Neumann nodal domains

Abstract

As an alternative to nodal domains in the usual sense, we propose a partition of the domain of a real-valued eigenfunction in by trajectories of the gradient, linking saddle points to extrema. Its most elementary properties are listed and exemplified. The main point is that the problem of avoided crossings is largely eliminated.

1. Introduction

The (Dirichlet) nodal domains of real-valued eigen-functions of the Laplacian in form a fascinating and much-studied subject, as evidenced by other papers in this issue. A notable feature of these domains is their instability under various perturbations because of ‘avoided crossings’ of their boundaries. The classic example [1], p. 451 involves the eigenfunctions of a square of side π. Each of these functions alone is divided by nodal lines into a ‘quilted’ or ‘checkerboard’ pattern of rectangular domains; but if m≠n the non-trivial linear combinations u_mn+au_nm, which are still eigenfunctions, have larger, meandering nodal domains. (For plots of such functions see, e.g. [2], fig. 2 and [3], figs 2 and 3.) The reason is that the originally intersecting nodal lines have split apart into curves that may closely approach each other (like the branches of a hyperbola) but do not intersect. In systems with less symmetry, this behaviour is the rule, not the exception. In fact, a famous theorem of Uhlenbeck [4] shows that intersections of (Dirichlet) nodal curves are very rare.

When nodal curves do cross, the point of intersection is (generically) a saddle point of the eigenfunction u. (The local behaviour of eigenfunctions can be studied in polar coordinates around the point in question [5].) When a small perturbation destroys the nodal crossing, the saddle point does not disappear; it merely occurs at a level other than u=0. This observation raises the possibility of replacing the nodal curves with a network of curves connecting the saddle points, which would survive the perturbation with a similar (for small perturbations, identical) topology. In terms of genericity [4], the existence of a critical point at imposes two requirements on the Taylor-series coefficients of u (namely, ∇u(x,y)=0), which can be expected to be satisfied at various isolated points, whereas to be an intersection of nodal curves imposes a third condition, u(x,y)=0, thereby making the existence of such a point into a rare coincidence.

Unfortunately, because the perturbed saddle points do not all occur at the same level (value of u), the connecting curves cannot be identified by the values of u at points thereon. The only distinguished curves departing from a saddle point are those of steepest descent or ascent. Such curves have tangent vectors everywhere parallel to ∇u, and hence u satisfies the Neumann boundary condition along them. We therefore call them ‘Neumann nodal curves’ and the resulting subdomains of the domain of u we call ‘Neumann nodal domains’. Unfortunately, there are many curves satisfying the Neumann condition that are not part of this network, because they do not pass through saddle points; this makes the construction of the Neumann nodal partition more difficult than that of the Dirichlet one. On the other hand, a Neumann nodal curve, unless it terminates at the boundary, is sure to reach an extremum of u, so the extrema are brought into the picture, unlike in the Dirichlet case.

Our hope is that the stability of the Neumann nodal partition (cf. figure 3) will make it more useful than the Dirichlet nodal partition for some purposes. In [3], a certain asymptotic equipartition property for vibrational energy was shown for one-dimensional eigenfunctions and hence for separable systems in two or more dimensions (viz., the energy of wave motion in every Dirichlet nodal domain of an eigenfunction is asymptotically (at high frequency) the same, and moreover is equally divided between kinetic and potential energy). Such a property clearly cannot hold when adjacent domains can suddenly merge because of the disappearance of a nodal crossing. Our original motivation was the hope that such a theorem might still be true for the Neumann partition; the numerical results in §4 are not encouraging in this regard, but our computations and analysis have not truly reached the asymptotic region. Meanwhile, much interest in the Dirichlet partition has arisen from the possibility of distinguishing isospectral systems by counting nodal domains [6]. There the instability is presumably an advantage (making the number of domains more variable), so we are not promoting the Neumann partition in that context.

Figure 3.

Graphs of proper Neumann nodes (solid) and Dirichlet nodes (dashed): (a) for v₃₂ on the domain [π/6,5π/6]×[π/6,5π/6], and (b) for v₇₆ on the domain [5π/14,9π/14]×[5π/14,9π/14]. The columns are (from left to right) for τ=π/8, τ=π/4 and τ=3π/8. In the middle panels, at upper left and lower right, boundaries have coalesced, destroying two nodal domains. The saddle points are at the X-like crossings of the proper Neumann nodes. It can be seen here that the proper Neumann nodal crossings never separate as the Dirichlet ones have (as soon as τ≠0,π/2). In this sense, the Neumann partition is more stable. (Online version in colour.)

