An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrodinger operator $-\Delta+V(x)$ in $R^n$ with periodic potential near the edges of the spectrum. A well known conjecture says that generically (with respect to perturbations of the periodic potential) the extrema are attained by a single branch of the dispersion relation, are isolated, and have non-degenerate Hessian (i.e., dispersion relations are graphs of Morse functions). The important notion of effective masses in solid state physics, as well as Liouville property, Green's function asymptotics, etc. hinge upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples exist in some discrete situations. We start establishing the following dichotomy: the non-degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters. The known counterexample has only two free parameters. This might be too tight for genericity to hold. We consider the maximal $Z^2$-periodic two-atomic nearest-cell interaction graph, with nine edges per unit cell and the discrete "Laplace-Beltrami" operator on it. We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph. We show three different approaches to the genericity, which might be suitable in various situations. It is also proven in this case that adding more parameters does not destroy the genericity result. We list all "bad" periodic subgraphs of the one we consider and discover that in all these cases genericity fails for "trivial" reasons only.

中文翻译：

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离散周期算子色散关系的一般性质

数学物理中的一个老问题涉及在光谱边缘附近具有周期性势能的薛定谔算子$-\Delta+V(x)$在$R^n$中的色散关系的结构。一个众所周知的猜想说，一般地（关于周期势的扰动）极值是由色散关系的一个分支获得的，是孤立的，并且具有非退化的 Hessian（即，色散关系是莫尔斯函数的图形） . 固体物理学中有效质量的重要概念，以及刘维尔性质、格林函数渐近性等都依赖于这一性质。证明这一猜想的进展缓慢。尝试研究离散问题是很自然的，其中色散关系（在适当的坐标中）是代数的，而不是解析的。此类模型通常用于固态物理学中的计算（紧束缚模型）。唉，在一些离散情况下存在反例。我们开始建立以下二分法：极值的非简并性要么失败，要么在参数的适当代数子集的补集中成立。已知的反例只有两个自由参数。这对于通用性来说可能太紧了。我们考虑最大 $Z^2$-周期性双原子最近单元相互作用图，每个单元具有 9 条边，并在其上使用离散的“Laplace-Beltrami”算子。然后，我们使用计算和组合代数几何的方法来证明该图的通用性猜想。我们展示了三种不同的通用性方法，它们可能适用于各种情况。在这种情况下也证明了增加更多的参数不会破坏通用性结果。我们列出了我们考虑的那个周期子图的所有“坏”周期子图，并发现在所有这些情况下，通用性仅因“微不足道”的原因而失败。