On the boundary value problem for the Sturm-Liouville equation with the discontinuous coefficient (Q2847209)

The authors investigate the Sturm-Liouville operator given by NEWLINE\[NEWLINEH[y](x):=\frac{1}{\rho(x)}(-y''(x)+q(x)y(x))NEWLINE\]NEWLINE with separated boundary conditions, NEWLINE\[NEWLINEy'(0)-h y(0)=0, \qquad y'(\pi) + h_1 y(\pi)=0.NEWLINE\]NEWLINE Here, \(q \in L^2([0,\pi])\) and \(\rho\) is a piecewise constant function, \(\rho(x)\equiv 1\) for \(x\in [0,a]\) and \(\rho(x)\equiv\alpha^2\) for \(x\in (a,\pi]\), for some \(\alpha >0\), \(\alpha\neq 1\).NEWLINENEWLINEAfter deriving some integral representations for the eigensolutions of this operator, the authors give proofs of completeness and expansion theorems. The proofs use complex analytic methods, using the fact that eigenfunctions can be viewed as analytic functions of the spectral parameter. In particular, denoting by \(\omega_1(x,\lambda)\) the solution of \(Hy=\lambda^2 y\) with \(\omega_1(0,\lambda)=1\), \(\omega_1'(0,\lambda)=h\) and introducing \(\Delta(\lambda)=\omega_1(\pi,\lambda)+h_1 \omega_1'(\pi,\lambda)\), the function \(\Delta\) has as its set of zeros precisely the set of eigenvalues of the Sturm-Liouville operator.NEWLINENEWLINESome of the results can be compared to general results about Sturm-Liouville operators which are proved by spectral theoretic methods, e.g. in Chapter 9 of \textit{G. Teschl} [Mathematical methods in quantum mechanics. With applications to Schrödinger operators. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1342.81003)].