# On the eigenstructure of a Sturm-Liouville problem with an impedance boundary condition.

Abstract

Structural properties of the eigenvalues and eigenfunctions of a Sturm-Liouville problem with an impedance boundary condition are studied herein. This entails a Robin boundary condition with a complex boundary parameter. The thrust of the present investigation is provided by the mapping of Neumann to Dirichlet eigen-problems that obtains as the complex boundary parameter varies from zero to the point at infinity along a fixed direction in the complex plane. Various properties of the spectrum and modal functions are explored.

AMS Subject Classification: 34B24, 34L05, 34M25. Keywords: Sturm-Liouville problem, impedance boundary condition, non-self-adjoint boundary value problem.

1. Introduction

Herein, we explore the eigenstructure of the Sturm-Liouville boundary value problem (SL-BVP) with an impedance boundary condition (IBC)

u"(x) + [lambda] x u(x) = 0, 0 < x < L, u'(0) - [sigma] x u(0) = 0, u'(L) + [sigma] x u(L) = 0. (1.1)

The complex boundary parameter [sigma] = [[sigma].sub.R] + [??] x [[sigma].sub.I] is assumed to have the same value at both ends of the interval J = [0,L]. Attention will be restricted to this case as it is the one most relevant to the extension of the results for the eigenstructure of the equilateral triangle [4, 5] from real to complex values of [sigma] which furnishes the motivation for the present one-dimensional study.

[FIGURE 1 OMITTED]

Figure 1 illustrates the parallel plate waveguide problem which gives rise to the SL-BVP, Equation (1.1). Either an acoustic [8, pp. 485-496] or electromagnetic [3, pp. 81-84] time-harmonic wave of angular frequency [omega] is propagating in the z-direction with no field variation in the y-direction. The complex material parameter [sigma] is related to the wall impedance which accounts for field penetration into the walls of the waveguide. After the SL-BVP has been solved for the eigenvalue [lambda] and the corresponding eigenfunction (mode) u(x), the real physical field may be reconstructed as

U(x, z, t) = Re{u(x) x [e.sup.[??]{([omega]t - [gamma]z)}, [gamma] := [square root of - [lambda]]. (1.2)

This same SL-BVP arises in the study of the vibrating string with sidewise displacement of the end supports [6, pp. 133-134].

Much of the current mathematical literature devoted to SL-BVPs is concerned primarily with the self-adjoint case [1, 10]. However, the SL-BVP described by Equation (1.1) is non-self-adjoint for complex values of [sigma]. Fortunately, a classic reference on the non-self-adjoint case is provided by [2, pp. 298-305] where it is shown that this problem reduces to the study of the solutions of a single transcendental equation.

In what follows, we will derive and exhaustively study this transcendental equation. The principal focus will be on what happens to the eigenstructure of the Neumann problem ([sigma] = 0) as [sigma] proceeds along rays emanating from the origin toward the point at infinity in the complex plane. We will find that, for the most part, there is a natural homotopy connecting the Neumann modes to those of a corresponding Dirichlet problem (IBC-Dirichlet modes). However, we will show that, under appropriate conditions, there will be two Neumann eigenvalues that have no Dirichlet counterpart. The determination of the precise nature of these latter eigenvalues together with that of their corresponding eigenfunctions ("missing modes") is the primary concern of the present investigation.

2. Solution of the Sturm-Liouville Problem

Ignoring the boundary conditions, the general solution to Equation (1.1) is

u(x) = cos (v[pi]x/L - [delta]), [lambda] = [(v[pi]/L).sup.2]. (2.1)

Application of the boundary condition at x = 0 yields

tan ([delta]) = [sigma]L/v[pi], (2.2)

while imposition of the boundary condition at x = L yields

tan (v[pi] - [delta]) = [sigma]L/v[pi]. (2.3)

Equations (2.2) and (2.3) may be combined to produce the transcendental equation

(2[delta] + n[pi]) tan ([delta]) = [sigma]L, v = 2[delta]/[pi] + n; (2.4)

with n an integer.

Observe that in Equation (1.1) when [sigma] [right arrow] 0 we recover the Neumann problem, u'(0) = 0 = u'(L), whose solution, u(x) = [N.sub.n](x) := cos (n[pi]x/L), is obtained from Equation (2.4) with [delta] = 0 [??] v = n. Thus, we may profitably view [sigma] as a continuation parameter which provides a homotopy extending from this well understood problem to that of the impedance boundary condition. Throughout the ensuing development we will avail ourselves of this important observation.

Furthermore, note that if the normal derivative remains bounded then [sigma] [right arrow] [infinity] yields the Dirichlet problem, u(0) = 0 = u(L), whose solution, u(x) = [+ or -][D.sub.[n[+ or -]1](x) := [+ or -] sin ((n [+ or -] 1)[pi]x/L), is obtained from Equation (2.4) with [delta] = [+ or -][pi] 2) v = n [+ or -] 1. In that case, the homotopy may be further extended to lead from a Neumann mode to a corresponding Dirichlet mode. As will subsequently be shown, for the impedance boundary condition this is usually, although by no means always, the case.

Introduction of z : = [delta] + n[pi]/2 into Equation (2.4) reduces it to

z tan (z) = [sigma]L/2, (2.5)

if n is even and

z cot (z) = -[sigma]L/2, (2.6)

if n is odd. These two cases may then be separately studied from the graphical representation of their respective complex transformations [8, p. 909].

However, in the case of the equilateral triangle [4, 5] which is our ultimate goal, such a reduction is not available. Thus, Equation (2.4) itself will be numerically approximated using MATLAB. This will permit us to trace out trajectories in the complex [delta]-plane or, equivalently, in the complex v-plane as the complex boundary parameter [sigma] is varied.

[FIGURE 2 OMITTED]

As shown in Figure 2, beginning with [sigma] = 0 (the Neumann problem), we will allow [sigma] to move along a ray inclined at an angle [theta] to the horizontal toward the point at infinity. We may then track the trajectory of each Neumann eigenvalue in order to determine whether it eventually approaches a Dirichlet eigenvalue (in which case it will be called an IBC-Dirichlet mode) or migrates to infinity (in which case it will be called a "missing mode"). We will eventually find that these are the only two possible types of asymptotic behavior.

3. SL-BVP/IBC Solution Properties

Before constructing a taxonomy of the asymptotic nature of the eigenstructure of our SL-BVP with an IBC as [sigma] [right arrow] [infinity] (at which time, the mysterious icons on Figure 2 will become intelligible), it behooves us to first catalog some important properties of the eigenvalues and eigenfunctions of Equation (1.1).

We begin with the observation that, without loss of generality, we may restrict our attention to the case Im([sigma]) [greater than or equal to] 0 since, by taking complex conjugates in Equation (2.4), we find that [sigma] [??] [bar.[sigma]] [??] [delta] [??] [bar.[delta]] [??] v [??] [bar.v]. Thus, in the event that Im([sigma]) < 0, we can obtain trajectories in either the [delta]-plane or v-plane by reflection about the real axis of the corresponding trajectories for [bar.[sigma]].

Furthermore, taking complex conjugates in Equation (1.1) itself reveals that [sigma] [??] [bar.[sigma]] [??] [lambda] [??] [bar.[lambda]] =: [mu], u(x) [??] [bar.u](x) =: v(x) where [mu] and v(x) form the solution to the adjoint boundary value problem

v'(x) + [mu] x v(x) = 0, 0 < x < L,

v'(0) - [bar.[sigma]] x v(0) = 0,

v'(L) + [bar.[sigma]] x v(L) = 0. (3.1)

Since the SL-BVP is non-self-adjoint for complex [sigma], the eigenvalues and eigenfunctions can be complex and, most importantly, eigenfunctions corresponding to distinct eigenvalues are not necessarily orthogonal with respect to the complex inner product < f(x); g(x) >:= [[integral.sup.L.sub.0] f(x)[bar.g](x) dx. However, we do have the biorthogonality relationship involving the eigenfunctions of the boundary value problem, Equation (1.1), and those of of the adjoint boundary value problem, Equation (3.1),

< [u.sub.p](x), [v.sub.q](x) >:= [[integral].sup.L.sub.0] [u.sub.p](x)[[bar.v].sub.q](x) dx = 0. (3.2)

In turn, this provides us with the eigenfunction expansion

f(x) = [[infinity].summation over (k=1)] < f(x), [v.sub.k](x) > [u.sub.k](x), (3.3)

where the series converges in the mean for f(x) [member of] [L.sup.2](0, L) [2, pp. 310-312].

We have already seen via Equations (2.5) and (2.6) that the modes naturally partition according to the parity of n. This fact is underscored by

Theorem 1 The even/odd numbered modes are symmetric/antisymmetric, respectively, on the interval [0, L] for all values of [sigma].

Proof. Equation (3) may be recast as [u.sub.n](x) = cos (v[pi]/L (x - L/2) + n[pi/ 2). Thus, if n is even/odd then [u.sub.n](x) is a cosine/sine, respectively, centered at x = L/2.

