Computing eigenvalues and Hermite interpolation for Dirac systems with eigenparameter in boundary conditions

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper we use the derivative sampling theorem ‘Hermite interpolations’ to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Using computable error bounds, we obtain eigenvalue enclosures. Examples with tables and illustrative figures are given. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc-method in Annaby and Tharwat (BIT Numer. Math. 47:699-713, 2007) and explain that the Hermite interpolations method gives remarkably better results.MSC:34L16, 94A20, 65L15.


Introduction
Let σ >  and PW  σ be the Paley-Wiener space of all L  (R)-entire functions of exponential type σ . Assume that f (t) ∈ PW  σ ⊂ PW  σ . Then f (t) can be reconstructed via the Hermitetype sampling series where S n (t) is the sequences of sinc functions cf. []. Both types of errors normally appear in numerical techniques that use interpolation procedures. In the following we summarize these estimates. The truncation error associated with (.) is defined to be where f N (t) is the truncated series It is proved in [] that if f (t) ∈ PW  σ and f (t) is sufficiently smooth in the sense that there exists k ∈ Z + such that t k f (t) ∈ L  (R), then, for t ∈ R, |t| < Nπ/σ , we have where the constants E k and ξ k,σ are given by The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be where f ( nπ σ ) and f ( nπ σ ) are approximate samples of f ( nπ σ ) and f ( nπ σ ), respectively. Let us assume that the differences ε n := f ( nπ σ )f ( nπ σ ), ε n := f ( nπ σ )f ( nπ σ ), n ∈ Z, are bounded by a positive number ε, i.e., |ε n |, |ε n | ≤ ε. If f (t) ∈ PW  σ satisfies the natural decay conditions  < ω ≤ , then for  < ε ≤ min{π/σ , σ /π, / √ e}, we have, [], e( + σ ) + (π/σ )A + M f ρ(ε) where the convergence is absolute and uniform on R and it is uniform on compact sets of C, cf.
Therefore f (t) ∈ PW  σ , i.e., f (t) also has an expansion of the form (.). However, f (t) can be also obtained by the term-by-term differentiation formula of (.) see [, p.] for convergence. Thus the use of Hermite interpolations will not cost any additional computational efforts since the samples f ( nπ σ ) will be used to compute both f (t) and f (t) according to (.) and (.), respectively. Consider the Dirac system which consists of the system of differential equations and the boundary conditions where r  (·), r  (·) ∈ L  (, ) and (.) http://www.boundaryvalueproblems.com/content/2013/1/36 The eigenvalue problem (.)-(.) will be denoted by (r, α, β, α , β ) when (α  , α  ) = (, ) = (β  , β  ). It is a Dirac system when the eigenparameter λ appears linearly in both boundary conditions. The classical problem when α  = α  = β  = β  = , which we denote by (r, α, β, , ), is studied in the monographs of Levitan and Sargsjan [, ]. Annaby and Tharwat [] used Hermite-type sampling series (.) to compute the eigenvalues of problem (r, α, β, , ) numerically. In [], Kerimov proved that (r, α, β, α , β ) has a denumerable set of real and simple eigenvalues with ±∞ as the limit points. Similar results are established in [] for the problem when the eigenparameter appears in one condition, i.e., when α  = α  = , (β  , β  ) = (, ) or equivalently when (α  , α  ) = (, ) and β  = β  = , where also sampling theorems have been established. These problems will be denoted by (r, α, β, , β ) and (r, α, β, α , ), respectively. The aim of the present work is to compute the eigenvalues of (r, α, β, α , β ) and (r, α, β, , β ) numerically by the Hermite interpolations with an error analysis. This method is based on sampling theorem, Hermite interpolations, but applied to regularized functions hence avoiding any (multiple) integration and keeping the number of terms in the Cardinal series manageable. It has been demonstrated that the method is capable of delivering higher-order estimates of the eigenvalues at a very low cost; see []. In Sections  and , we derive the Hermite interpolation technique to compute the eigenvalues of Dirac systems with error estimates. We briefly derive some necessary asymptotics for Dirac systems' spectral quantities. The last section contains three worked examples with comparisons accompanied by figures and numerics with the Lagrange interpolation method.
