Computation of the eigenpairs of two-parameter Sturm-Liouville problems with three-point boundary conditions. (English) Zbl 1056.34089

The authors present a method to compute the eigenpairs of the two-parameter Sturm-Liouville problem with three-point boundary conditions given by \[ -y'' + q(y) = (\mu_1^2 w_1 + \mu_2^2 w_2) y,\;0 < x < 1\quad \text{and}\quad y(0) = y(\gamma) = y(1) = 0, \] where \(w_1, w_2\) are positive and in \(C^2 [0, 1], q \in L^1[0, 1]\) and \(0 < \gamma < 1\).
To this end, they consider the intersection of the eigencurves \(y(1; \mu_1,\mu_2) = 0\) and \(y(\gamma; \mu_1, \mu_2) = 0\), where \(y(x; \mu_1, \mu_2)\) is the solution of the initial value problem \[ y'' + (\mu_1^2 w_1 + \mu_2^2 w_2) y = q(y),\quad 0 < x < 1, \] with \(y(0) = 0, y'(0) = 1\).
They prove that as a function of \((\mu_1, \mu_2)\), \(\tilde y (x; \mu_1, \mu_2) := y(x; \mu_1, \mu_2) \alpha (x; \mu_1, \mu_2)\), lies in the Paley-Wiener space \(PW_{2 \sigma_1 (x), 2 \sigma_2 (x)}\), where \[ \alpha (x; \mu_1, \mu_2) = \text{sinc} \left( \sqrt {4x \int_0^x [\mu_1^2 w_1 (\tau) + \mu_2^2 w_2 (\tau) ] \,d \tau } \right), \] \(\text{sinc} (z) = z^{-1} \sin z\), \(\sigma_i (x) = 2 [ x \int_0^x w_i (\xi) \,d \xi ]^{1/2}\) for \(i = 1, 2\), and \(PW_{\beta_1, \beta_2}\) is the set of those entire functions \(h(z_1, z_2)\) such that \(| h(z_1, z_2) | \leq C \exp (\beta_1 | z_1 | + \beta_2 | z_2 | )\) and \(\int_{- \infty}^\infty \int_{-\infty}^\infty | h (z_1, z_2) | \,d z_1\, d z_2 < \infty\). Then they apply the Whitaker-Shannon-Kotel’nikov sampling theorem which states for \(f \in PW_{\beta_1, \beta_2}\), \[ f(\mu_1, \mu_2) = \sum_{j = - \infty}^\infty \sum_{k = - \infty}^\infty f(\mu_{1j}, \mu_{2k}) \text{sinc} (\beta_1 (\mu_1 - \mu_{1j})) \text{sinc} (\beta_2 (\mu_2 - \mu_{2k})), \] the convergence is uniformly on compact subsets of \(\mathbb C^2\) and in \(L_{d \mu_1 d \mu_2} (\mathbb R^2)\). That is, for \(c \in (0, 1], \tilde y (c; \mu_1, \mu_2)\) can be recovered from its samples at the lattice points \(\tilde y (c; \mu_{1j}, \mu_{2k})\) with \((\mu_{1j}, \mu_{2k}) = (j \pi /(2 \sigma_1 (c)), k \pi /(2 \sigma_2 (c)))\), \((j, k) \in \mathbb{Z}^2\).
In this paper, the authors demonstrate the above-mentioned computational result for \( q \equiv 0, w_1 \equiv 1, w_2 = e^{2x}\).


34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B05 Linear boundary value problems for ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory