Interface asymptotics of Eigenspace Wigner distributions

# Interface asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator

Boris Hanin and Steve Zelditch Department of Mathematics, Texas A&M and Facebook AI Research, NYC Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
###### Abstract.

Eigenspaces of the quantum isotropic Harmonic Oscillator on have extremally high multiplicites and the eigenspace projections have special asymptotic properties. This article gives a detailed study of their Wigner distributions . Heuristically, if , is the ‘quantization’ of the energy surface , and should be like the delta-function on ; rigorously, tends in a weak* sense to . But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of in the interior of ; interface Airy scaling asymptotics in tubes of radius around , with either in the interior or exterior of the energy ball; and exponential decay rate sin the exterior of the energy surface.

Research partially supported by NSF grant DMS-1810747.

B. Hanin]bhanin@tamu.edu

S. Zelditch]zelditch@math.northwestern.edu

July 20, 2019

## 1. Introduction

 Wℏ,EN(ℏ)(x,ξ):=∫RdΠℏ,EN(ℏ)(x+v2,x−v2)e−iℏv⋅ξdv(2πh)d (1)

of the eigenspace projections for the isotropic Harmonic Oscillator

 ˆHℏ=d∑j=1(−ℏ22∂2∂x2j+x2j2)onL2(Rd,dx). (2)

The notation in (1) is as follows: the spectrum of (2) consists of the eigenvalues

 EN(ℏ)=ℏ(N+d/2),(N=0,1,2,…) (3)

with multiplicities given by the composition function of and (i.e. the number of ways to write as an ordered sum of non-negative integers). That is, the eigenspaces

 Vℏ,EN(ℏ):={ψ∈L2(Rd):ˆHℏψ=EN(ℏ)ψ}, (4)

have dimensions given by

 dimVℏ,EN(ℏ)=p(N,d)≃Nd−1. (5)

Due to extreme degeneracy of the spectrum of (2) when , the eigenspace projections

 Πℏ,EN(ℏ):L2(Rd)→Vℏ,EN(ℏ) (6)

have very special properties reflecting the periodicity of the classical Hamiltonian flow and of the Schrödinger propagator (see Definition 1.1 for notation). The focus of this article is on the Wigner distributions of individual eigenspace projections.

###### Definition 1.1.

The Wigner distributions of the eigenspace projections are defined by,

 Wℏ,EN(ℏ)(x,ξ)=∫RdΠℏ,EN(ℏ)(x+v2,x−v2)e−iℏv⋅ξdv(2πh)d. (7)

When , i.e.

 ℏ=ℏN(E):=EN+d2, (8)

the Wigner distribution of a single eigenspace projection (7) is the ‘quantization’ of the energy surface of energy and should therefore be localized at the classical energy level , where . We denote the (energy) level sets by

 ΣE={(x,ξ)∈T∗Rd:H(x,ξ):=12(||x||2+||ξ||2)=E}. (9)

The Hamiltonian flow of is periodic, and its orbits form the complex projective space where is the equivalence relation of belonging to the same Hamilton orbit. Due to this periodicity, the projections (6) are semi-classical Fourier integral operators (see [GU12, HZZ15]). This is also true for the Wigner distributions (7). Their properties are basically unique to the isotropic oscillator (2) and that is the motivation to single them out in this article111See Section 1.8 for a more precise statement.. These properties are visible in Figure 2 depicting the graph of .

Wigner distributions were introduced in [W32] as phase space densities. Heuristically, the Wigner distribution (6) is a kind of probability density in phase space of finding a particle of energy at the point . This is not literally true, since is not positive: it oscillates with heavy tails inside the energy surface (9), has a kind of transition across and then decays rapidly outside the energy surface. The purpose of this paper is to give detailed results on the concentration and oscillation properties of these Wigner distributions in three phase space regimes, depending on the position of with respect to . The main results of this article are:

1. Theorem 1.5, giving the interior Bessel asymptotics for ;

2. Theorem 1.4, giving the interface Airy scaling asymptotics of the Wigner distributions (7) in tubes of radius around , with either in the interior or exterior of the energy ball;