4. The Case of Real [sigma]

In the case of real [sigma], the eigenstructure of the SL-BVP with an IBC defined by Equation (1.1) has been exhaustively treated in [9, pp. 90-98]. As a springboard for the study of the case of complex [sigma] in the next section, we review here the highlights of those results.

When [sigma] is real, the problem is self-adjoint so that the eigenvalues are real and the eigenfunctions can also be chosen to be real. Moreover, eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to the real inner product < f(x); g(x) >:= [[integral.sup.L.sub.0] f(x)g(x) dx. Because of their very different behaviors, we treat the cases [sigma] [greater than or equal to] 0 and [sigma] < 0 separately. We will eventually see that, taken together, they display the characteristic behaviors exhibited in the general case of complex [sigma].

In what follows, we denote the dependence of the eigenvalues and eigenfunctions upon [sigma] by [[lambda].sub.n]([sigma]) = [(v([sigma])[pi]/L).sup.2] and un(x; [sigma]), respectively, for n = 0, 1, .... The subscript n is chosen so that when [sigma] = 0 they reduce to the corresponding values for the Neumann problem, [d.sub.n](0) = 0 [??] [v.sub.n](0) = n and [u.sub.n](x; 0) = [N.sub.n](x) := cos(n[pi]x/L). Also, the Dirichlet problem has the same eigenvalues [[lambda].sub.n](0) but with the restriction n = 1, 2, ... and its corresponding eigenfunctions are denoted by un(x; 0) = [D.sub.n](x) := sin(n[pi]x/L).

4.1. The Case [sigma] [greater than or equal to] 0

[FIGURE 3 OMITTED]

The case of the radiation boundary condition ([sigma] > 0) is by far the simplest in that the eigenvalues are not only real but are in fact all positive. Moreover, n < [[upsilon].sub.n]([sigma]) < n + 1. It also possesses the simplest asymptotic behavior in that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.1)

That is, the Neumann mode [N.sub.n](x) "morphs" analytically into the Dirichlet mode [D.sub.n+1](x) as [sigma] ranges from 0 to [infinity]. This is illustrated in Figure 3 which displays this homotopy between fundamental modes (n = 0) for 0 [less than or equal to] [sigma] [less than or equal to] -[infinity].

4.2. The Case [sigma] < 0

The case of the absorbing boundary condition ([sigma] < 0) is more complicated in that the eigenvalues while still real are no longer all positive. However, for n = 2, 3, ... we have n - 1 < [v.sub.n]([sigma]) < n. These so-called IBC-Dirichlet modes possess the simple asymptotic behavior described by

[FIGURE 4 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.2)

That is, the Neumann mode [N.sub.n](x) "morphs" analytically into the Dirichlet mode - [D.sub.n-1](x) as [sigma] ranges from 0 to -[infinity]. This is illustrated in Figure 4 which displays this homotopy between [N.sub.2](x) and -[D.sub.1](x) for 0 [greater than or equal to] [sigma] [greater than or equal to] -[infinity].

[FIGURE 5 OMITTED]

This leaves open the case of the "missing modes" n = 0, 1. Since there are no Dirichlet modes for n = -1, 0 we clearly do not have the simple asymptotic behavior described by Equation (4.2). Thus we are confronted with the question: "What happens to the missing n = 0, 1 modes as [sigma] [right arrow] -[infinity]?". The solution to the mystery of the missing modes naturally decomposes into two special cases each of which we now explore separately.

[FIGURE 6 OMITTED]

For n = 0, [v.sub.0]([sigma]) is pure imaginary and, consequently, [[lambda].sub.0]([sigma]) is negative. Specifically, for [sigma] [right arrow] -[infinity],

[[delta].sub.0]([sigma]) [approximately equal to] - [sigma]L/2 x [??] [v.sub.0]([sigma]) [approximately equal to] - [sigma]L/[pi] x [??] [??] [u.sub.0](x; [sigma]) [approximately equal to] cosh (-[sigma]x + [sigma]L/2). (4.3)

This asymptotic expression for [u.sub.0](x; [sigma]) becomes unbounded as [sigma] [right arrow] -[infinity]. However, if we first scale it by its value at an endpoint, cosh ([sigma]L/2), we find that this normalized mode approaches 1 at the two endpoints and 0 elsewhere. Such singular limiting behavior, necessary since this does not approach a Dirichlet mode, is on prominent display in Figure 5.

For n = 1, [[nu].sub.1]([sigma]) initially decreases along with [sigma] until it vanishes when [sigma] = - 2/L. Specifically,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

However, if we first scale [u.sub.1](x; [sigma]) by its value at the endpoint x = 0, cos ([delta]), we find that this normalized mode approaches the straight line [u.sub.1](x; -2/L) = 1 - 2/L x x.

As [sigma] continues to decrease we have, for [sigma] < - 2/L, pure imaginary [[delta].sub.1]([sigma])+ [pi]/2 thereby producing a [v.sub.1]([sigma]) which is also pure imaginary and, consequently, [[lambda].sub.1]([sigma]) becomes negative. Specifically, for [sigma] [right arrow] -[infinity],

[[delta].sub.1]([sigma]) [approximately equal to] - [pi]/2 - [sigma]L/2 [??] [??] x [v.sub.1]([sigma]) [approximately equal to] - [sigma]L/[pi] x [??] [u.sub.1](x) [approximately equal to] [??] x sinh ([sigma]x - [sigma]L/2). (4.5)

This asymptotic expression for [u.sub.1](x; [sigma]) becomes unbounded as [sigma] [right arrow] -[infinity]. However, if we first scale it by its value at the endpoint x = 0, [??] x sinh (- [sigma]L/2), we find that this normalized mode approaches [+ or -]1 at the left-/right-hand endpoint, respectively, and 0 elsewhere. The resulting unbounded derivative, necessary since this does not approach a Dirichlet mode, is evident in Figure 6.

5. The Case of Complex [sigma]

[FIGURE 7 OMITTED]

In the case of complex [sigma], as previously noted, the SL-BVP defined by Equation (1.1) is non-self-adjoint and the eigenstructure is consequently complex. Defining the residual function of Equation (2.4) as

[[rho].sub.n]([delta]; [sigma]) = |(2[delta] + n[pi]) tan ([delta]) - [sigma]L|; (5.1)

we note that the sought-after eigenvalues are determined by its local minima. Next, define

[sigma] = r x [e.sup.x[theta]]; with [theta] fixed and 0 [less than or equal to] r [less than or equal to] + [infinity]. (5.2)

Figure 7, which is a contour plot of [[rho].sub.n], displays the resultant trajectory in the [delta]-plane for n = 0 as r varies with [theta] = [pi]/4. As occurred in the real case with [sigma] [greater than or equal to] 0, [delta] varies from 0 to [pi]/2 only now it makes an excursion into the complex plane rather than traveling along the real axis. The attendant complex mode morphing from [N.sub.0](x) to [D.sub.1](x) is made explicit by Figure 8.

Figure 9 displays the corresponding trajectories, this time in the [nu]-plane, for the the first four modes over the full range of values 0 [less than or equal to] [theta] [less than or equal to] [pi]. As is evident from each of these plots, for some values of [theta] mode morphing occurs, i.e [[lambda].sub.n]([sigma]) [right arrow] [[lambda].sub.n[+ or -]1](0), (IBC-Dirichlet modes) while for other values of [theta] we observe |[[lambda].sub.n]([sigma])| [right arrow] [infinity] (missing modes). We are going to devote our remaining efforts to clarifying this asymptotic behavior.

[FIGURE 8 OMITTED]

In order to achieve this, we will require detailed knowledge of the properties of trajectories in the v-plane. We commence with

Theorem 2 1. Let [[delta].sub.n](r) = [[delta].sup.R.sub.n] (r) + [??] x [[delta].sup.I.sub.n](r) and [v.sub.n](r) = [v.sup.R.sub.n] (r) + [??] x [v.sup.I.sub.n](r). Then, the trajectories [[delta].sub.n](r) and [v.sub.n](r) have the same slope.

(a) If n [not equal to] 0 then the trajectory [v.sub.n](r) makes an angle [theta] measured counterclockwise from the real axis at [v.sub.n](0) = n.

(b) If n = 0 then the trajectory [v.sub.0](r) makes an angle [theta]/2 measured counterclockwise from the real axis at [v.sub.0](0) = 0.

2. As r [right arrow] [infinity], either [v.sub.n](r) [right arrow] n [+ or -] 1 or |[v.sub.n](r)| [right arrow] [infinity].

(a) if [v.sub.n](r) [right arrow] n [+ or -] 1 then the trajectory [v.sub.n](r) makes an angle [pi] - [theta] measured counterclockwise from the real axis at [v.sub.n](1) = n [+ or -] 1.

(b) if |[v.sub.n](r)| [right arrow] [infinity] then the trajectory [v.sub.n](r) goes off to infinity at an angle [theta] - [pi]/2 measured counterclockwise from the real axis as r [right arrow] [infinity].

