Approximation of smooth periodic potentials by algebraic-geometrical potentials

Let $U$ be a smooth real-valued double-periodic potential on ${\Bbb C}^1={\Bbb R}^2$ with periods $1$ and $\tau$. Algebraic-geometrical potentials are the finite-gap potentials corresponding to Bloch varieties $\Gamma$ of finite genus.

Approximation property. Any smooth potential can be approximated by algebraic-geometrical ones with the same periods.

This approximation property would provide a different route to a strict proof that the Bloch variety is invariant under the Modified Novikov–Veselov hierarchy for all smooth potentials. The paper indicates that the invariance is proved for algebraic-geometrical potentials, while extending the proof to arbitrary smooth potentials requires additional analytic information about Bloch functions near infinity.