Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic
We study the scaling asymptotics of the eigenspace projection kernels of the isotropic Harmonic Oscillator of eigenvalue in the semi-classical limit . The principal result is an explicit formula for the scaling asymptotics of for in a neighborhood of the caustic as The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as . In previous work we proved that the density of zeros of Gaussian random eigenfunctions of have different orders in the Planck constant in the allowed and forbidden regions: In the allowed region the density is of order while it is in the forbidden region. Our main result on nodal sets is that the density of zeros is of order in an -tube around the caustic. This tube radius is the ‘critical radius’. For annuli of larger inner and outer radii with we obtain density results which interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff -dimensional measure of the intersection of the nodal set with the caustic is of order .
S. Zelditch]email@example.com P. Zhou]firstname.lastname@example.org B. Hanin]email@example.com
July 17, 2018
This article is concerned with the scaling asymptotics of eigenspace projections of the isotropic Harmonic Oscillator
and their applications to nodal sets of random Hermite eigenfunctions when . It is well-known that the spectrum of consists of the eigenvalues
The semi-classical limit at the energy level is the limit as with fixed , so that only takes the values
We denote the corresponding eigenspaces by
The eigenspace projections are the orthogonal projections
An important feature of eigenfunctions of Schrödinger operators is that as and with fixed eigenvalue , they are rapidly oscillating in the classically allowed region
and exponentially decaying in the classically forbidden region
with an Airy type transition along the caustic
This reflects the fact that a classical particle of energy is confined to In the forbidden region, eigenfunctions exhibit exponential decay as , measured by the Agmon distance to the caustic. We refer to [Ag, HS] for background. In dimension one, eigenfunctions have no zeros in the forbidden region, but in dimensions they do. In the allowed region, nodal sets of eigenfunctions behave in a similar way to nodal sets on Riemannian manifolds [Jin], but in the forbidden region they are sparser. The only results at present on forbidden nodal sets seem to be those of [HZZ, CT]. This article contains the first results on the behavior of nodal sets in the transition region around the caustic. The scaling asymptotics of zeros around the caustic is analogous in many ways to the scaling asymptotics of eigenvalues of random Hermitian matrices around the edge of the Wigner distribution in [TW], and as will be seen, the scaled Airy kernel of [TW] is the same as the scaled eigenspace projections when (see Remark 1).
When the eigenspaces have dimension and it is a classical fact (based on WKB or ODE techniques) that Hermite functions and more general Schrödinger eigenfunctions exhibit Airy asympotics at the caustic (turning points). See for instance [Sz, O, T, Th, FW]. The main purpose of this article is to formulate and prove a generalization of these Airy asymptotics to all dimensions for the isotropic Harmonic Oscillator. Instead of considering individual eigenfunctions, we consider the scaling asymtptoics of the eigenspace projection kernels (3) with in an -tube around . Our main result gives scaling asymptotics for the eigenspace projection kernels (3) around a point of the caustic. To state the result, we introduce some notation. Let be a point on the caustic for . Points in an neighborhood of may be expressed as with . The caustic is a -sphere whose normal direction at is , so the normal component of is when , where . We also put for the tangential component, and identify . By rotational symmetry, we may assume , so that .
Above, is the Airy function, and is a weighted Airy function, defined for by
where is the usual contour for Airy function, running from to on the right half of the complex plane (see Appendix A for a brief review of the Airy function).
To our knowledge, this is the first result on Airy scaling asymptotics of Schrödinger eigenfunctions in dimensions . Theorem 1.1 is proved in Section 3. The on-diagonal result (6) is proved first in Proposition 3.3 because it is the important case for the applications to nodal sets. It is not obvious that (5) reduces to (6) when , but this is proved by combining (7) with Lemma A.2 on products of Airy functions. The case of general is obtained by a simple rescaling as in Section 1.
The isotropic Harmonic Oscillator is special even among Harmonic Oscillators because of the maximally high multiplicity of eigenvalues, and there is no direct generalization of Theorem 1.1 to eigenspace projections of other Schrödinger operators. However, we expect that the scaling asymptotics generalize if we replace eigenspaces by spectral projections for small intervals in the spectrum (work in progress). To explain the unique features of (1), we recall (Section 2.1) that is spanned by Hermite functions of degree in variables and
The high multiplicities are due to the -invariance of the isotropic Harmonic Oscillator, and the periodicity of the classical Hamiltonian flow. As a result, the quantum propagator is (essentially) periodic, and the Mehler formula (20) expresses the propagator as an integral over the circle. This expression is used to obtain the scaling asymptotics of for near . For general Harmonic Oscillators with incommensurate frequencies the eigenvalues have multiplicity one and the eigenspace projections are of a very different type. It is for this reason that we only consider the isotropic Harmonic Oscillator in this article.