3. Proposition 1.7, giving the exponential decay rate in the exterior of the energy surface.

Of these results, the scaling asymptotics around is the most novel and is part of a more general program to analyse spectral scaling asymptotics around various types of interfaces. In a subsequent article [HZ19A], we consider Wigner distributions of more general spectral projections for ‘windows’ of eigenvalues of varying widths and centers. When one sums over large enough windows, the scaling asymptotics should be universal, and that is the subject of [HZ19B]. Wigner distributions of individual eigenspace projections of the isotropic harmonic oscillator have special asymptotics and are not universal. For general Schrödinger operarators, whose classical Hamiltonian flows are non-periodic and whose eigenspaces are not of ‘maximal multiplicity’, the generalization of individual eigenspace projections of (2) is spectral projections for a window of width around an energy level. In general, even for generic Harmonic oscillators, one would get an asymptotic expansion in terms of periodic orbits. Since all orbits of the classical isotropic oscillator are periodic, the asymptotics may be stated without reference to periodic orbits.

### 1.1. Statement of results

The first result is an exact formula for the Wigner distributions (7) of the eigenspace projections for the isotropic Harmonic oscillator in terms of Laguerre functions (see Appendix 5.2 and [T] for background on Laguerre functions).

###### Proposition 1.2.

The Wigner distribution of Definition 1.1 is given by,

 Wℏ,EN(ℏ)(x,ξ)=(−1)N(πℏ)de−2H/ℏL(d−1)N(4H/ℏ),H=H(x,ξ)=|x|2+|ξ|22, (10)

where is the associated Laguerre polynomial of degree and type .

In dimension the formula was proved in [O, JZ]. The authors did not find the formula in the literature for , but essentially the same formula is proved in [T, Theorem 1.3.5] for matrix elements of the Heisenberg group; by [F, (1.89)] the latter are related to the Wigner distributions by the change of variables and a multiplication by . We follow in Section 3.2 the technique in [T] to give a brief derivation of Proposition 1.2 as a corollary of Proposition 1.9 below. A proof that is a radial function on using its symmetry properties and no special calculations is given in Section 2.3.

The second result is a weak* limit result for normalized Wigner distributions.

###### Proposition 1.3.

Let be a semi-classical symbol of order zero and let be its Weyl quantization. Then, as , with ,

where is Liouville measure on and .

Thus, in the sense of weak* convergence. But this limit is due to the oscillations inside the energy ball; the pointwise asymptotics are far more complicated. The proof is given in Section 4.7 as a corollary of the pointwise asymptotics of the Wigner function (or, more precisely, its proof).

### 1.2. Interface asymptotics for Wigner distributions of individual eigenspace projections

Our first main result gives the scaling asymptotics for the Wigner function of the projection onto the -eigenspace of when lies in an neighborhood of the energy surface

###### Theorem 1.4.

Fix . Assume and let as in (8). Suppose satisfies

 H(x,ξ)=E+u(ℏ2E)2/3,u∈R,H(x,ξ)=∥x∥2+∥ξ∥22 (11)

with . Then,

 (12)

When we also have the estimate

 Wℏ,EN(ℏ)(x,ξ)=2(2πℏ)d(ℏ2E)1/3Ai(u/E)(1+O((1+|u|)3/2|u|ℏ2/3)).

Here, is the Airy function. The Airy scaling of is illustrated in Figure 2. We give two proofs of Theorem 1.4. The first is based on special functions; the asymptotics are obtained from Proposition 1.2 and results on Laguerre functions due to Olver [O], Franzen-Wong [FW] (see Proposition 4.1). The second proof is self-contained and uses semi-classical parametrices for (see Section 4.1).

The assumption (11) may be stated more invariantly that lies in the tube of radius around defined by the gradient flow of with respect to the Euclidean metric on . The asymptotics are illustrated in Figure 2. Due to the behavior of the Airy function , these formulae show that in the semi-classical limit , , concentrates on the energy surface surface , is oscillatory inside the energy ball and is exponentially decaying outside the ball. There is a more complicated but more complete result which requires some notation and defined in Section 4.1; we defer the statement since it is too lengthy to define all the notation here.

