finitegap

Finitegap, often written finite-gap or finite gap, refers to a class of potentials in one-dimensional spectral problems and, more generally, to certain Lax operators arising in integrable systems, whose spectral data comprise a finite number of spectral bands.

For the Schrödinger operator L = -d^2/dx^2 + u(x), a potential u is finite-gap if the spectrum of L on the real line is a union of a finite number g+1 of closed intervals, [E_0, E_1], [E_2, E_3], ..., separated by gaps. The spectrum is purely absolutely continuous, and Floquet-Bloch theory applies. The finite-gap property is preserved under the flows of the Korteweg-de Vries (KdV) hierarchy, and such potentials yield quasi-periodic, exactly solvable solutions.

The finite-gap construction is inherently algebro-geometric. To a finite-gap potential one associates a hyperelliptic Riemann surface of genus g, defined by μ^2 = ∏_{j=0}^{2g+1} (E - E_j). A Baker-Akhiezer function with prescribed analytic properties on this curve serves as a universal tool to generate the potential, which can be expressed in terms of theta functions on the curve’s Jacobian. Conversely, given the branch points E_j, the associated u(x) can be recovered via the Krichever correspondence, producing explicit, finite-band solutions.

In integrable systems, finite-gap potentials provide stationary and time-evolving solutions to equations such as the KdV and nonlinear Schrödinger equations. The finite-gap method, developed by Novikov, Its, Matveev, Dubrovin, and others, is a cornerstone of the algebro-geometric approach to nonlinear wave phenomena and has applications in mathematical physics and spectral theory.