ABSTRACT In this paper, we establish new renormalized oscillation theorems for discrete symplectic eigenvalue problems with Dirichlet boundary conditions. These theorems present the number of finite eigenvalues of the problem in the arbitrary interval using the number of focal points of a transformed conjoined basis associated with Wronskian of two principal solutions of the symplectic system evaluated at the endpoints a and b. We suppose that the symplectic coefficient matrix of the system depends nonlinearly on the spectral parameter and that it satisfies certain natural monotonicity assumptions. In our treatment, we admit possible oscillations in the coefficients of the symplectic system by incorporating their non-constant rank with respect to the spectral parameter.

中文翻译：

---

谱参数非线性相关的辛特征值问题的重整化振荡理论

摘要 在本文中，我们为具有 Dirichlet 边界条件的离散辛特征值问题建立了新的重整化振荡定理。这些定理使用与在端点 a 和 b 处评估的辛系统的两个主解的 Wronskian 相关联的变换联合基的焦点数，呈现了任意区间中问题的有限特征值的数量。我们假设系统的辛系数矩阵非线性地依赖于谱参数，并且它满足某些自然单调性假设。在我们的处理中，我们通过结合它们相对于谱参数的非常数秩来承认辛系统系数的可能振荡。