### 1.3. Interior Bessel and Trigonometric asymptotics

In addition to the Airy asymptotics in an -tube around , classical asymptotics of Laguerre functions (see [AT15, AT15b] for recent results and references) show that when (i.e. is a dsitance at most form the origin in phase space), exhibits Bessel asymptotics and trigonometric asymptotics farther into the interior of (i.e. when ). We record the precise asymptotic statements in Theorem 1.5 below. To state it we define for ,

 A(t)=12[√t−t2+sin−1√t],t∈[0,1].

For the is replaced by and the by . The function appears in the uniform asymptotic expansions of Laguerre polynomials (see [FW, (2.7)]). Also, let be the Bessel function (of the first kind) of index .

###### Theorem 1.5.

Fix and suppose For each write

 HE:=H(x,ξ)E=∥x∥2+∥ξ∥22E,νE:=4Eℏ.

Fix Uniformly over , there is an asymptotic expansion,

 Wℏ,EN(ℏ)(x,ξ)=2(2πℏ)d[Jd−1(νEA(HE))A(HE)d−1α0(HE)+O(ν−1E∣∣∣Jd(νEA(HE))A(HE)d∣∣∣)]. (13)

In particular, uniformly over in a compact subset of we find

 Wℏ,EN(ℏ)(x,ξ)=(2πℏ)−d+1/2PH,Ecos(ξℏ,E,H)+O(ℏ−d+3/2), (14)

where we’ve set

 ξℏ,E,H=−π4−2Hℏ(H−1E−1)1/2+2Eℏcos−1(H1/2E)

and

 PE,H:=(πE1/2(H−1E−1)1/4(HE)d/2)−1.

We prove Theorem 1.5 in Section 4.4. We also give an independent semiclassical proof of (14) in Section 4.5. Note that when is near Hence, although Theorem 1.5 does not strictly apply when is a distance form the origin, the relation (13) suggests that if the distance from to the origin is on the order , the Wigner function behaves like a normalized version of up to constant factors (this fact can also be see by directly setting in Proposition 1.2). The small ball behavior of is investigated in Sections 1.4 and 1.6 below (see also Figure 3).

### 1.4. Small ball integrals

The interior Bessel asymptotics do not encompass the behavior of in shrinking balls around . In that case, we have,

###### Proposition 1.6.

For sufficiently small and for any ,

 (15)

where is a smooth radial cut-off that is identically on the ball of radius and is identically outside the ball of radius

### 1.5. Exterior asymptotics

If , then concentrates on and is exponentially decaying in the complement . The precise statement is,

###### Proposition 1.7.

Suppose that and let . Then, there exists so that

 |Wℏ,EN(ℏ)(x,ξ)|≤C1ℏ−d+12e−2Eℏ[√H2E−HE−cosh−1√HE].

Moreover, as , there exists so that

 |Wℏ,EN(ℏ)(x,ξ)|≤C2ℏ−d+12e−2H(x,ξ)ℏ.

This result is proved in Section 4.3 as a simple consequence of classical asymptotics of Laguerre functions reviewed in Section 4.2.

### 1.6. Supremum at ρ=0

The origin is the point at which has its global maximum (see Figure 3). We have,

 Wℏ,EN(ℏ)(0,0)=(−1)N(πℏ)dLd−1N(0)=(−1)N(πℏ)dΓ(N+d)Γ(N+1)Γ(d)≃(−1)NπdCdℏ−dNd−1. (16)

The last statement follows from the explicit formula (see e.g. [T, (1.1.39)]).

The fact that is a local maximum can be seen from the eigenvalue equation (30) below; since it is radial function, a negative value of its Laplacian at is equivalent to its Hessian being negative definite. The fact that it is a global maximum follows from Proposition 1.2 and the known properties of Laguerre polynomials. We briefly discuss why the maximum is so large in Section 2.7.

On the complement of the ball , the Wigner distribution is much smaller than at its maximum. The following is proved by combining the estimates of Theorem 1.4 , Theorem 1.5 and Proposition 1.7.

###### Proposition 1.8.

For any ,

 sup(x,ξ):H(x,ξ)≥ε|Wℏ,EN(ℏ)(x,ξ)|≤Cℏ−d+13.