2. Definitions

Let be open and connected, and let be a smooth function.

Definition 2.1 (Neumann nodal curve) —

Consider , where and is connected and open; α is a Neumann nodal curve if and only if (i) α is continuous, (ii) α is differentiable at all t for which ∇u(α(t))≠0, and (iii) at all such t, for some .

Definition 2.2 (Proper Neumann nodal curve) —

A Neumann nodal curve is a proper Neumann nodal curve if and only if S is a closed interval [s₀,s₁], α(s₀) is a saddle point of u, α(s₁) is a critical point of u, and ∀s∈S such that s₀<s<s₁, α(s) is not a critical point of u.

Definition 2.3 (Proper Neumann node) —

If is a proper Neumann nodal curve of u, then the image α[S] is a proper Neumann node of u.

Definition 2.4 (Neumann nodal domain) —

Let N_u⊆Ω be the union of all proper Neumann nodes of u, and let Q be the set of all connected subsets of Ω\N_u. A set D⊆Ω is a Neumann nodal domain if and only if D∈Q and ∀q∈Q, D⊄q.

3. Analytical results

From definition 2.1, we know for a proper Neumann nodal curve that and . So, curve α can be found by solving

3.1

In studying Neumann nodes it is natural (but not necessary [2]) to consider problems with Neumann boundary conditions. So, as a first example, we consider the eigenfunctions of the Laplacian operator ∇² in the square [0,π]×[0,π] with Neumann boundary conditions. Let the eigenfunction u_λ correspond to the eigenvalue λ, and let the eigenfunctions u_mn correspond to the eigenvalue λ_mn=m²+n², so that now

3.2

Plugging u_mn into equation (3.1) and letting x(t)=t gives

3.3

for a curve through a point (x₀,y₀). There is an interesting difference in these curves for the cases m=n and m≠n as illustrated in figure 1. In the first case, the curves intersect at the extrema transversely, but in the second case, they intersect tangentially. In both cases, the curves intersect at the saddle points at right angles. These turn out to be general features of proper Neumann nodal intersections.

Figure 1.

The proper Neumann nodes (solid) and Dirichlet nodes (dashed) for (a) u₂₂ and (b) u₂₃. (Online version in colour.)

Return now to the general case. The solutions to elliptic partial differential equations are smooth enough that we can approximate them locally with a Taylor expansion. For a small enough neighbourhood around a non-degenerate critical point, the function's behaviour will be dominated by its quadratic form at the critical point. Without loss of generality, we consider a function f whose non-degenerate critical point is at (0,0) and whose mixed partial derivatives vanish there. The function's quadratic form is then

3.4

where f_xx and f_yy are the remaining second-order partial derivatives of f (which are non-zero by definition of ‘non-degenerate’).

We construct a Neumann nodal curve through a point (x₀,y₀) that is near, but not equal, to (0,0). Then a non-singular change of independent variable can be made to assure that c(t)=1 in definition 2.1. (Because of the absence of linear terms in (3.4), we would run into trouble if we tried to do that right down to the critical point. As our analysis and figures show, the usual existence-and-uniqueness theorem need not apply at that singular point of the system; solution curves can be tangent at a point.) Thus, the equation to find the Neumann nodal curves α becomes

3.5

Solving this yields four possible solutions for α that have promise of being pairwise continuations of each other:

3.6

By suitably choosing (x₀,y₀), each of these solutions can be made to pass through (0,0). The freedom to have allows them to have non-vanishing limiting tangent vectors there.

There are three possible outcomes to this construction, of which figure 2 shows the interesting two. The first, and the least interesting, case is that f_xx(0,0)=f_yy(0,0). In this case, the curves come out of the critical point in arbitrary directions determined by x₀ and y₀. Since the direction is arbitrary, intersections at such critical points have almost no restrictions on them. The second possibility comes when again f_xx(0,0) and f_yy(0,0) have the same sign, but f_xx(0,0) ≠ f_yy(0,0). In this case, the curves generically are tangent at (0,0) (horizontal or vertical depending on the direction of the inequality), but an isolated trajectory perpendicular to the others may exist (for x₀=0 in the case where f_xx is the larger as seen in the left panel of figure 2). Both cases where f_xx(0,0) and f_yy(0,0) have the same sign are extrema of f, which act as attractors for the Neumann nodal curves, in the sense that all sufficiently nearby nodal curves reach the extremum.