Proof.

1. Since [v'.sub.n](r) = 2/[pi] x [[delta]'.sub.n](r), both trajectories have slope [[delta].sup.I'.sub.n](r)=[[delta].sup.R'.sub.n](r).

[FIGURE 9 OMITTED]

(a) r = 0 [??] [sigma] = 0 so that Equation (2.4) produces [[delta].sub.n](0) = 0) [[nu].sub.n](0) = n. Substitution of Equation (5.2) into Equation (5.3) with subsequent differentiation with respect to r yields

[[delta]'.sub.n](r) = Le{[theta] [cos.sup.2] ([[delta].sub.n](r)) sin (2[[delta].sub.n](r)) + 2[[delta].sub.n](r) + n[pi] : (5.3)

Thus, [[delta]'.sub.n](0) = [Le.sup.[??][theta]]/n[pi] [??] tan [empty set] := [[delta].sup.I'.sub.n] (0)/[[delta].sup.R'.sub.n](0) = tan [theta] [??] [empty set] = [theta].

(b) For n = 0, Equation (5.3) produces [[delta]'.sub.0](0) = [infinity] so that the slope tan [empty set] := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is indeterminate. However, by Equation (5.3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Thus, tan [empty set] = tan ([theta] - [empty set]) [??] [empty set] = [theta]/2.

2. As r = |[sigma]| [right arrow] [infinity], Equation (2.4) clearly implies that either [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2, in which case [v.sub.n](r) [right arrow] n [+ or -] [infinity], or |[[delta].sub.n](r)| [right arrow] [infinity], in which case |[v.sub.n](r)| [right arrow] [infinity].

(a) If [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2 then [[nu].sub.n](r) [right arrow] n [+ or -] 1 and [[delta]'.sub.n](r) [right arrow] 0 by Equations (2.4) and (5.3), respectively. Thus, as r [right arrow] [infinity], [[delta]'.sub.n](r) [right arrow] [[epsilon].sub.1] [e.sup.[??][empty set]] and [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2 [+ or -][[epsilon].sub.2][e.sup.[??][empty set]] where [[empty set].sub.1], [[epsilon].sub.2] [right arrow] 0. Comparison of the arguments of both sides of Equation (5.3) reveals that [empty set] = [theta] + 2[empty set] [??] [empty set] = -[theta].

(b) If |[[delta].sub.n](r)| [right arrow] [infinity] then tan ([[delta].sub.n](r)) [right arrow] [??]. Inserting this into Equation (2.4) and equating real and imaginary parts produces 2[[delta].sup.R.sub.n] [right arrow] rL sin ([theta]) and -2[[delta].sup.I.sub.n] [right arrow] rL cos ([theta]). Thus, [[delta].sup.I.sub.n](r)/ [[delta].sup.R.sub.n] (r) [right arrow] -cot ([theta]) = tan (theta] - [phi]/2).

Returning now to Figure 9, note the following important structural features. For modes n = 0 and n = 1, there is a critical angle [[theta].sup.-.sub.n] such that for [theta] < [[theta].sup.-.sub.n] we have the mode morphing [N.sub.n](x) [??] [D.sub.n+1](x) while, for [theta] > [[theta].sup.-.sub.n], mode n is missing in the previously defined sense as r [right arrow] 1. For all other modes n [less than or equal to] 2, there are two critical angles [[theta].sup.-.sub.n] and [[theta].sup.+.sub.n]. For [theta] < [[theta].sup.-.sub.n] we have the mode morphing [N.sub.n](x) [??] [D.sub.n+1](x) and for [theta] > [[theta].sup.+.sub.n] we have the mode morphing [N.sub.n](x) [??] -[D.sub.n-1](x) as r [right arrow] [infinity]. In the intermediate regime [[theta].sup.-.sub.n] < [theta] < [[theta].sup.+.sub.n], mode n is missing as r [right arrow] [infinity].

Our next result concerns the trajectories at these critical angles.

Theorem 3 1. The critical trajectories all possess a corner located in the [delta]-plane at the roots of the function

[[tau].sub.n]([delta]) := sin (2[delta]) + 2[delta] + n[pi]: (5.4)

The smallest root with Re([[delta].sup.-.sub.n]) > 0 corresponds to [[theta].sup.-.sub.n] and the smallest root with Re([[delta].sup.+.sub.n]) < 0 corresponds to [[theta].sup.+.sub.n].

2. All of these corners of the critical trajectories are 90[delta].

3. In the special case, [theta] = -[pi], n = 1, the corner is located at [delta] = [pi]/2 with a corresponding [sigma] = - 2/L.

Proof.

1. Clearly, there can be no transition from a bounded mode-morphing trajectory to the unbounded trajectory of a missing mode without a singularity exhibiting a corner. At such a corner, the derivative [[delta]'.sub.n](r) must either vanish or become unbounded. Inspection of Equation (5.3) reveals that the derivative does not vanish since Im([[delta].sub.n](r) > 0) for r > 0. Further perusal of Equation (5.3) shows that the derivative becomes unbounded if and only if [[delta].sub.n](r) is located at a zero of the denominator, i.e. at a root of Equation (5.4). These critical values [[delta].sup.[+ or -].sub.n] occur at branch points [7, pp. 404-408] of Equation (2.5) if n is even and of Equation (2.6) if n is odd. Observe that once [[delta].sup.[+ or -].sub.n] is known then Equation (2.4) may be used to find the critical angle and modulus from [[sigma].sup.[+ or -].sub.n] = [r.sup.[+ or -].sub.n] [e.sup.[??][[theta].sup.[+ or -].sub.n]].

2. The corner will be 90[delta] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is pure imaginary since, in that and only that case, the tangent vector is multiplied by a pure imaginary as we pass through the corner thereby producing the posited rotation by [+ or -]90[delta]. But,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

by l'Hopital's Rule. Thus, [L.sup.2] = -1 [??] L = [+ or -][??].

3. Set [theta] = -[pi] and n = 1. Then, since the eigenvalue is real, the corner on the critical trajectory occurs as it passes through [v.sub.1] = 0 [??] [[delta].sub.1] = -[pi]/2. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] by l'Hopital's Rule.

[FIGURE 10 OMITTED]

Returning once again to Figure 9, note that the corner on the critical trajectory corresponding to [[theta].sup.[+ or -].sub.n] is a bifurcation point where one path leads to n"1 and the other veers off to infinity at an angle of [[theta].sup.[+ or -].sub.n] -[pi]/2 (Theorem 2, Part 2b). Turning attention to Figure 10 which is an amalgam of Figure 9, the top frame for the even pinumbered modes and the bottom frame for the odd numbered modes, observe that the boundaries of adjacent modal regions formed from these bifurcated trajectories align, the result in both cases being a corresponding partition of the [nu]-plane.

Furthermore, the trajectories corresponding to adjacent modes overlap but do not intersect in the sense that they never come in contact for the same value of [sigma]. Specifically we have the following

[FIGURE 11 OMITTED]

Theorem 4 1. A trajectory emanating from n never intersects a trajectory emanating from n [+ or -] 1 for the same value of [sigma].

2. A trajectory emanating from n intersects a trajectory emanating from n [+ or -] 2 forthe same value of [sigma] only at their common branch (bifurcation) point. In fact, [[delta].sup.+.sub.n] = [[delta].sup.-.sub.n-2] - [pi] and [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2]. At the common branch point there is a modal deficiency and along the bifurcated trajectory there is modal ambiguity.

Proof.

1. According to Equation (2.4), [v.sub.n] = [v.sub.n+1] if and only if [[delta].sub.n] = [[delta].sub.n+1] + [pi]/2 and tan ([[delta].sub.n+1] + [pi]/2) = tan ([[delta].sub.n+1]). But, this would require that - cot ([[delta].sub.n+1]) = tan ([[delta].sub.n+1]) which is impossible.

2. From Equation (5.4), we have

[[tau].sub.n-2]([delta]) = sin (2([delta] - [pi])) + 2([delta] - [pi]) + n[pi] = [[tau].sub.n]([delta] - [pi]), (5.5)

which implies that the roots of [[tau].sub.n]([delta]) are the roots of [[tau].sub.n-2]([delta]) shifted to the left by [pi]. Thus, [[delta].sup.+.sub.n] = [[delta].sup.-.sub.n-2] - [pi] and [[theta].sup.+.sub.n = [[theta].sup.-.sub.n-2]. Now, Equation (5.3) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (5.6)

so that [[delta].sub.n](r) and [[delta].sub.n-2](r) - [pi] satisfy the same differential equation with different initial conditions. By the fundamental Existence and Uniqueness Theorem for ordinary differential equations [2, pp. 1-11], they can only intersect at a singularity and such an intersection is equivalent to [v.sub.n-2](r) = [v.sub.n](r). At this common branch point, modes n - 2 and n coalesce thereby producing a modal deficiency. Beyond the branch point, there is modal ambiguity in that it is not clear which mode to associate with which bifurcation branch.