The supremum in this region is achieved in at satisfying (11) where is the global maximum of .

### 1.7. Outline of proofs

The main object in the study of the eigenspace projections of Definition 1.1 is the Wigner function of the propagator. By ‘propagator’ is meant the solution operator of the Cauchy problem for the Schrödinger equation

 iℏ∂∂tu=ˆHℏu,

and its Schwartz kernel is thus given by

 Uh(t,x,y)=e−ihtˆHh(x,y). (17)

The special feature of the isotropic oscillator is that we may express individual eigenspace projections as Fourier components of the Propagator . On the level of Schwartz kernels,

 Πh,EN(ℏ)(x,y)=∫π−πUℏ(t−iε,x,y)e−ih(t−iε)EN(ℏ)dt2π,

where we recall that The Wigner transform is linear on the level of Schwartz kernels and hence

 Wℏ,EN(ℏ)=∫π−πUℏ(t−iε,x,y)e−ih(t−iε)EN(ℏ)dt2π, (18)

where is the Wigner function of the propagator.

###### Proposition 1.9.

The Wigner distribution of is given by

 Uℏ(t,x,ξ)=1(2πℏ)dcos(t/2)−dexp(−2iHℏsin(t/2)cos(t/2)),H=H(x,ξ):=||x||2+||ξ||22

Proposition 1.9 is the basic tool underlying the scaling asymptotics of Wigner functions of eigenspace projections. The Wigner distribution is well-defined as a distribution (see Section 3.3) but not as a locally function. is a distribution in for fixed with Dirac mass singularities at . Despite the singularities it has scaling asymptotics in different phase space regimes. Combined with (18) it gives scaling asymptotics of

### 1.8. Prior and related results

The literature on Wigner distributions and on the isotropic Harmonic Oscillator is vast. In dimension one, some of the results of this article are proved in [JZ], and others are stated and to some extent proved in [Ber91], along with more detailed asymptotics near interfaces. Somewhat surprisingly, the results on Wigner distributions of spectral projections have not previously been generalized to higher dimensions , even for the model case (2) of the isotropic harmonic oscillator. However, there do exist known relations between Wigner functions and Laguerre functions [T, F] and we explain how to use known asymptotics of Laguerre functions (from [FW]) to obtain interface asymptotics results on Wigner functions. But our ultimate aim is to generalize the results to more general Schrödinger operators, and therefore we also give semi-classical analysis arguments .

It has long been known in dimension [Ber] that Wigner distribution of spectral projections (corresponding to individual eigenfunctions) of quite general Schrödinger operators exhibit Airy scaling asymptotics in shells around the corresponding energy curve. In dimension , Berry [Ber] obtained the expression for the Wigner function of an eigenfunction of any Hamiltonian,

 Wℏ(q,p)=CA(q,p)1/6Ai(−[3A(q,p)2ℏ]2/3)πℏ23[Iq(2)Ip(1)−Ip(2)I1(q)]12.

Here, is the area between the chord and the arc of the boundary between . Of course, in dimension one the Schrödinger operator is an ODE and the techniques available there do not generalize to higher dimensions. See also [O] for many results in the one-dimensional case.

In [HZZ16], the authors studied the configuration space spectral projections kernels for a fixed energy level for the isotropic Harmonic oscillator in . The allowed region in configuration space is the projection where is the natural projection. The interface is the caustic , i.e. the projection of the points of where . Theorem 1.1 of [HZZ16] gives Airy scaling asymptotics of for in an neighborhood of . Theorem 1.4 of the present article is a lift of this result to phase space.

In [ZZ16], Zelditch-Zhou studied scaling asymptotics around of the so-called Husimi distribution of rather than the Wigner distribution. The Husimi distribution is the covariant symbol (value on the diagonal) of the conjugate of the eigenspace projection by the Bargmann transform to the holomorphic Bargmann-Fock space. The Husimi distribution is the density obtained by holomorphic quantization of the eigenspace projection and is a kind of Gaussian coherent state centered on with Gaussian decay in a tube of radius in the normal directions. Thus, the phase space interface scaling around is very different in the two representations. The exact relation between the two phase space distributions was given by Cahill-Glauber [CG69I, (7.32)] (see also [CG69II, (6.32)] and [Oz, (5.32)]), who showed that the Bargmann-Fock (Husimi distribution) is a Gaussian convolution of the Wigner distribution.

