We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to \(-\,\infty \), a narrow cluster of finitely many eigenvalues tends to \(-\,\infty \), while the eigenvalues above this cluster remain bounded from below. Certain “rogue” eigenvalues break away from this cluster and tend even faster toward \(-\,\infty \). The spectrum can be visualized as the intersection points of two objects in the plane—a spiral curve depending on the Schrödinger potential, and a set of curves depending on the branching factor, the diameter of the tree, and the Robin parameter.

通过与提供图形角度的正交多项式的关系，我们研究了有限正则度量树上Schrödinger算子的谱。随着Robin顶点参数趋向于\（-\，\ infty \），具有有限多个特征值的狭窄簇趋于\（-\，\ infty \），而该簇上方的特征值仍从下面限制。某些“流氓”特征值脱离了这个簇，并且趋向于更快地趋向\（-\，\ infty \）。可以将频谱可视化为平面中两个对象的交点-一条螺旋曲线（取决于Schrödinger势），以及一组曲线（取决于分支因子，树的直径和Robin参数）。