On Darboux-Treibich-Verdier potentials. (English) Zbl 1242.34152
In the article, certain spaces of potentials of the Schrödinger operator \(-{d^2\over dz^2}+ u(z)\) are studied. The potential \(u(z)\) is called finite-gap if the operator has a finite number of gaps in its spectrum. A linear subspace \(V\) spanned by \(n\) meromorphic functions is called finite-gap if it contains an infinite subset \(K\subset V\) consisting of finite-gap potentials, such that \(K\) is dense in \(V\) in the Zariski topology. The subspace \(V\) is maximal if it is not contained in a larger finite-gap space.
The main result of the article claims that there are only three maximal finite-gap subspaces, of dimensions \(n= 5\), \(3\) and \(2\). The most general five-dimensional subspace is spanned by \(1\), the classical Weierstrass elliptic function and its shifts by three half-periods. It contains an infinite number of finite-gap Darboux-Treibich-Verdier potentials.
The crucial property of the finite-gap operators under consideration is the Painlevé-Kowalevskaya property: all solutions of the corresponding Schrödinger equation \[ -\psi''+ u(z)\psi= \lambda\psi \] are meromorphic in the whole complex plane for all \(\lambda\). The finite-gap potentials are also known to have only second-order poles with zero residues.
The author also notes that the set of all elliptic finite-gap potentials is much bigger than the Darboux-Treibich-Verdier family and its effective description is still to be done.
The main result of the article claims that there are only three maximal finite-gap subspaces, of dimensions \(n= 5\), \(3\) and \(2\). The most general five-dimensional subspace is spanned by \(1\), the classical Weierstrass elliptic function and its shifts by three half-periods. It contains an infinite number of finite-gap Darboux-Treibich-Verdier potentials.
The crucial property of the finite-gap operators under consideration is the Painlevé-Kowalevskaya property: all solutions of the corresponding Schrödinger equation \[ -\psi''+ u(z)\psi= \lambda\psi \] are meromorphic in the whole complex plane for all \(\lambda\). The finite-gap potentials are also known to have only second-order poles with zero residues.
The author also notes that the set of all elliptic finite-gap potentials is much bigger than the Darboux-Treibich-Verdier family and its effective description is still to be done.
Reviewer: Renat Gontsov (Moskva)
MSC:
| 34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |
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