论文标题
在重量变化的微分方程的本本特征扩展上
On eigenfunction expansions of differential equations with degenerating weight
论文作者
论文摘要
令$ a $为对称操作员。通过使用边界三重态的方法，我们可以根据Nevanlinna参数$τ$参数化所有出口空间扩展名$ \ wt a = \ wt a = \ wt a^$ $ a $，带有离散的频谱$ \ s（\ wt a）$，并表征$ \ wt的$ \ wt a $ a $ a $ the $ a $ the $ trivation Bargect a $ trivation Bourcation Bourcaikearke Bargreagation Bargect Bourcaige Bearnection a $ \ wt。接下来，我们将这些结果应用于$ 2R $ -TH订单微分方程$ l [y] = \ d（x）y $的特征值问题，以$ [a，b），\; - \ infty <a <a <a <b \ leq \ infty，$受$ $ - $ dependende的边界条件，并带有整个操作员函数$ C_0（ol）$和$ C_1（ol）$，它们形成Nevanlinna Pair $（C_0，C_1）$。权重$ \ d（x）$是非负的，可能会在某些间隔$（\ a，\ b）\ subset \ ci $上消失。 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）$在$ \ li $中收敛。此外，我们给出了一种在此扩展中计算本本征$ y_k $的明确方法，并在$ y $上指定边界条件，这意味着y的特征功能扩展的均匀收敛。这些结果开发了针对正权重$ \ d $的情况获得的已知结果，以及$ $ $ $ deepended的边界条件的更限制的类别。
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.