At the beginning of this article, it is stated that the isotropic quantum oscillator (2) is the unique Schrödinger operator with its exceptionally high eigenvalue multiplicities (degeneracies). Suppose that on is a Schrödinger operator with a potential of quadratic growth, and suppose that it has the same eigenvalue multiplicities as the un-perturbed isotropic oscillator (2). Then it is a ‘maximally degenerate’ Schrödinger operator in the sense of [Z96]. As proved in that article, it must have periodic classical Hamiltonian flow and then, by results of Weinstein, Widom and Guillemin on compact manifolds (see [G78] for background) and of Chazaraint for Schrödinger operators on , [Ch80], the eigenvalues concentrate in ‘clusters’ around the arithmetic progression ; a recent article studying perturbations of (2) is [GU12]. Maximal degeneracy means that every cluster has just one distinct eigenvalue. We are not aware of a proof that there exist no ‘maximally degenerate’ perturbations of the semi-classial Schrödinger operator (2) but it seems that the techniques of [Z96] might yield the result that (2) is the unique maximally degenerate analytic Schrödinger operator on with a quadratic growth potential. We plan to consider this problem elsewhere and do not discuss it further here.

## 2. Background

It is well-known that the spectrum of in the isotropic case consists of the eigenvalues (3), and one has the spectral decomposition

 L2(Rd,dx)=⨁N∈NVℏ,EN(ℏ),ˆHℏ|Vh,EN(ℏ)=ℏ(N+d2) (19)

An orthonormal basis of eigenfunctions of (2) is given by scaled Hermite functions,

 φα,h(x)=h−d/4pα(x⋅h−1/2)e−x2/2h, (20)

where is a dimensional multi-index and is the product of the hermite polynomials (of degree ) in one variable. The eigenvalue of is given by

 Hhφα,h=h(|α|+d/2)φα,h. (21)

The multiplicity of the eigenvalue is the partition function of , i.e. the number of with a fixed value of . The spectral projections (6) are given by

 Πℏ,EN(ℏ)(x,y):=∑|α|=Nφα,hN(x)φα,hN(y). (22)

### 2.1. Semi-classical scaling

To clarify the roles of and to facilitate comparisons to references such as [F] which do not employ the semi-classical scaling (22), we record the following notation throughout the article.

• For (2) with , the eigenvalues are with corresponding eigenspaces (see (3)). We denote by the orthogonal projections (again, with

 Πℏ,EN(ℏ)(x,y)=ℏ−d/2ΠN(ℏ−1/2x,ℏ−1/2y).
• The Wigner distribution of is denoted by,

 WN(x,ξ):=∫RdΠN(x+v2,x−v2)eivpdv(2π)d. (23)
• The Wigner distribution of is

 Wℏ,EN(ℏ)(x,ξ)=∫RdΠℏ,EN(ℏ)(x+v2,x−v2)e−iℏv⋅ξdv(2πℏ)d.

It is related to that of by

 Wℏ,EN(ℏ)(x,ξ)=ℏ−dWN(x/ℏ1/2,ξ/ℏ1/2). (24)

### 2.2. Weyl pseudo-differential operators, metaplectic covariance and Wigner distributions

A semi-classical Weyl pseudo-differential operator is defined by the formula,

 Opwh(a)u(x)=∫Rd∫Rdaℏ(12(x+y),ξ)eiℏ⟨x−y,ξ⟩u(y)dydξ.

We refer to [F, Zw] for background.

A key property of Weyl quantization is the so-called metaplectic covariance. Let denote the symplectic group and let denote the metaplectic representation of its double cover. Then, where denotes translation by .