The modal deficiency and ambiguity established in the previous theorem must be taken into account when utilizing the eigenfunction expansion of Equation (3.3). Also, this theorem alleviates the need to calculate [[delta].sup.+.sub.n] and [[theta].sup.+.sub.n] since they are obtainable from [[delta].sup.-.sub.n-2] and [[theta].sup.-.sub.n-2], respectively.

[FIGURE 12 OMITTED]

Figure 11 shows the level curves of |[[tau].sub.0]([delta])| (see Equation (5.4)) where the branch point [[delta].sup.-.sub.0] is on prominent display. We next study [[delta].sup.-.sub.n] as n varies.

Theorem 5 1. As n [right arrow] 1, [[sigma].sup.-n] [right arrow] [-ln ((3 + 2n)[pi]) + [??] x (n + 3/2)[pi]] = L.

2. For 0 [less than or equal to] [theta] [less than or equal to] [pi]/2, all modes are IBC-Dirichlet modes (i.e. there are no missing modes).

Proof.

1. As n [right arrow] 1, Re([[delta].sup.-.sub.n]) [right arrow] [(3[??]/4).sup.-] and [[delta].sup.[I.sup.-.sub.n]] := Im([[delta].sup.-.sub.n]) [right arrow] + [infinity]. Thus, tan ([[delta].sup.[I.sup.-.sub.n]]) [right arrow] [??] and, from [[tau].sub.n]([[delta].sup.-.sub.n]) = 0, [[delta].sup.[I.sup.-.sub.n]] [right arrow] 1/2 ln ((3 + 2n)[pi]). Solving for [[sigma].sup.-.sub.n] in Equation (2.4) yields [[sigma].sup.-.sub.n] [right arrow] [-ln ((3 + 2n)[pi]) + [??] x (n + 3=2)[pi]]/L.

2. Since [[theta].sup.-.sub.n] := arg [[sigma].sup.-.sub.n], we have [[theta].sup.-.sub.n] = [tan.sup.-1] [-(n + 3/2)[pi]= ln ((3 + 2n)[pi])] [right arrow] [tan.sup.-1] (-[infinity]) = [([pi]/2).sup.+]. Therefore, all of the critical angles lie in the range [pi]/2 < [[theta].sup.-.sub.n] [less than or equal to] [pi].

5.1. The Case Re([sigma]) [greater than or equal to] 0

Because of the second part of Theorem 5, the asymptotic behavior for Re([sigma]) [greater than or equal to] 0 is especially simple. With reference to Figure 12, there is a complete modal homotopy from each of the Neumann modes to a corresponding Dirichlet mode. Specifically, [N.sub.n](x) [??] [D.sub.n+1](x) for all n. This asymptotic behavior, illustrated by Figure 8 for n = 0 and [theta] = [pi]=4, is directly analogous to the case of [sigma] real and positive except that now the homotopy passes through the complex plane.

5.2. The Case Re([sigma]) < 0

[FIGURE 13 OMITTED]

The asymptotic behavior for Re([sigma]) < 0, is directly analogous to the case of [sigma] real and negative in that there are always two missing modes with the remainder being IBC-Dirichlet modes. However, which modes are missing is now determined by the value of [theta]. Specifically, if [[theta].sup.-.sub.n] < [theta] < [[theta].sup.-.sub.n-1] then modes n and n + 1 will be missing. This is shown graphically in Figure 13. Table 1 lists many branch points [[delta].sup.-.sub.n] together with their corresponding critical angles [[theta].sup.-.sub.n].

5.2.1 IBC-Dirichlet Modes

If [theta] < [[theta].sup.-.sub.n] then [N.sub.n](x) [??] [D.sub.n+1](x) analogous to the case of the mode morphing displayed in the case of Re([sigma]) [greater than or equal to] 0. However, if [theta] > [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2] then [N.sub.n](x) [??] -[D.sub.n-1](x) characteristic of the mode morphing displayed in the case of [sigma] real and negative. This latter mode-morphing behavior is on display in Figure 14 for n = 2 and [theta] = 3[pi]=4.

5.2.2 Missing Modes

If [[theta].sup.-.sub.n] < [theta] < [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2] then mode n is missing. The resulting rightward procession of missing modes as [theta] is lowered from [pi] to [pi]/2 is shown in Figure 15. The missing modes for [pi]=2 < [theta] < [pi] exhibit a peculiar singular behavior as r = |[sigma]| [right arrow] [infinity].

Theorem 6 For [pi]/2< [theta] < [pi], all missing modes oscillate between [+ or -]1 at the endpoints and approach zero elsewhere.

Proof. Simply let |[delta] = [[delta].sub.R] + [??] x [[delta].sub.I]| [right arrow] [infinity] in Equation (2.1). At the endpoints, [u.sub.n] [right arrow] [+ or -]([e.sup.[[delta].sub.I]]/2) x [cos ([[delta].sub.R]) - [??] x sin ([[delta].sub.R])] and normalization by [e.sup.[[delta].sub.I]]/2 reveals the oscillatory behavior at the endpoints as well the approach to zero for interior points.

This oscillatory singular behavior is in evidence in Figures 16 (even numbered modes) and 17 (odd numbered modes).

6. Conclusion

The bulk of this paper has been devoted to exploring the asymptotic nature of the eigenstructure of the SL-BVP with an IBC, Equation (1.1), as [sigma] [right arrow] 1. Our main results may be summarized as follows (see Figure 2):

Theorem 7 (Asymptotic Behavior of SL-BVP/IBC Eigenstructure) Consider the SL-BVP with an IBC described by Equation (1.1) with [sigma](r) = [re.sup.[??][theta]] for fixed [theta] and 0 [less than or equal to] r [less than or equal to] [infinity].

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

1. If 0 [less than or equal to] [theta] [less than or equal to] [pi]/2 then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for all n.

2. If [pi]/2 < [theta] [less than or equal to] [pi] then there exists n([theta]) such that

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k < n([theta]) - 1,

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k > n([theta]),

(c) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k = n([theta]) - 1, n([theta]).

These observations are directly applicable to the rectangular waveguide [8, pp. 503-509].

Furthermore, these results lead naturally to the question of the corresponding asymptotic nature of the eigenstructure of the Laplacian on an equilateral triangle with an impedance boundary condition. The special cases of the radiation boundary condition [4] and the absorbing boundary condition [5] have already received exhaustive treatment. The one-dimensional results of the present paper seem to indicate that these two special cases exhibit the full spectrum of possible asymptotic behavior as [sigma] [right arrow] [infinity]. The final installment of this series of papers, Eigenstructure of the Equilateral Triangle, Part V: The Impedance Boundary Condition, will address this problem.

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

Acknowledgements

The author thanks Mrs. Barbara A. McCartin for her indispensable aid in constructing the figures. This paper is dedicated to the memory of our beloved Mother, Dorothy Frances (Kelly) McCartin, on this 25th Anniversary of her departure from her family. Gone but not forgotten!

References

[1] Amrein W. O., Hinz A. M. and Pearson D. B., Sturm-Liouville Theory: Past and Present, Birkhauser, Basel, 2005.

[2] Coddington E. A. and Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, NY, 1955.

[3] Mahmoud S. F., Electromagnetic Waveguides: Theory and Applications, Peter Peregrinus Ltd., London, UK, 1991.

[4] McCartin B. J., 2004, Eigenstructure of the Equilateral Triangle, Part III: The Robin Problem, International Journal of Mathematics and Mathematical Sciences, Vol. 2004(16), pp. 807-825.

[5] McCartin B. J., 2007, Eigenstructure of the Equilateral Triangle, Part IV: The Absorbing Boundary Condition, International Journal of Pure and Applied Mathematics, Vol.37(3).

[6] Morse P. M., Vibration and Sound, Acoustical Society of America, Melville, NY, 1976.

[7] Morse P. M. and Feshbach H., Methods of Theoretical Physics, Part I, McGraw-Hill, New York, NY, 1953.

[8] Morse P. M. and Ingard K. U., Theoretical Acoustics, McGraw-Hill, New York, NY, 1968.

[9] Strauss W. A., Partial Differential Equations: An Introduction, Wiley, New York, NY, 1992.

[10] Zettl A., Sturm-Liouville Theory, American Mathematical Society, Providence, RI, 2005.

Brian J. McCartin

Applied Mathematics, Kettering University

1700 West Third Avenue, Flint, MI 48504-4898 USA

E-mail: bmccarti@kettering.edu

Structural properties of the eigenvalues and eigenfunctions of a Sturm-Liouville problem with an impedance boundary condition are studied herein. This entails a Robin boundary condition with a complex boundary parameter. The thrust of the present investigation is provided by the mapping of Neumann to Dirichlet eigen-problems that obtains as the complex boundary parameter varies from zero to the point at infinity along a fixed direction in the complex plane. Various properties of the spectrum and modal functions are explored.

AMS Subject Classification: 34B24, 34L05, 34M25. Keywords: Sturm-Liouville problem, impedance boundary condition, non-self-adjoint boundary value problem.

