$ (\partial^2_y - \partial^2_x + V(x))K(x, y) = 0. $$

The crucial fact is

$$ \frac{d}{dx} K(x, x) = V(x). $$

One can further derive the following equation

$$ K(x, y) + \Omega(x, y) + \int_{0}^{\infty} K(x, t)\Omega(t, y) \, dt = 0, \quad \text{for all } x > y, $$

where \( \Omega(x, y) \) is a function constructed from the S-matrix and information of bound states. This is called the Gel'fand–Levitan equation.

Thus, the scenario of the reconstruction of \( V(x) \) is as follows: From the scattering matrix and the bound states, one constructs \( \Omega(x, y) \). Solving for \( K(x, y) \) gives us \( K \), and the potential \( V(x) \) is obtained by the equation:

$$ V(x) = \frac{d}{dx} K(x, x). $$

What is the hidden mechanism? This is truly an ingenious trick, and it is not easy to find the key fact behind their theory. It was Kay and Moses who studied an algebraic aspect of the Gel'fand–Levitan method.

.. excerpt from

Inverse Spectral Theory: Part I

by Hiroshi Isozaki

Department of Mathematics
Tokyo Metropolitan University
Hachioji, Minami-Osawa 192-0397
Japan
E-mail: isozakih@comp.metro-u.ac.jp