Fundamental solutions and eigenvalues of a boundary-value problem with transmission conditions (Q1881696)

The authors consider two regular Sturm-Liouville operators \(\eta_{1}(u):=-u^{\prime \prime }+q_{1}u\) for \(-1\leq x\leq 0,\) and \(\eta _{2}(u):=-u^{\prime \prime }+q_{2}u\) for \(0\leq x\leq 1\), that are linked by the boundary conditions \(u(0-)=h_{1}u(0+)\) and \(u^{\prime }(0-)=h_{2}u^{\prime }(0+)\). The remaining boundary conditions are \(u(-1)=0\) and \(\lambda bu(1)+(\lambda -a)u^{\prime }(1)=0.\) The condition at \(x=0\) is called transmission as the properties of the solution to the left of the origin are transmitted or linked with the solution to the right of zero. It is shown that the operator is selfadjoint if \(h_{1}h_{2}>0\) and that the product \(h_{1}h_{2}\) indeed appears in the denominator of the characteristic function.