Besides the edge asymptotics of [TW], the asymptotic behavior of in a - neighborhood of the caustic is reminiscent of the scaling asymptotics of the Szegő projector in [BSZ] and of spectral projections on Riemannian manifolds [CH], which both have universal scaling limits. However, the presence of allowed and forbidden regions is a new feature of Schrödinger operators that does not occur for Laplacians or in the complex setting. If one rescales in an nieghborhood of a point in the allowed region, one would obtain results analogous to those for Laplacians on Riemannian manifolds in [CH]. But the -scaling asymptotics along the caustic are of a fundamentally different nature. Although there are several studies of Airy asymptotics of Wigner functions in dimension one around the caustic in phase space (originating in [Be]), we are not aware of any prior studies of the scaling asymptotics of the spectral projections kernels along the caustic in configuration space. It is an important aspect of Schrödinger equations that deserves to be studied in generality. In [HZZ3] we study the Wigner distribution of (3) and its Airy scaling asymptotics around the phase space energy surface , which gives a higher dimensional generalization of [Be].
1.1. Random Hermite eigenfunctions
where and . Here the coefficients are i.i.d. normal random variables and is an orthonormal basis of consisting of multivariable Hermite functions (see Section 2.1). Equivalently, we use the basis to identify and then endow with the standard Gaussian measure. The Schwartz kernel of the eigenspace projection (3) is the covariance (or two-point) function of
The random Hermite functions are a centered Gaussian field. Their properties are therefore completely determined by
As a first application of Theorem 1.1, we determine the expected mass of -normalized random Hermite eigenfunctions in a metric tube of radius around the caustic. Here is the distance from to . We define the -mass-squared of an -normalized eigenfunction in the -tube by
Let , . Then the expected mass-squared of an -normalized Hermite eigenfunction in are given by
The mass density in the -tube is ; integration over the thin tube introduces the additional volume factor . The proof is given in §3.5.
norms of Hermite eigenfunctions are studied in [Th, KT] (see also their references) and pointwise bounds on (non-dilated) Hermite functions are given near the caustic in [Sz] in dimension 1 (see Lemma 1.5.1 of [Th]). In dimension 1, the maximum of the th (unscaled) Hermite function is achieved at a point close to the caustic (turning points). This motivates the question of how mass of eigenfunctions builds up around the caustic for general in all dimensions. The same question for arises in the study of nodal volumes. Eigenfunctions concentrated on a single trajectory are extremals for low norms. The above Corollary shows that the mass density is constant when and decays for for a random Hermite function.
For it is shown in [Th], Lemma 1.5.2 that the sup norm of normalized Hermite functions is . As reviewed in §2.1, the sup norm of the semi-classically scaled eigenfunctions of this article (18) equal times the sup norm of the unscaled Hermite functions . If we fix the energy level and set as , then , which agrees with the mass density formula above.
1.2. Applications to nodal sets of random Hermite eigenfunctions
One of our principal motivations to study the scaling asymptotics of the eigenspace projections is to understand the transition between the behavior of nodal sets of random eigenfunctions in the allowed and forbidden regions. In particular, we study in
Theorem 1.4 the average density of the nodal set
near Let us denote by the random measure of integration over with respect to the Euclidean hypersurface measure (the co-dimension Hausdorff measure ) of the nodal set. Thus for any measurable set ,
Its expectation is the average density or distribution of zeros and is the measure on defined by
where is the Gaussian from which is sampled. In recent work [HZZ], the authors showed that has a different order as in the allowed and forbidden regions.
Theorem 1.3 ([Hzz] 2013).
Fix The measure has a density with respect to Lebesgue measure given by
where the implied constants in the ‘’ symbols are uniform on compact subsets of the interiors of and , and where and depend only on
Our main result on nodal sets (Theorem 1.4) gives scaling asymptotics for the average nodal density that ‘interpolate’ between (11) and (12). The computer graphics of Bies-Heller [BH] (reprinted as Figure 1 in [HZZ]) show that the nodal set in near the caustic consists of a large number of highly curved nodal components apparently touching the caustic while the nodal set in near consists of fewer and less curved nodal components all of which touch the caustic. The scaling limit of the density of zeros in a shrinking neighborhood of the caustic, or in annular subdomains of and at shrinking distances from the caustic are given Theorems 1.4 and 1.5. The varying density of zeros in near the caustic proved there is the new phenomenon related to nodal sets at issue in this article. For the expected density of zeros in the -neighborhood of the caustic, we apply the Kac-Rice formula to Theorem 1.1; for the expected density of zeros in the -neighborhood of the caustic, where , we invoke the Kac-Rice formula together with a non-standard stationary phase method (see Propositions 6.1 and 6.5).