1. Introduction

Herein, we explore the eigenstructure of the Sturm-Liouville boundary value problem (SL-BVP) with an impedance boundary condition (IBC)

u"(x) + [lambda] x u(x) = 0, 0 < x < L, u'(0) - [sigma] x u(0) = 0, u'(L) + [sigma] x u(L) = 0. (1.1)

The complex boundary parameter [sigma] = [[sigma].sub.R] + [??] x [[sigma].sub.I] is assumed to have the same value at both ends of the interval J = [0,L]. Attention will be restricted to this case as it is the one most relevant to the extension of the results for the eigenstructure of the equilateral triangle [4, 5] from real to complex values of [sigma] which furnishes the motivation for the present one-dimensional study.

[FIGURE 1 OMITTED]

Figure 1 illustrates the parallel plate waveguide problem which gives rise to the SL-BVP, Equation (1.1). Either an acoustic [8, pp. 485-496] or electromagnetic [3, pp. 81-84] time-harmonic wave of angular frequency [omega] is propagating in the z-direction with no field variation in the y-direction. The complex material parameter [sigma] is related to the wall impedance which accounts for field penetration into the walls of the waveguide. After the SL-BVP has been solved for the eigenvalue [lambda] and the corresponding eigenfunction (mode) u(x), the real physical field may be reconstructed as

U(x, z, t) = Re{u(x) x [e.sup.[??]{([omega]t - [gamma]z)}, [gamma] := [square root of - [lambda]]. (1.2)

This same SL-BVP arises in the study of the vibrating string with sidewise displacement of the end supports [6, pp. 133-134].

Much of the current mathematical literature devoted to SL-BVPs is concerned primarily with the self-adjoint case [1, 10]. However, the SL-BVP described by Equation (1.1) is non-self-adjoint for complex values of [sigma]. Fortunately, a classic reference on the non-self-adjoint case is provided by [2, pp. 298-305] where it is shown that this problem reduces to the study of the solutions of a single transcendental equation.

In what follows, we will derive and exhaustively study this transcendental equation. The principal focus will be on what happens to the eigenstructure of the Neumann problem ([sigma] = 0) as [sigma] proceeds along rays emanating from the origin toward the point at infinity in the complex plane. We will find that, for the most part, there is a natural homotopy connecting the Neumann modes to those of a corresponding Dirichlet problem (IBC-Dirichlet modes). However, we will show that, under appropriate conditions, there will be two Neumann eigenvalues that have no Dirichlet counterpart. The determination of the precise nature of these latter eigenvalues together with that of their corresponding eigenfunctions ("missing modes") is the primary concern of the present investigation.

2. Solution of the Sturm-Liouville Problem

Ignoring the boundary conditions, the general solution to Equation (1.1) is

u(x) = cos (v[pi]x/L - [delta]), [lambda] = [(v[pi]/L).sup.2]. (2.1)

Application of the boundary condition at x = 0 yields

tan ([delta]) = [sigma]L/v[pi], (2.2)

while imposition of the boundary condition at x = L yields

tan (v[pi] - [delta]) = [sigma]L/v[pi]. (2.3)

Equations (2.2) and (2.3) may be combined to produce the transcendental equation

(2[delta] + n[pi]) tan ([delta]) = [sigma]L, v = 2[delta]/[pi] + n; (2.4)

with n an integer.

Observe that in Equation (1.1) when [sigma] [right arrow] 0 we recover the Neumann problem, u'(0) = 0 = u'(L), whose solution, u(x) = [N.sub.n](x) := cos (n[pi]x/L), is obtained from Equation (2.4) with [delta] = 0 [??] v = n. Thus, we may profitably view [sigma] as a continuation parameter which provides a homotopy extending from this well understood problem to that of the impedance boundary condition. Throughout the ensuing development we will avail ourselves of this important observation.

Furthermore, note that if the normal derivative remains bounded then [sigma] [right arrow] [infinity] yields the Dirichlet problem, u(0) = 0 = u(L), whose solution, u(x) = [+ or -][D.sub.[n[+ or -]1](x) := [+ or -] sin ((n [+ or -] 1)[pi]x/L), is obtained from Equation (2.4) with [delta] = [+ or -][pi] 2) v = n [+ or -] 1. In that case, the homotopy may be further extended to lead from a Neumann mode to a corresponding Dirichlet mode. As will subsequently be shown, for the impedance boundary condition this is usually, although by no means always, the case.

Introduction of z : = [delta] + n[pi]/2 into Equation (2.4) reduces it to

z tan (z) = [sigma]L/2, (2.5)

if n is even and

z cot (z) = -[sigma]L/2, (2.6)

if n is odd. These two cases may then be separately studied from the graphical representation of their respective complex transformations [8, p. 909].

However, in the case of the equilateral triangle [4, 5] which is our ultimate goal, such a reduction is not available. Thus, Equation (2.4) itself will be numerically approximated using MATLAB. This will permit us to trace out trajectories in the complex [delta]-plane or, equivalently, in the complex v-plane as the complex boundary parameter [sigma] is varied.

[FIGURE 2 OMITTED]

As shown in Figure 2, beginning with [sigma] = 0 (the Neumann problem), we will allow [sigma] to move along a ray inclined at an angle [theta] to the horizontal toward the point at infinity. We may then track the trajectory of each Neumann eigenvalue in order to determine whether it eventually approaches a Dirichlet eigenvalue (in which case it will be called an IBC-Dirichlet mode) or migrates to infinity (in which case it will be called a "missing mode"). We will eventually find that these are the only two possible types of asymptotic behavior.

3. SL-BVP/IBC Solution Properties

Before constructing a taxonomy of the asymptotic nature of the eigenstructure of our SL-BVP with an IBC as [sigma] [right arrow] [infinity] (at which time, the mysterious icons on Figure 2 will become intelligible), it behooves us to first catalog some important properties of the eigenvalues and eigenfunctions of Equation (1.1).

We begin with the observation that, without loss of generality, we may restrict our attention to the case Im([sigma]) [greater than or equal to] 0 since, by taking complex conjugates in Equation (2.4), we find that [sigma] [??] [bar.[sigma]] [??] [delta] [??] [bar.[delta]] [??] v [??] [bar.v]. Thus, in the event that Im([sigma]) < 0, we can obtain trajectories in either the [delta]-plane or v-plane by reflection about the real axis of the corresponding trajectories for [bar.[sigma]].

Furthermore, taking complex conjugates in Equation (1.1) itself reveals that [sigma] [??] [bar.[sigma]] [??] [lambda] [??] [bar.[lambda]] =: [mu], u(x) [??] [bar.u](x) =: v(x) where [mu] and v(x) form the solution to the adjoint boundary value problem

v'(x) + [mu] x v(x) = 0, 0 < x < L,

v'(0) - [bar.[sigma]] x v(0) = 0,

v'(L) + [bar.[sigma]] x v(L) = 0. (3.1)

Since the SL-BVP is non-self-adjoint for complex [sigma], the eigenvalues and eigenfunctions can be complex and, most importantly, eigenfunctions corresponding to distinct eigenvalues are not necessarily orthogonal with respect to the complex inner product < f(x); g(x) >:= [[integral.sup.L.sub.0] f(x)[bar.g](x) dx. However, we do have the biorthogonality relationship involving the eigenfunctions of the boundary value problem, Equation (1.1), and those of of the adjoint boundary value problem, Equation (3.1),

< [u.sub.p](x), [v.sub.q](x) >:= [[integral].sup.L.sub.0] [u.sub.p](x)[[bar.v].sub.q](x) dx = 0. (3.2)

In turn, this provides us with the eigenfunction expansion

f(x) = [[infinity].summation over (k=1)] < f(x), [v.sub.k](x) > [u.sub.k](x), (3.3)

where the series converges in the mean for f(x) [member of] [L.sup.2](0, L) [2, pp. 310-312].

We have already seen via Equations (2.5) and (2.6) that the modes naturally partition according to the parity of n. This fact is underscored by

Theorem 1 The even/odd numbered modes are symmetric/antisymmetric, respectively, on the interval [0, L] for all values of [sigma].

Proof. Equation (3) may be recast as [u.sub.n](x) = cos (v[pi]/L (x - L/2) + n[pi/ 2). Thus, if n is even/odd then [u.sub.n](x) is a cosine/sine, respectively, centered at x = L/2.

4. The Case of Real [sigma]

In the case of real [sigma], the eigenstructure of the SL-BVP with an IBC defined by Equation (1.1) has been exhaustively treated in [9, pp. 90-98]. As a springboard for the study of the case of complex [sigma] in the next section, we review here the highlights of those results.

When [sigma] is real, the problem is self-adjoint so that the eigenvalues are real and the eigenfunctions can also be chosen to be real. Moreover, eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to the real inner product < f(x); g(x) >:= [[integral.sup.L.sub.0] f(x)g(x) dx. Because of their very different behaviors, we treat the cases [sigma] [greater than or equal to] 0 and [sigma] < 0 separately. We will eventually see that, taken together, they display the characteristic behaviors exhibited in the general case of complex [sigma].

