具有退化权的微分方程的特征函数展开式
On eigenfunction expansions of differential equations with degenerating weight
摘要：
设$A$是一个对称算子。通过边界三元组方法，我们用一个Nevanlinna参数$τ$来参数化$A$的所有出口空间扩展$\wt A=\wt A^*$和离散谱$\s(\wt A)$，并用抽象边界条件来刻画$\wt A$的Shtraus族.接下来，我们应用这些结果的特征值问题的$2r$ -阶微分方程$l[y]= ł\D(x)y$的区间$[a,b), \; -\infty <a<b\leq \infty,$受到$ł$依赖分离边界条件与整个运营商功能$C_0(ł) $和$C_1(ł)$，形成一个Nevanlinna对$(C_0,C_1)$。权重$\D(x)$是非负的，并且可能在某些间隔$(\a,\b)\subset \cI$上消失。我们证明了当方程的极小算子具有离散谱时（特别是拟正则方程），特征值问题的特征值集是$\bR$的无限子集，没有有限个极限点，且每个函数$y\in\LI$都允许特征函数展开式$y(x)=\sum_{k=1}^\infty y_k (x)$收敛于$\LI$ .此外，我们给出了一个明确的方法计算本征函数$y_k$在此扩展和指定的边界条件$y$暗示一致收敛的本征函数展开的y。这些结果发展了已知的情况下获得的正权重$\D$和更严格的类$ł$依赖的边界条件。
英文摘要：
Let $A$ be a symmetric operator. By using the method of boundary triplets we parameterize in terms of a Nevanlinna parameter $τ$ all exit space extensions $\wt A=\wt A^*$ of $A$ with the discrete spectrum $\s(\wt A)$ and characterize the Shtraus family of $\wt A$ in terms of abstract boundary conditions. Next we apply these results to the eigenvalue problem for the $2r$-th order differential equation $l[y]= ł\D(x)y$ on an interval $[a,b), \; -\infty <a<b\leq \infty,$ subject to $ł$-depending separated boundary conditions with entire operator-functions $C_0(ł) $ and $C_1(ł)$, which form a Nevanlinna pair $(C_0,C_1)$. The weight $\D(x)$ is nonnegative and may vanish on some intervals $(\a,\b)\subset \cI$. We show that in the case when the minimal operator of the equation has the discrete spectrum (in particular, in the case of the quasiregular equation) the set of eigenvalues of the eigenvalue problem is an infinite subset of $\bR$ without finite limit points and each function $y\in\LI$ admits the eigenfunction expansion $y(x)=\sum_{k=1}^\infty y_k (x)$ converging in $\LI$. Moreover, we give an explicit method for calculation of eigenfunctions $y_k$ in this expansion and specify boundary conditions on $y$ implying the uniform convergence of the eigenfunction expansion of y. These results develop the known ones obtained for the case of the positive weight $\D$ and the more restrictive class of $ł$-depending boundary conditions.
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