The nodal set of near the caustic consists of a mixture of components from the forbidden nodal set and from the allowed nodal set (see Figure 1). To be more precise, if is non-zero, then there do not exist nodal domains contained entirely in , where the potential is greater than the energy , because forces and to have the same sign in . In a nodal domain we may assume , but then is a positive subharmonic function in and cannot be zero on without vanishing identically. Hence, every nodal component which intersects must also intersect and therefore . As indicated in the computer graphics of [BH], some of these components remain in a very small tube around and some stretch far out into and most of these (apparently) stretch out to infinity. The density results of Theorem 1.3 suggest that there should only exist on average order of of the latter, not enough to explain the size of around the caustic proved in Theorem 1.4 below (see also Remark 4). Hence, one expects the main contribution to the nodal density near to come from nodal components living mainly in which cross .
To state our first result precisely, we fix , where , and study the rescaled ensemble
and the associated hypersurface measure
Our main result gives the asymptotics of when is in terms of the weighted Airy functions (see (7)).
Theorem 1.4 (Nodal set in a shrinking ball around a caustic point).
Fix and , i.e. . For any bounded measurable
and is the symmetric matrix
where . The implied constant in the error estimate from (13) is uniform when varies in compact subsets of .
The leading term in is -independent and positive everywhere since the matrix as a linear operator has nontrivial range. Indeed, as shown in Proposition 4.1, for all integers , hence term in (14) is nonzero. The matrix in (14) is a rank projection onto the direction. Hence, since the dimension , it cannot cancel out the second term.
Theorem 1.4 says that if and for some bounded measurable then
which shows that the average (unscaled) density of zeros in a tube around grows like as
The choice of radius is dictated by the scaling asymptotics of the associated covariance (2-point) function (10), which are stated in Theorem 1.1 and proved in §3. The rescaled random Hermite functions converge to an infinite dimensional Gaussian ensemble of solutions of a scaled eigenvalued problem, which is identified in Section 3.6.
As mentioned above, the scaling asymptotics of zeros around the caustic, especially in the radial (normal) direction, is analogous to the scaling asyptotics of eigenvalues of random Hermitian matrices around the edge of the spectrum. But the scaled radial distribution of zeros of random Hermite eigenfunctions does not seem to be a determinantal process, while eigenvalues of random Hermitian matrices is determinantal.
1.3. Sub-critical shrinking of balls around caustics points
So far, we have considered in detail the rescaling of the eigenspace projections and nodal sets in a -tube around the caustic. In [HZZ] we have studied the bevavior within the allowed and forbidden regions at fixed (-independent) distance from the caustic. We refer to such regions as the ‘bulk’. In section 6, we fill in the gaps between the caustic tube and the ‘bulk’ in the allowed and forbidden regions. That is, we consider shrinking annuli around the caustic which lie outside the -tube in a sequence of rings around the caustic. In this way, we obtain scaling results that interpolate between the bulk results of [HZZ] and the caustic scaling results above. This is the purpose of studying sub-critical rescaling exponents, i.e. the nodal set around where and .
Fix and , and . Then the rescaled expected distribution of zeros has a density with respect to Lebesgue measure given by
where the implied constants in the ‘’ symbols are uniform for in a compact subset of , and are positive dimensional constants.
The two error terms come from the non-standard stationary phase expansion used to prove Theorem 1.5: the comes from approximating the critical point of the phase function, while the marks the failure of the stationary phase expansion as approaches .
1.4. Nodal set intersections with the caustic
Our next result measures the density of intersections of the nodal set with the caustic. This is much simpler than measuring the density in shrinking tubes or annuli, since it is not necessary to rescale the covariance kernel (10). For an open set we consider
When this means to count the number of nodal intersections with the caustic. Since the Gaussian measure is invariant and the caustic is a sphere, the average nodal density along the caustic is constant.
