On the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces
Let \(I_ j = (a_ j,b_ j)\), \(j=1,2,\dots,q\) be open intervals in \(\mathbb{R}\), and denote by \(T_ 0(M_ j)\), \(T_ 1(M_ j)\) the minimal and maximal operators respectively generated by a formally symmetric \(n\)th- order differential expression \(M_ j\) in \(L^ 2(I_ j)\). Set \(M=(M_ 1,\dots,M_ q)\) and define the direct sums \(T_ 1(M)=\bigoplus^ q_{j=1} T_ 1(M_ j)\), \(T_ 0(M) = \bigoplus^ q_{j=1} T_ 0(M_ j)\) and \(H = \bigoplus^ q_{j=1} L^ 2(I_ j)\). Then \(T_ 0(M)\) is a closed symmetric operator in \(H\) with adjoint \(T_ 1(M)\) and deficiency indices \(d^ + = \sum^ q_{j=1} d^ +_ j\), \(d^ - = \sum^ q_{j=1} d^ -_ j\), where \((d^ +_ j,d^ -_ j)\) are the deficiency indices of \(T_ 0(M_ j)\). It is assumed that \(d^ +_ j = d^ -_ j\) and hence \(d^ + = d^ -\), so that \(T_ 0(M)\) has self- adjoint extensions. The author obtains a characterization of all these self-adjoint extensions in terms of solutions of \(M_ jy = \pm iy\) which are square integrable near the end-points \(a_ j\), \(b_ j\) of the intervals \(I_ j\). It is shown that there are self-adjoint extensions which can not be expressed as direct sums of operators in \(L^ 2(I_ j)\), a phenomenon already observed by \textit{W. N. Everitt} and \textit{A. Zettl} [Rocky Mt. J. Math. 16, 497-516 (1986; Zbl 0624.34020)]. Other recent contributions to this problem are [\textit{W. N. Everitt} and \textit{A. Zettl}, Proc. Lond. Math. Soc., III. Ser. 64, 524-544 (1992; Zbl 0723.34022)] and [\textit{S. E. Ibrahim}, Rocky Mt. J. Math. 22, No. 3, 877- 915 (1992)].
scientific article; zbMATH DE number 147078
scientific article