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Avoided intersections of nodal lines

We consider real eigen-functions of the Schr\"odinger operator in 2-d. The nodal lines of separable systems form a regular grid, and the number of nodal crossings equals the number of nodal domains. In contrast, for wave functions of non integrable systems nodal intersections are rare, and for random waves, the expected number of intersections in any finite area vanishes. However, nodal lines display characteristic avoided crossings which we study in the present work. We define a measure for the avoidance range and compute its distribution for the random waves ensemble. We show that the avoidance range distribution of wave functions of chaotic systems follow the expected random wave distributions, whereas for wave functions of classically integrable but quantum non-separable wave functions, the distribution is quite different. Thus, the study of the avoidance distribution provides more support to the conjecture that nodal structures of chaotic systems are reproduced by the predictions of the random waves ensemble.


I. INTRODUCTION
The morphology of the nodal sets of wave functions depends crucially on whether the underlying classical dynamics is integrable or chaotic. This was first proposed in [1] and was followed by the study of various features of the nodal lines, such as the distribution of its curvatures [2]. Recently, the counting statistics of nodal domains for integrable and chaotic systems were investigated in [3], and it was shown that in the chaotic case, the statistics follow the predictions derived by assuming that the wave functions are random superpositions of plane waves [4,5]. Local effects due to boundary conditions, and the corresponding modifications of the random wave ensembles were also studied [6,7].
The interest in the properties of the nodal set is not confined to the physics literature only. Most of the mathematics literature on this subject is concerned with solutions of the Helmholtz equation in the interior of compact domains in R 2 with Dirichlet or Neumann boundary conditions. Courant [8,9] and later Pleijel [10] pioneered these studies, and computed an upper bound on the number of nodal domains. Krahn [11] provided a lower bound for the area of a nodal domain. Other authors gave estimates on e.g., the length of the nodal sets [12], and addressed the general properties of the nodal network [13].
Uhlenbeck's theorem [14] states that the nodal lines of "generic" wave functions do not intersect (this statement will be further discussed in the next section). An important class of exceptions of this rule, are the eigen-functions of separable systems, where the nodal lines form a grid, and the number of intersections equals the number of nodal domains.
The purpose of the present work is to provide a quantitative measure of the degree by which nodal lines avoid each other. We shall associate the avoidance range to each avoided crossing, and will compute their distribution for the random waves ensemble. The avoidance range vanishes for an intersection, and therefore, the avoidance range distribution for separable functions is proportional to a Dirac δ at zero. We also show that eigen-functions of classically chaotic systems display an avoidance range distribution which is similar to the one obtained for a random waves ensemble, while for an intermediate system, the avoidance distribution differs substantially from either of the extreme distributions mentioned above. The nodal structure for a few representative wave functions can be seen in figure (1).
The wave function of a rectangle (a) displays a perpendicular grid of nodal lines typical of separable systems. An equilateral triangle (b) is classically integrable, but not separable, The paper is organized in the following way: We shall begin the next section by discussing the nodal structure of general solutions of the Helmholtz equation. We shall then identify avoided crossings of the nodal lines, and define the corresponding avoidance range. The avoidance range distribution will be written down explicitly. In section (III) we shall define the random wave ensemble and compute explicitly the expected distribution of the avoidance ranges. The resulting expression will be compared with the avoidance distributions obtained numerically for high lying eigen-functions of a chaotic domain. For Dirichlet problems, the boundary belongs to the nodal set, where nodal intersections occur at a density which is approximately 2 intersections per wavelength. This property cannot be reproduced by the uniform random wave ensembles, which are expected to account for bulk properties. This problem, in the context of the avoided crossing distribution, is discussed at the end of section III.