Fix and define the constant
where are defined in (7). Then, as for any open set
In particular, if , where
The proof is to use a Kac-Rice formula for the expected number of intersections of the nodal set with the caustic. The relevant covariance kernel is the restriction of in (14) to the tangent plane of the caustic (hence the radial component drops out). It is analogous to the formula in [TW] (Proposition 3.2) for the expected number of intersections of nodal lines with the boundary of a plane domain, and we therefore omit its proof.
1.5. Outline of the proofs
As mentioned above, the proofs of Theorems 1.4, 1.5, and 1.6 are based on a detailed analysis of the Kac-Rice formula (Lemma 2.2 in §2.4), which gives a formula for the average density of zeros at in terms of a covariance matrix depending only on and its derivatives
restricted to the diagonal. The kernels behave differently depending on the position of relative to the caustic and have contour integral representations of the following type
where is the usual Airy contour (Appendix A), is a parameter, and we simplified the phase function and the amplitude by keeping only the leading term. See (27) for the precise formulas. The critical points of the simplified phase function are . If , we would have two separate critical points, which were analyzed in our previous paper [HZZ]. If , then as the two critical points starts to move closer, with , and each corresponds with a Gaussian bump of width . Hence the two bumps will start overlapping when . In more details, the integral near each critical point can be roughly evaluated as follows
where is a normalization constant. The point of the above sketch computation is to show that due to the singularity of , the error term is enhanced to from the naive expectation .
Hence the break down of the expansion at . Another way to see the break down of stationary phase at , which we do not go into detail in the paper
Hence we see at , the factor in the exponent is unity, hence we do not have a small parameter to do asymptotic expansion, and have to express the result in terms of the weighted Airy functions defined in (7).
To get rid of factors, we will set for the rest of the article, and drop the subscript. The general case can be obtained by the following replacement
For general , the dilation factor in Theorem 1.5 should be changed to . We will also abbreviate throughout with the understanding that is fixed so that
We follow the notation of [HZZ] and refer there for background on the isotropic Harmonic Oscillator, its spectrum and its spectral projections. We also refer to that article for background on the Kac-Rice formula and other fundamental notions on Gaussian random Hermite eigenfunctions. In this section, we recall some of the basic definitions and facts that are used in the proofs of the main results.
Fix . An orthonormal basis of the eigenspace (2) is given by
where is a dimensional multi-index with and is the product of the Hermite polynomials (of degree ) in one variable, with the normalization that . The eigenvalue of is given by
The multiplicity of the eigenvalue is the partition function of , i.e. the number of with a fixed value of . Hence
For further background and notation we refer to [HZZ].
2.2. Mehler Formula for the propagator
The Mehler formula gives an explicit expression for the Schwartz (Mehler) kernel
of the propagator, The Mehler formula [F] reads
where and . The right hand side is singular at It is well-defined as a distribution, however, with understood as . Indeed, since has a positive spectrum the propagator is holomorphic in the lower half-plane and is the boundary value of a holomorphic function in .
2.3. Spectral projections
We also use that the spectrum of is easily related to the integers . The operator with the same eigenfunctions as and eigenvalues is often called the number operator, . If we replace by then the spectral projections are simply the Fourier coefficients of . In [HZZ], we used the related formula,
where, as before, The integral is independent of . Using the Mehler formula (20) we obtain a rather explicit integral representation of (10). If we introduce a new complex variable , then the above integral can be written as
where the contour is a circle traversed counter-clockwise.
2.4. The Kac-Rice Formula
As with Theorem 1.3, the proofs of Theorems 1.4, 1.5 and 1.6 are based on the Kac-Rice formula [AW, Thm. 6.2, Prop. 6.5] for the average density of zeros. The Kac-Rice formula is the formula for the pushforward of the Gaussian measure on the random Hermite functions under the evalution maps . It is valid at as long as the so-called 1-jet spanning property holds at Namely, is surjective, or equivalently, the covariance matrix of values and gradients is invertible. We now verify that the condition for the validity of the Kac-Rice formula holds.
For any , the 1-jet evaluation map
is surjective, where is defined in Eq (2). Equivalently, if is equipped with a standard Gaussian measure induced by the inner product on , then its pushforward under is a non-degenerate Gaussian measure on .
Fix an orthonormal basis of , then can be written as a matrix , where . Showing is surjective is equivalent to showing has rank , or is a non-degenerate square matrix. By definition, is the covariance matrix of the Gaussian measure . Hence, the two statements in the proposition are equivalent.
Recall that is the Gaussian random variable valued in with measure . We then express in polar coordinates where and . The first observation is that is block diagonal if we break it up into its radial part and angular part,