In what follows, we denote the dependence of the eigenvalues and eigenfunctions upon [sigma] by [[lambda].sub.n]([sigma]) = [(v([sigma])[pi]/L).sup.2] and un(x; [sigma]), respectively, for n = 0, 1, .... The subscript n is chosen so that when [sigma] = 0 they reduce to the corresponding values for the Neumann problem, [d.sub.n](0) = 0 [??] [v.sub.n](0) = n and [u.sub.n](x; 0) = [N.sub.n](x) := cos(n[pi]x/L). Also, the Dirichlet problem has the same eigenvalues [[lambda].sub.n](0) but with the restriction n = 1, 2, ... and its corresponding eigenfunctions are denoted by un(x; 0) = [D.sub.n](x) := sin(n[pi]x/L).

4.1. The Case [sigma] [greater than or equal to] 0

[FIGURE 3 OMITTED]

The case of the radiation boundary condition ([sigma] > 0) is by far the simplest in that the eigenvalues are not only real but are in fact all positive. Moreover, n < [[upsilon].sub.n]([sigma]) < n + 1. It also possesses the simplest asymptotic behavior in that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.1)

That is, the Neumann mode [N.sub.n](x) "morphs" analytically into the Dirichlet mode [D.sub.n+1](x) as [sigma] ranges from 0 to [infinity]. This is illustrated in Figure 3 which displays this homotopy between fundamental modes (n = 0) for 0 [less than or equal to] [sigma] [less than or equal to] -[infinity].

4.2. The Case [sigma] < 0

The case of the absorbing boundary condition ([sigma] < 0) is more complicated in that the eigenvalues while still real are no longer all positive. However, for n = 2, 3, ... we have n - 1 < [v.sub.n]([sigma]) < n. These so-called IBC-Dirichlet modes possess the simple asymptotic behavior described by

[FIGURE 4 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.2)

That is, the Neumann mode [N.sub.n](x) "morphs" analytically into the Dirichlet mode - [D.sub.n-1](x) as [sigma] ranges from 0 to -[infinity]. This is illustrated in Figure 4 which displays this homotopy between [N.sub.2](x) and -[D.sub.1](x) for 0 [greater than or equal to] [sigma] [greater than or equal to] -[infinity].

[FIGURE 5 OMITTED]

This leaves open the case of the "missing modes" n = 0, 1. Since there are no Dirichlet modes for n = -1, 0 we clearly do not have the simple asymptotic behavior described by Equation (4.2). Thus we are confronted with the question: "What happens to the missing n = 0, 1 modes as [sigma] [right arrow] -[infinity]?". The solution to the mystery of the missing modes naturally decomposes into two special cases each of which we now explore separately.

[FIGURE 6 OMITTED]

For n = 0, [v.sub.0]([sigma]) is pure imaginary and, consequently, [[lambda].sub.0]([sigma]) is negative. Specifically, for [sigma] [right arrow] -[infinity],

[[delta].sub.0]([sigma]) [approximately equal to] - [sigma]L/2 x [??] [v.sub.0]([sigma]) [approximately equal to] - [sigma]L/[pi] x [??] [??] [u.sub.0](x; [sigma]) [approximately equal to] cosh (-[sigma]x + [sigma]L/2). (4.3)

This asymptotic expression for [u.sub.0](x; [sigma]) becomes unbounded as [sigma] [right arrow] -[infinity]. However, if we first scale it by its value at an endpoint, cosh ([sigma]L/2), we find that this normalized mode approaches 1 at the two endpoints and 0 elsewhere. Such singular limiting behavior, necessary since this does not approach a Dirichlet mode, is on prominent display in Figure 5.

For n = 1, [[nu].sub.1]([sigma]) initially decreases along with [sigma] until it vanishes when [sigma] = - 2/L. Specifically,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

However, if we first scale [u.sub.1](x; [sigma]) by its value at the endpoint x = 0, cos ([delta]), we find that this normalized mode approaches the straight line [u.sub.1](x; -2/L) = 1 - 2/L x x.

As [sigma] continues to decrease we have, for [sigma] < - 2/L, pure imaginary [[delta].sub.1]([sigma])+ [pi]/2 thereby producing a [v.sub.1]([sigma]) which is also pure imaginary and, consequently, [[lambda].sub.1]([sigma]) becomes negative. Specifically, for [sigma] [right arrow] -[infinity],

[[delta].sub.1]([sigma]) [approximately equal to] - [pi]/2 - [sigma]L/2 [??] [??] x [v.sub.1]([sigma]) [approximately equal to] - [sigma]L/[pi] x [??] [u.sub.1](x) [approximately equal to] [??] x sinh ([sigma]x - [sigma]L/2). (4.5)

This asymptotic expression for [u.sub.1](x; [sigma]) becomes unbounded as [sigma] [right arrow] -[infinity]. However, if we first scale it by its value at the endpoint x = 0, [??] x sinh (- [sigma]L/2), we find that this normalized mode approaches [+ or -]1 at the left-/right-hand endpoint, respectively, and 0 elsewhere. The resulting unbounded derivative, necessary since this does not approach a Dirichlet mode, is evident in Figure 6.

5. The Case of Complex [sigma]

[FIGURE 7 OMITTED]

In the case of complex [sigma], as previously noted, the SL-BVP defined by Equation (1.1) is non-self-adjoint and the eigenstructure is consequently complex. Defining the residual function of Equation (2.4) as

[[rho].sub.n]([delta]; [sigma]) = |(2[delta] + n[pi]) tan ([delta]) - [sigma]L|; (5.1)

we note that the sought-after eigenvalues are determined by its local minima. Next, define

[sigma] = r x [e.sup.x[theta]]; with [theta] fixed and 0 [less than or equal to] r [less than or equal to] + [infinity]. (5.2)

Figure 7, which is a contour plot of [[rho].sub.n], displays the resultant trajectory in the [delta]-plane for n = 0 as r varies with [theta] = [pi]/4. As occurred in the real case with [sigma] [greater than or equal to] 0, [delta] varies from 0 to [pi]/2 only now it makes an excursion into the complex plane rather than traveling along the real axis. The attendant complex mode morphing from [N.sub.0](x) to [D.sub.1](x) is made explicit by Figure 8.

Figure 9 displays the corresponding trajectories, this time in the [nu]-plane, for the the first four modes over the full range of values 0 [less than or equal to] [theta] [less than or equal to] [pi]. As is evident from each of these plots, for some values of [theta] mode morphing occurs, i.e [[lambda].sub.n]([sigma]) [right arrow] [[lambda].sub.n[+ or -]1](0), (IBC-Dirichlet modes) while for other values of [theta] we observe |[[lambda].sub.n]([sigma])| [right arrow] [infinity] (missing modes). We are going to devote our remaining efforts to clarifying this asymptotic behavior.

[FIGURE 8 OMITTED]

In order to achieve this, we will require detailed knowledge of the properties of trajectories in the v-plane. We commence with

Theorem 2 1. Let [[delta].sub.n](r) = [[delta].sup.R.sub.n] (r) + [??] x [[delta].sup.I.sub.n](r) and [v.sub.n](r) = [v.sup.R.sub.n] (r) + [??] x [v.sup.I.sub.n](r). Then, the trajectories [[delta].sub.n](r) and [v.sub.n](r) have the same slope.

(a) If n [not equal to] 0 then the trajectory [v.sub.n](r) makes an angle [theta] measured counterclockwise from the real axis at [v.sub.n](0) = n.

(b) If n = 0 then the trajectory [v.sub.0](r) makes an angle [theta]/2 measured counterclockwise from the real axis at [v.sub.0](0) = 0.

2. As r [right arrow] [infinity], either [v.sub.n](r) [right arrow] n [+ or -] 1 or |[v.sub.n](r)| [right arrow] [infinity].

(a) if [v.sub.n](r) [right arrow] n [+ or -] 1 then the trajectory [v.sub.n](r) makes an angle [pi] - [theta] measured counterclockwise from the real axis at [v.sub.n](1) = n [+ or -] 1.

(b) if |[v.sub.n](r)| [right arrow] [infinity] then the trajectory [v.sub.n](r) goes off to infinity at an angle [theta] - [pi]/2 measured counterclockwise from the real axis as r [right arrow] [infinity].

Proof.

1. Since [v'.sub.n](r) = 2/[pi] x [[delta]'.sub.n](r), both trajectories have slope [[delta].sup.I'.sub.n](r)=[[delta].sup.R'.sub.n](r).

