Some inverse spectral problems for vectorial Sturm-Liouville equations (Q2764799)

The author investigates boundary problems generated by the system of equations NEWLINE\[NEWLINE y''(x)+(\lambda I_n -Q(x))y(x)=0 NEWLINE\]NEWLINE on the interval \([0,1]\) subject to homogeneous boundary conditions at zero and at unity. Here, \(Q(x)\) is a real symmetric \(n\times n\)-matrix-valued function. The aim is to generalize known results by Borg, Levinson, Hochstadt and Lieberman, Gesztesy and Simon to the case of \(n>1\). In particular, the author proves that if \(Q(x)\) is even, i.e. \(Q(x)=Q(1-x)\) and each of the Dirichlet eigenvalues is of multiplicity \(n\), then \(Q(x)=q(x)I_n\), where \(q(x)\) is a scalar-valued function.