2 Treatment of (r, α, β, α , β ) In this section we derive approximate values of the eigenvalues of (r, α, β, α , β ). Recall that (r, α, β, α , β ) has a denumerable set of real and simple eigenvalues, cf. []. Let ϕ(·, λ) = (ϕ  (·, λ), ϕ  (·, λ)) be a solution of (.) satisfying the following initial: Here A denotes the transpose of a matrix A. Since ϕ(·, λ) satisfies (.), then the eigenvalues of the problem (r, α, β, α , β ) are the zeros of the function Similarly to [, p.], ϕ  (·, λ) and ϕ  (·, λ) satisfy the system of integral equations where T i and T i , i = , , are the Volterra operators defined by (.) http://www.boundaryvalueproblems.com/content/2013/1/36 For convenience, we define the constants Define h  (·, λ) and h  (·, λ) to be As in [] we split (λ) into two parts via The analyticity of S(λ) as well as estimate (.) are not adequate to prove that S(λ) lies in a Paley-Wiener space. To solve this problem, we will multiply S(λ) by a regularization factor. Let θ >  and m ∈ Z + , m ≥ , be fixed. Let F θ,m (λ) be the function We choose θ sufficiently small for which |θλ| < π . More specifications on m, θ will be given latter on. Then F θ,m (λ), see [], is an entire function of λ which satisfies the estimate What we have just proved is that F θ,m (λ) belongs to the Paley-Wiener space PW  σ with σ =  + mθ . Since F θ,m (λ) ∈ PW  σ ⊂ PW  σ , then we can reconstruct the functions F θ,m (λ) via the following sampling formula: Since all eigenvalues are real, then from now on we restrict ourselves to λ ∈ R. Since are not known explicitly. So, we approximate them by solving numerically N +  initial value problems at the nodes Using standard methods for solving initial problems, we may assume that for |n| < N , for a sufficiently small ε. From (.) we can see that F θ,m (λ) satisfies the condition (.) when m ≥  and therefore whenever  < ε ≤ min{π/σ , σ /π, / √ e}, we have Here In the following, we use the technique of [], where only the truncation error analysis is considered, to determine enclosure intervals for the eigenvalues; see also [, ]. Let λ * be an eigenvalue with |θλ * | < π , that is, Then it follows that ) has computable upper bound, we can define an enclosure for λ * by solving the following system of inequalities: Its solution is an interval containing λ * , and over which the graph Using the fact that uniformly over any compact set, and since λ * is a simple root, we obtain, for large N and sufficiently small ε, in a neighborhood of λ * . Hence the graph of G(λ)+( sin θλ θλ ) -m F θ,m,N (λ) intersects the graphs -| sin θλ θλ | -m (T N,m-,σ (λ) + A(ε)) and | sin θλ θλ | -m (T N,m-,σ (λ) + A(ε)) at two points with abscissae a -(λ * , N, ε) ≤ a + (λ * , N, ε) and the solution of the system of inequalities (.) is the interval I ε,N := aλ * , N, ε , a + λ * , N, ε and in particular λ * ∈ I ε,N . Summarizing the above discussion, we arrive at the following lemma which is similar to that of [] for Sturm-Liouville problems.
Proof Since (λ N ) -N (λ N ) = (λ N ) -(λ * ), then from (.) and after replacing λ by λ N , we obtain Using the mean value theorem yields that for some ζ ∈ J ε,N : Since the eigenvalues are simple, then for sufficiently large N inf ζ ∈I ε,N | (ζ )| >  and we get (.). The rest of the proof follows from the fact that N (λ) converges uniformly to (λ) in R and A(ε) →  when ε → .
Thus, R m,θ (λ) belongs to the Paley-Wiener space PW  σ with σ =  + mθ . Since R θ,m (λ) ∈ PW  σ ⊂ PW  σ , then we can reconstruct the functions R θ,m (λ) via the following sampling formula: Since all eigenvalues are real, then from now on we restrict ourselves to λ ∈ R. Since The samples {R θ,m ( nπ σ )} N n=-N and {R θ,m ( nπ σ )} N n=-N , in general, are not known explicitly. So, we approximate them by solving numerically N +  initial value problems at the nodes Using standard methods for solving initial problems, we may assume that for |n| < N , for a sufficiently small ε. From (.) we can see that R θ,m (λ) satisfies the condition (.) when m ≥  and therefore whenever  < ε ≤ min{π/σ , σ /π, / √ e}, we have where there is a positive constant M R θ ,m for which, cf. (.), As in the above section, we have the following lemma.
In the following section, we have taken θ = / (Nm), where σ =  + mθ , in order to avoid the first singularity of ( sin θλ N θλ N ) - .

Examples
This We also indicate the effect of the parameters m and θ by several choices. Each example is exhibited via figures that accurately illustrate the procedure near to some of the approximated eigenvalues. More explanations are given below. Recall that a ± (λ) and b ± (λ) are defined by respectively. Recall also that the enclosure intervals I ε,N := [a -, a + ] and I ε,N := [b -, b + ] are determined by solving respectively. We would like to mention that Mathematica has been used to obtain the exact values for the three examples where eigenvalues cannot be computed concretely. Mathematica is also used in rounding the exact eigenvalues, which are square roots.

Example  The Dirac system
is a special case of the problem treated in the previous section with r  (x) = r  (x) = x  , α  = β  = , α  = β  = β  =  and β  = -. The characteristic function is The function K(λ) will be        Table 9 For