[FIGURE 9 OMITTED]

(a) r = 0 [??] [sigma] = 0 so that Equation (2.4) produces [[delta].sub.n](0) = 0) [[nu].sub.n](0) = n. Substitution of Equation (5.2) into Equation (5.3) with subsequent differentiation with respect to r yields

[[delta]'.sub.n](r) = Le{[theta] [cos.sup.2] ([[delta].sub.n](r)) sin (2[[delta].sub.n](r)) + 2[[delta].sub.n](r) + n[pi] : (5.3)

Thus, [[delta]'.sub.n](0) = [Le.sup.[??][theta]]/n[pi] [??] tan [empty set] := [[delta].sup.I'.sub.n] (0)/[[delta].sup.R'.sub.n](0) = tan [theta] [??] [empty set] = [theta].

(b) For n = 0, Equation (5.3) produces [[delta]'.sub.0](0) = [infinity] so that the slope tan [empty set] := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is indeterminate. However, by Equation (5.3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Thus, tan [empty set] = tan ([theta] - [empty set]) [??] [empty set] = [theta]/2.

2. As r = |[sigma]| [right arrow] [infinity], Equation (2.4) clearly implies that either [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2, in which case [v.sub.n](r) [right arrow] n [+ or -] [infinity], or |[[delta].sub.n](r)| [right arrow] [infinity], in which case |[v.sub.n](r)| [right arrow] [infinity].

(a) If [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2 then [[nu].sub.n](r) [right arrow] n [+ or -] 1 and [[delta]'.sub.n](r) [right arrow] 0 by Equations (2.4) and (5.3), respectively. Thus, as r [right arrow] [infinity], [[delta]'.sub.n](r) [right arrow] [[epsilon].sub.1] [e.sup.[??][empty set]] and [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2 [+ or -][[epsilon].sub.2][e.sup.[??][empty set]] where [[empty set].sub.1], [[epsilon].sub.2] [right arrow] 0. Comparison of the arguments of both sides of Equation (5.3) reveals that [empty set] = [theta] + 2[empty set] [??] [empty set] = -[theta].

(b) If |[[delta].sub.n](r)| [right arrow] [infinity] then tan ([[delta].sub.n](r)) [right arrow] [??]. Inserting this into Equation (2.4) and equating real and imaginary parts produces 2[[delta].sup.R.sub.n] [right arrow] rL sin ([theta]) and -2[[delta].sup.I.sub.n] [right arrow] rL cos ([theta]). Thus, [[delta].sup.I.sub.n](r)/ [[delta].sup.R.sub.n] (r) [right arrow] -cot ([theta]) = tan (theta] - [phi]/2).

Returning now to Figure 9, note the following important structural features. For modes n = 0 and n = 1, there is a critical angle [[theta].sup.-.sub.n] such that for [theta] < [[theta].sup.-.sub.n] we have the mode morphing [N.sub.n](x) [??] [D.sub.n+1](x) while, for [theta] > [[theta].sup.-.sub.n], mode n is missing in the previously defined sense as r [right arrow] 1. For all other modes n [less than or equal to] 2, there are two critical angles [[theta].sup.-.sub.n] and [[theta].sup.+.sub.n]. For [theta] < [[theta].sup.-.sub.n] we have the mode morphing [N.sub.n](x) [??] [D.sub.n+1](x) and for [theta] > [[theta].sup.+.sub.n] we have the mode morphing [N.sub.n](x) [??] -[D.sub.n-1](x) as r [right arrow] [infinity]. In the intermediate regime [[theta].sup.-.sub.n] < [theta] < [[theta].sup.+.sub.n], mode n is missing as r [right arrow] [infinity].

Our next result concerns the trajectories at these critical angles.

Theorem 3 1. The critical trajectories all possess a corner located in the [delta]-plane at the roots of the function

[[tau].sub.n]([delta]) := sin (2[delta]) + 2[delta] + n[pi]: (5.4)

The smallest root with Re([[delta].sup.-.sub.n]) > 0 corresponds to [[theta].sup.-.sub.n] and the smallest root with Re([[delta].sup.+.sub.n]) < 0 corresponds to [[theta].sup.+.sub.n].

2. All of these corners of the critical trajectories are 90[delta].

3. In the special case, [theta] = -[pi], n = 1, the corner is located at [delta] = [pi]/2 with a corresponding [sigma] = - 2/L.

Proof.

1. Clearly, there can be no transition from a bounded mode-morphing trajectory to the unbounded trajectory of a missing mode without a singularity exhibiting a corner. At such a corner, the derivative [[delta]'.sub.n](r) must either vanish or become unbounded. Inspection of Equation (5.3) reveals that the derivative does not vanish since Im([[delta].sub.n](r) > 0) for r > 0. Further perusal of Equation (5.3) shows that the derivative becomes unbounded if and only if [[delta].sub.n](r) is located at a zero of the denominator, i.e. at a root of Equation (5.4). These critical values [[delta].sup.[+ or -].sub.n] occur at branch points [7, pp. 404-408] of Equation (2.5) if n is even and of Equation (2.6) if n is odd. Observe that once [[delta].sup.[+ or -].sub.n] is known then Equation (2.4) may be used to find the critical angle and modulus from [[sigma].sup.[+ or -].sub.n] = [r.sup.[+ or -].sub.n] [e.sup.[??][[theta].sup.[+ or -].sub.n]].

2. The corner will be 90[delta] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is pure imaginary since, in that and only that case, the tangent vector is multiplied by a pure imaginary as we pass through the corner thereby producing the posited rotation by [+ or -]90[delta]. But,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

by l'Hopital's Rule. Thus, [L.sup.2] = -1 [??] L = [+ or -][??].

3. Set [theta] = -[pi] and n = 1. Then, since the eigenvalue is real, the corner on the critical trajectory occurs as it passes through [v.sub.1] = 0 [??] [[delta].sub.1] = -[pi]/2. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] by l'Hopital's Rule.

[FIGURE 10 OMITTED]

Returning once again to Figure 9, note that the corner on the critical trajectory corresponding to [[theta].sup.[+ or -].sub.n] is a bifurcation point where one path leads to n"1 and the other veers off to infinity at an angle of [[theta].sup.[+ or -].sub.n] -[pi]/2 (Theorem 2, Part 2b). Turning attention to Figure 10 which is an amalgam of Figure 9, the top frame for the even pinumbered modes and the bottom frame for the odd numbered modes, observe that the boundaries of adjacent modal regions formed from these bifurcated trajectories align, the result in both cases being a corresponding partition of the [nu]-plane.

Furthermore, the trajectories corresponding to adjacent modes overlap but do not intersect in the sense that they never come in contact for the same value of [sigma]. Specifically we have the following

[FIGURE 11 OMITTED]

Theorem 4 1. A trajectory emanating from n never intersects a trajectory emanating from n [+ or -] 1 for the same value of [sigma].

2. A trajectory emanating from n intersects a trajectory emanating from n [+ or -] 2 forthe same value of [sigma] only at their common branch (bifurcation) point. In fact, [[delta].sup.+.sub.n] = [[delta].sup.-.sub.n-2] - [pi] and [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2]. At the common branch point there is a modal deficiency and along the bifurcated trajectory there is modal ambiguity.

Proof.

1. According to Equation (2.4), [v.sub.n] = [v.sub.n+1] if and only if [[delta].sub.n] = [[delta].sub.n+1] + [pi]/2 and tan ([[delta].sub.n+1] + [pi]/2) = tan ([[delta].sub.n+1]). But, this would require that - cot ([[delta].sub.n+1]) = tan ([[delta].sub.n+1]) which is impossible.

2. From Equation (5.4), we have

[[tau].sub.n-2]([delta]) = sin (2([delta] - [pi])) + 2([delta] - [pi]) + n[pi] = [[tau].sub.n]([delta] - [pi]), (5.5)

which implies that the roots of [[tau].sub.n]([delta]) are the roots of [[tau].sub.n-2]([delta]) shifted to the left by [pi]. Thus, [[delta].sup.+.sub.n] = [[delta].sup.-.sub.n-2] - [pi] and [[theta].sup.+.sub.n = [[theta].sup.-.sub.n-2]. Now, Equation (5.3) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (5.6)

so that [[delta].sub.n](r) and [[delta].sub.n-2](r) - [pi] satisfy the same differential equation with different initial conditions. By the fundamental Existence and Uniqueness Theorem for ordinary differential equations [2, pp. 1-11], they can only intersect at a singularity and such an intersection is equivalent to [v.sub.n-2](r) = [v.sub.n](r). At this common branch point, modes n - 2 and n coalesce thereby producing a modal deficiency. Beyond the branch point, there is modal ambiguity in that it is not clear which mode to associate with which bifurcation branch.

The modal deficiency and ambiguity established in the previous theorem must be taken into account when utilizing the eigenfunction expansion of Equation (3.3). Also, this theorem alleviates the need to calculate [[delta].sup.+.sub.n] and [[theta].sup.+.sub.n] since they are obtainable from [[delta].sup.-.sub.n-2] and [[theta].sup.-.sub.n-2], respectively.

[FIGURE 12 OMITTED]

Figure 11 shows the level curves of |[[tau].sub.0]([delta])| (see Equation (5.4)) where the branch point [[delta].sup.-.sub.0] is on prominent display. We next study [[delta].sup.-.sub.n] as n varies.