II. AVOIDED CROSSINGS OF NODAL LINES AND THE AVOIDANCE RANGE
We consider real solutions of the Helmholtz equation in a domain Ω ∈ R 2 which can be expressed as linear superpositions of regular solutions of the equation in the entire plane.
They can be expressed in terms of either plane or cylindrical waves, and the two representations are equivalent. In the plane waves representation the wave function is written as with wave vectors (k n = −k −n , |k n | 2 = k 2 ) directed at an angle θ n and coefficients which ensure that the series (1) converges absolutely (we require a * n = a −n to render Ψ(r) real). Using the expansion of a plane wave in cylindrical coordinates we get with α l = ∞ n=−∞ a n e ilθn , and α * l = (−1) l α −l .
The infinite sums in both (1) and (2) can be truncated, when one considers domains of finite area. Semiclassical arguments show that the necessary number of terms is L ≈ Lk/π, where L is the perimeter of the domain. For reasons which will become clear in the sequel, we prefer to use the cylindrical wave representation in the present work.
The expansion (2) refers to a particular choice of the origin. Using Graf's addition theorem [15] the origin can be shifted to r, and the wave function retains its form, where with α l (0) = α l , and the angle φ is measured from the direction defined by r. The translated coefficients α m (r) = β m (r) + iγ m (r) are related to the wave-function and its derivatives computed at the point r: In the close vicinity of r, where kρ < 1, and to second order in kρ where φ l are the phases of α l (r). If β 0 (r) = 0, r is a nodal point. It cannot be an isolated zero since the second term vanishes on the line segment through r which is oriented at the direction π 2 − φ 1 . Hence, the nodal set consists of lines. Two nodal lines intersect at r if both β 0 (r) = 0 and α 1 (r) = 0, while α 2 (r) = 0. The intersection is perpendicular since cos 2(φ + φ 2 ) vanishes along two perpendicular lines which intersect at r. For the time being we shall continue the discussion assuming that α 2 (r) = 0. The more general case will be commented on at the end of this section.
An avoided crossing occurs at r when α 1 (r) = 0 and |α 2 (r)| > |β 0 (r)| > 0. In other words, when r is a saddle point of the wave function. This can be easily seen by writing the equation of the zero set of (7) in terms of the local coordinates ρ = (ξ, η), This is a hyperbola (ellipse) if |α 2 (r)| is larger (smaller) than |β 0 (r)|. At an avoided crossing, the scaled distance between the two branches is 5 This is the avoidance range associated with the avoided crossing at r.
A few comments are in order: i. At a nodal intersection z = 0.
iii. An equivalent expression for z(r) in terms of Ψ(r) and its Cartesian derivatives reads iv. Consider an elliptic critical point of the wave function. In the quadratic approximation the area of the elliptic nodal domain is This area is always larger than 4πk −2 . Krahn's theorem [11] gives j 2 0,1 πk −2 as the maximal lower bound to the area of any nodal domain, where j 0,1 ≈ 2.405 is the first zero of the Bessel function J 0 (x). The lower bound 4πk −2 is smaller but not very far from Krahn's exact value and thus consistent.
So far we considered the intersections of two nodal lines. However, higher order intersections may occur. In general, if the first non vanishing coefficient at r is α q , then r is a nodal point of order q, where q nodal lines intersect at angles π q . The higher q, the more rare are the intersections, since more conditions are to be satisfied by the coefficients. This explains Uhlenbeck's theorem [14] mention above. From now on we shall discuss the most common intersections with q = 2, and comment about the higher order intersections whenever necessary.
Up to now we discussed individual avoided crossings, and defined the associated avoidance ranges. In the next section we shall consider the distribution of the avoidance ranges of a wave function in the domain of its definition, and compute its mean for random waves ensembles.
Numerically computed I(z) are shown in figure 2 for the three wave functions plotted in

A. Avoidance range distribution for the random waves ensemble
One of the main goals of the present work is to show that the properties of the nodal set of chaotic billiards, as detected by the distribution of avoidance ranges, are reproduced by the distributions computed for the random wave ensembles. Because wave functions are subject to boundary conditions, it is expected that the predictions of the isotropic random wave ensemble used e.g. in [3,5], are relevant only to the bulk of the domain, and will do poorly in the λ = ( 2π k ) vicinity of the boundary. This was observed and discussed in [3]. The isotropic random wave ensemble is the ensemble of wave functions (2) where the real parameters β l and γ l are independent, identically distributed random Gaussian variables with zero mean and unit variance for all |l| ≥ 1. Because γ 0 = 0 the variance of β 0 is twice that of all the others. The ensemble average of a function f will be denoted by f .
The local coefficients α l (r) = β l (r) + iγ l (r) were derived from the original ones by a unitary transformation. Hence, they are also independent identically distributed Gaussian variables, and the ensemble averages of the number of critical points (12), the number of saddle points (14) and the mean distribution of avoidance ranges (15) can be computed by considering Gaussian integrations with respect to the variables β 0 , β 1 , γ 1 , β 2 and γ 2 . A straightforward integration gives, where |Ω| is the area of the domain. The average number of saddle points is a half of the number of critical points. The other half are the points were the wave function has either a minimum or a maximum.
The mean number of saddle points with an avoidance range less than or equal to z can be also computed, The normalization gives trivially the mean number of saddle points in the ensemble. Therefore we can define the probability with a corresponding density These results served to test the conjecture that the bulk properties of the nodal sets Another distribution which we compared to the prediction of the random waves ensemble is the number of saddle points. Normalizing the number of saddle points by the prediction of the isotropic random wave ensemble, we observe as k increases that the numerical computation approach the predicted value (see figure 4).
The systematic deviation observed at finite k is due to boundary effects. Following [6] we compute the effect of an infinite straight Dirichlet line, which is reproduced by the wave ensemble Ψ(r) = 2 ∞ n=1 c n sin(nθ) J n (kr) , where the c n are real coefficients taken as independent random Gaussian distributed variables.
Not entering into the details of the calculation, the computed mean density of saddle points approach the bulk expression (16) as the distance from the Dirichlet line increases.
The integrated density in the perpendicular direction shows a global deficiency of saddle points relative to the bulk value. It diverges logarithmically as a function of the distance from the Dirichlet line [6].
We compared the results of the ensemble (20) to the number of saddle points counted for highly exited eigen-states of a chaotic billiard with sufficiently smooth boundaries. The integral along the boundary multiplies the density by the perimeter of the billiard L. The perpendicular integral must be truncated in view of the logarithmic divergence mentioned above. A sensible choice of the truncation distance is R = |Ω|/2. The resulting estimate for the mean number of saddle points is with σ 1 ≈ 0.014 and σ 2 ≈ 2.0. The deficiency of saddle points is explained by the effect of the Dirichlet boundaries that affects the statistics. The dashed line in figure 4 represents the equation (21) and the agreement is definitely improving. However, the domain of low values k < 500 is not in complete agreement. This deficiency can be associated with the corners of the billiard, to its curvature, or to the finiteness of the Dirichlet line that the ensemble (20) can not reproduce. These possible causes will be studied elsewhere.
In conclusion we can say that the properties of the nodal set of chaotic wave functions which were investigated in this work, are very well reproduced by the isotropic random waves ensemble in the semiclassical domain. These findings are consistent with previous works on the subject, and add support to the random waves conjecture. at the Weizmann Institute.