Figure 2.

Examples of Neumann nodes of a quadratic form. (a) The case of u_xx and u_yy unequal but of the same sign. (In the case shown, u_xx is larger.) All of these solutions pass through the critical point. The solid curves represent a family of solutions that intersect tangentially at the critical point. The dashed line is a single unique solution that is perpendicular to the others. (b) The case of u_xx and u_yy of different sign. In this case, the curves shown do not pass through the critical point, but are converging to the single set of solutions that does, the dashed curves being closer than the solid ones. (Online version in colour.)

The third case is that (0,0) is a saddle point, so that f_xx(0,0) and f_yy(0,0) are different in sign. In this case, the curves are hyperbolas with the principal axes of the quadratic form as the asymptotes. Thus, generically a saddle point is a repeller, not an attractor. In the limiting case where α reaches the critical point, these hyperbolas become perpendicular lines.

These observations lead us to conclude the following:

Theorem 3.1 —

If Neumann nodal curves intersect at a non-degenerate extremum, and the Hessian matrix of the function has non-degenerate eigenvalues at that point, the curves must intersect either tangentially or perpendicularly (the latter being an unstable special case).

Theorem 3.2 —

If Neumann nodal curves intersect at a non-degenerate saddle point, they must do so perpendicularly.

Remark 3.3 —

An example of a degenerate saddle point occurs at the origin in a normal mode of a disc, with m≥3. Here, m Neumann nodal curves (alternating with m Dirichlet nodal curves) pass radially through the point.

4. Numerical results

Many situations cannot be examined analytically in a straightforward manner. We return to the degenerate eigenfunctions of the Laplacian in a square to see what happens to the Neumann nodal domains when the Dirichlet nodal domains are exhibiting intersection avoidance. The proper Neumann nodal curves of the functions

4.1

with u_mn as in (3.2), were found by the simple computational method of Euler integration out from the saddle points. (In that direction the integration is stable.) Figure 3 shows how the proper Neumann nodal domains change as τ is adjusted to move v_mn from u_mn to u_nm. At first glance, the Neumann nodal domains seem much more stable than their regular nodal counterparts, but there is a transition, shown by the middle images in figure 3, in which some of the domains shrink to zero area.

Figure 3 suggests that this feature persists for higher values of λ. In this situation, since the function v remains bounded, and we can choose the shrinking domain to be as small as we like by tuning t, the energy integral of this domain can be made as small as we like. If it is present for arbitrarily large λ, then we should have no expectation of asymptotic equipartition of energy.

5. Conclusion

In this note (based in part on [2]), we propose a substitute for the pattern of Dirichlet nodal curves in a real eigenfunction of the two-dimensional Laplacian. The idea is to connect the critical points of the function by paths of steepest descent, characterized as Neumann nodal curves because of the function's vanishing normal derivative in the orthogonal direction. These curves divide the region into Neumann nodal domains. The crucial property of this partition is its relative stability under perturbations of the eigenfunction (by a change of boundary condition or geometry or by considering a different eigenfunction for an eigenvalue with multiplicity). Here, we have presented only the merest beginning of the study of this potentially interesting structure.

The extension of the construction to more general operators (in particular, Schrödinger operators with potentials) is not difficult; note that most of the considerations in §§2 and 3 apply to any smooth function, not necessarily a Laplacian eigenfunction. The situation in higher dimensions (even for the Dirichlet nodal partition) is more complicated; we merely note that saddle points may (or may not) be more numerous than extrema [7].

It may be interesting to see how changes in the boundary conditions or geometry of the boundary affect the proper Neumann nodes. It may also be interesting to see what relationship, if any, exists between the counts of Neumann nodal domains and Dirichlet nodal domains. However, these questions are left for future exploration.

Funding statement

This research was supported in part by National Science Foundation Grants PHY-0554849 and PHY-0968269.