Theorem 5 1. As n [right arrow] 1, [[sigma].sup.-n] [right arrow] [-ln ((3 + 2n)[pi]) + [??] x (n + 3/2)[pi]] = L.

2. For 0 [less than or equal to] [theta] [less than or equal to] [pi]/2, all modes are IBC-Dirichlet modes (i.e. there are no missing modes).

Proof.

1. As n [right arrow] 1, Re([[delta].sup.-.sub.n]) [right arrow] [(3[??]/4).sup.-] and [[delta].sup.[I.sup.-.sub.n]] := Im([[delta].sup.-.sub.n]) [right arrow] + [infinity]. Thus, tan ([[delta].sup.[I.sup.-.sub.n]]) [right arrow] [??] and, from [[tau].sub.n]([[delta].sup.-.sub.n]) = 0, [[delta].sup.[I.sup.-.sub.n]] [right arrow] 1/2 ln ((3 + 2n)[pi]). Solving for [[sigma].sup.-.sub.n] in Equation (2.4) yields [[sigma].sup.-.sub.n] [right arrow] [-ln ((3 + 2n)[pi]) + [??] x (n + 3=2)[pi]]/L.

2. Since [[theta].sup.-.sub.n] := arg [[sigma].sup.-.sub.n], we have [[theta].sup.-.sub.n] = [tan.sup.-1] [-(n + 3/2)[pi]= ln ((3 + 2n)[pi])] [right arrow] [tan.sup.-1] (-[infinity]) = [([pi]/2).sup.+]. Therefore, all of the critical angles lie in the range [pi]/2 < [[theta].sup.-.sub.n] [less than or equal to] [pi].

5.1. The Case Re([sigma]) [greater than or equal to] 0

Because of the second part of Theorem 5, the asymptotic behavior for Re([sigma]) [greater than or equal to] 0 is especially simple. With reference to Figure 12, there is a complete modal homotopy from each of the Neumann modes to a corresponding Dirichlet mode. Specifically, [N.sub.n](x) [??] [D.sub.n+1](x) for all n. This asymptotic behavior, illustrated by Figure 8 for n = 0 and [theta] = [pi]=4, is directly analogous to the case of [sigma] real and positive except that now the homotopy passes through the complex plane.

5.2. The Case Re([sigma]) < 0

[FIGURE 13 OMITTED]

The asymptotic behavior for Re([sigma]) < 0, is directly analogous to the case of [sigma] real and negative in that there are always two missing modes with the remainder being IBC-Dirichlet modes. However, which modes are missing is now determined by the value of [theta]. Specifically, if [[theta].sup.-.sub.n] < [theta] < [[theta].sup.-.sub.n-1] then modes n and n + 1 will be missing. This is shown graphically in Figure 13. Table 1 lists many branch points [[delta].sup.-.sub.n] together with their corresponding critical angles [[theta].sup.-.sub.n].

5.2.1 IBC-Dirichlet Modes

If [theta] < [[theta].sup.-.sub.n] then [N.sub.n](x) [??] [D.sub.n+1](x) analogous to the case of the mode morphing displayed in the case of Re([sigma]) [greater than or equal to] 0. However, if [theta] > [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2] then [N.sub.n](x) [??] -[D.sub.n-1](x) characteristic of the mode morphing displayed in the case of [sigma] real and negative. This latter mode-morphing behavior is on display in Figure 14 for n = 2 and [theta] = 3[pi]=4.

5.2.2 Missing Modes

If [[theta].sup.-.sub.n] < [theta] < [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2] then mode n is missing. The resulting rightward procession of missing modes as [theta] is lowered from [pi] to [pi]/2 is shown in Figure 15. The missing modes for [pi]=2 < [theta] < [pi] exhibit a peculiar singular behavior as r = |[sigma]| [right arrow] [infinity].

Theorem 6 For [pi]/2< [theta] < [pi], all missing modes oscillate between [+ or -]1 at the endpoints and approach zero elsewhere.

Proof. Simply let |[delta] = [[delta].sub.R] + [??] x [[delta].sub.I]| [right arrow] [infinity] in Equation (2.1). At the endpoints, [u.sub.n] [right arrow] [+ or -]([e.sup.[[delta].sub.I]]/2) x [cos ([[delta].sub.R]) - [??] x sin ([[delta].sub.R])] and normalization by [e.sup.[[delta].sub.I]]/2 reveals the oscillatory behavior at the endpoints as well the approach to zero for interior points.

This oscillatory singular behavior is in evidence in Figures 16 (even numbered modes) and 17 (odd numbered modes).

6. Conclusion

The bulk of this paper has been devoted to exploring the asymptotic nature of the eigenstructure of the SL-BVP with an IBC, Equation (1.1), as [sigma] [right arrow] 1. Our main results may be summarized as follows (see Figure 2):

Theorem 7 (Asymptotic Behavior of SL-BVP/IBC Eigenstructure) Consider the SL-BVP with an IBC described by Equation (1.1) with [sigma](r) = [re.sup.[??][theta]] for fixed [theta] and 0 [less than or equal to] r [less than or equal to] [infinity].

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

1. If 0 [less than or equal to] [theta] [less than or equal to] [pi]/2 then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for all n.

2. If [pi]/2 < [theta] [less than or equal to] [pi] then there exists n([theta]) such that

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k < n([theta]) - 1,

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k > n([theta]),

(c) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k = n([theta]) - 1, n([theta]).

These observations are directly applicable to the rectangular waveguide [8, pp. 503-509].

Furthermore, these results lead naturally to the question of the corresponding asymptotic nature of the eigenstructure of the Laplacian on an equilateral triangle with an impedance boundary condition. The special cases of the radiation boundary condition [4] and the absorbing boundary condition [5] have already received exhaustive treatment. The one-dimensional results of the present paper seem to indicate that these two special cases exhibit the full spectrum of possible asymptotic behavior as [sigma] [right arrow] [infinity]. The final installment of this series of papers, Eigenstructure of the Equilateral Triangle, Part V: The Impedance Boundary Condition, will address this problem.

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

Acknowledgements

The author thanks Mrs. Barbara A. McCartin for her indispensable aid in constructing the figures. This paper is dedicated to the memory of our beloved Mother, Dorothy Frances (Kelly) McCartin, on this 25th Anniversary of her departure from her family. Gone but not forgotten!

References

[1] Amrein W. O., Hinz A. M. and Pearson D. B., Sturm-Liouville Theory: Past and Present, Birkhauser, Basel, 2005.

[2] Coddington E. A. and Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill, New York, NY, 1955.

[3] Mahmoud S. F., Electromagnetic Waveguides: Theory and Applications, Peter Peregrinus Ltd., London, UK, 1991.

[4] McCartin B. J., 2004, Eigenstructure of the Equilateral Triangle, Part III: The Robin Problem, International Journal of Mathematics and Mathematical Sciences, Vol. 2004(16), pp. 807-825.

[5] McCartin B. J., 2007, Eigenstructure of the Equilateral Triangle, Part IV: The Absorbing Boundary Condition, International Journal of Pure and Applied Mathematics, Vol.37(3).

[6] Morse P. M., Vibration and Sound, Acoustical Society of America, Melville, NY, 1976.

[7] Morse P. M. and Feshbach H., Methods of Theoretical Physics, Part I, McGraw-Hill, New York, NY, 1953.

[8] Morse P. M. and Ingard K. U., Theoretical Acoustics, McGraw-Hill, New York, NY, 1968.

[9] Strauss W. A., Partial Differential Equations: An Introduction, Wiley, New York, NY, 1992.

[10] Zettl A., Sturm-Liouville Theory, American Mathematical Society, Providence, RI, 2005.

Brian J. McCartin

Applied Mathematics, Kettering University

1700 West Third Avenue, Flint, MI 48504-4898 USA

E-mail: bmccarti@kettering.edu

Table 1: Branch Points and Critical Angles n [[delta].sup.-.sub.n] [[theta].sup.-.sub.n] 0 2.106196+1.125364[??] .7150[pi] 1 2.178042+1.384339[??] .6500[pi] 2 2.214676+1.551574[??] .6172[pi] 3 2.237591+1.676105[??] .5970[pi] 4 2.253497+1.775544[??] .5832[pi] 5 2.265277+1.858384[??] .5730[pi] 6 2.274400+1.929404[??] .5653[pi] 7 2.281699+1.991571[??] .5591[pi] 8 2.287689+2.046852[??] .5541[pi] 9 2.292704+2.096626[??] .5499[pi] 10 2.296970+2.141891[??] .5464[pi] 100 2.346069+3.229037[??] .5074[pi] 1,000 2.354804+4.373567[??] .5010[pi] 10,000 2.356019+5.524184[??] .5001[pi]

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Author: | McCartin, Brian J. |
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Publication: | Global Journal of Pure and Applied Mathematics |

Geographic Code: | 1USA |

Date: | Apr 1, 2007 |

Words: | 5846 |

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