Computing the spectrum of non-self-adjoint Sturm–Liouville problems with parameter-dependent boundary conditions

Abstract

This paper deals with the computation of the eigenvalues of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method.A few numerical examples among which singular ones will be presented to illustrate the merit of the method and comparison made with the exact eigenvalues when they are available.

Numerical computation of the eigenvalues of a discontinuous Dirac system using the sinc method with error analysis

“…Hence, it avoids any (multiple) integration and keeps the number of terms in the Cardinal series manageable. It has been demonstrated that the method is capable of delivering higher order estimates of eigenvalues at a very low cost, see [4,5,[13][14][15]17]. The space of all f such that the previous conditions hold is the Paley-Wiener space of band-limited functions with band width σ , which will be denoted by PW 2 σ .…”

A sinc-method computation for eigenvalues of Schrödinger operators with eigenparameter-dependent boundary conditions

“…Then it is stated that, [9, p. 230] the problem with its general setting has a discrete spectrum. This is not true, not only in the general setting of [9], but also even if the coefficient matrix satisfies the analyticity and growth condition stated in [10]. For instance, the problem…”

“…In addition, the error analysis of [9,10], neglects the amplitude error which affects the convergence rate, see the estimates below and the examples at the end of the paper. By the sinc method, see [15,19] we mean the use of the Whittaker-Kotel'nikovShannon (WKS) sampling theorem, which states that if f (μ) is entire in μ of exponential type σ , σ > 0, which belongs to L 2 (R) when restricted to R, then f (μ) can be reconstructed via the sampling (interpolation) representation…”

“…In [9] a very general problem is treated. The author investigated the situation when conditions (1.2)-(1.3) are replaced by the more general ones V 1 (y) = a 11 (μ)y(0) + a 12 (μ)y (0) + a 13 (μ)y(1) + a 14 (μ)y (1) = 0,…”

Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems

“…A similar problem was treated in [4], [11]. We pose the Neumann boundary conditions to illustrate the performance of the proposed method in the case when one has to use approximations of both transmutation operators T f and T 1/f and there is no zero coefficient in the expression (6.15).…”