In particular, acts on by translation of functions, using the identification defined by the standard complex structure . is a subgroup of the symplectic group and the complete symbol of (2) is invariant, so by metaplectic covariance, commutes with the metaplectic represenation of

### 2.3. Proof that the Wigner distribution is radial

The fact that the Wigner distribution (7) is radial can be seen from the symmetry of the isotropic oscillator without calculations. As mentioned above, the classical Hamiltonian is invariant under the standard action of the unitary group on . Since is a subgroup of the symplectic group, the action is quantized by the metaplectic representation on and commutes with and therefore with . By metaplectic covariance, the Wigner function is invariant under the lift of the action to , so that, all ,

 WU∗Πℏ,EN(ℏ)U(x,ξ)=Wℏ,EN(ℏ)(U(x,ξ)).

This gives a simple explanation of the fact that is a function of . In particular, the subgroup generated by the isotropic Harmonic Oscillator corresponds to the periodic Hamiltonian flow of the classical oscillator, and is invariant under this group.

### 2.4. Trace and Hilbert-Schmidt properties

It follows that . By using the identity

 ⟨Opw(a)f,f⟩=∫T∗Rda(x,ξ)Wf,f(x,ξ)dxdξ,

of [F, Proposition 2.5] for orthonormal basis elements of and summing over , one obtains the (well-known) identity,

 TrOpwh(a)Πℏ,EN(ℏ)=∫T∗Rda(x,ξ)Wℏ,EN(ℏ)(x,ξ)dxdξ. (25)

Further, the Wigner transform (1) taking kernels to Wigner functions is an isometry from Hilbert-Schmidt kernels on to their Wigner distributions on [F]. From (25) and this isometry, it is straightforward to check that,

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(i)∫T∗RdWℏ,EN(ℏ)(x,ξ)dxdξ=TrΠℏ,EN(ℏ)=dimVℏ,EN(ℏ)=(N+d−1d−1)(ii)∫T∗Rd∣∣Wℏ,EN(ℏ)(x,ξ)∣∣2dxdξ=TrΠ2ℏ,EN(ℏ)=dimVℏ,EN(ℏ)=(N+d−1d−1)(iii)∫T∗RdWℏ,EN(ℏ)(x,ξ)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Wℏ,EM(ℏ)(x,ξ)dxdξ=TrΠℏ,EN(ℏ)Πℏ,EM(ℏ)=0,forM≠N., (26)

In these equations, and is the composition function of (i.e. the number of ways to write as an ordered us of non-negative integers). Thus, the sequence,

 {1√dimVℏ,EN(ℏ)Wℏ,EN(ℏ)}∞N=1⊂L2(R2n)

is orthonormal.

In comparing (25), (26)(i)-(ii) one should keep in mind that is rapidly oscillating in with slowly decaying tails in the interior of , with a large ‘bump’ near and with maximum given by Proposition 1.8. Integrals (e.g. of ) against involve a lot of cancellation due to the oscillations. The square integrals in (ii) enhance the ‘bump’ and decrease the tails and of course are positive.

### 2.5. The Wigner transform

For any Schwartz kernel one may define the Wigner distribution of by

 WK,ℏ(x,ξ):=(2πℏ)−d∫RdKℏ(x+v2,x−v2)e−iℏvξdv(2πh)d, (27)

As mentioned in Section 2.4, the map from defines a unitary operator

 Wℏ:L2(Rd×Rd)→L2(T∗Rd),

which we will call the ‘Wigner transform’.

The unitary group acts on by conjugation,. where we identify with the associated Hilbert-Schmidt operator. Metaplectic covariance implies that,

 WℏU(g)=TgWℏ.

### 2.6. Eigenvalue equation

We employ the well-known semi-classical Moyal product on symbols of Weyl pseudo-differential operators is defined by [F, Sh]

 Opwℏ(a)∘Opwℏ(b)=Opwℏ(a∗ℏb).

The Wigner distribution is the Weyl symbol of the eigenspace projection .

The eigenvalue equation,

 ^HℏΠℏ,EN(ℏ)=EN(ℏ)Πℏ,EN(ℏ)

gives

 H∗ℏWℏ,EN(ℏ)=Wℏ,EN(ℏ)∗ℏH=EN(ℏ)Wℏ,EN(ℏ). (28)

It is proved in Section 2.3 that is a radial function on , the explicit formula being given in Proposition 1.2. Up to a scalar fixed by the trace identities, is the only radial solution of these equations. Indeed, the equations (28) are equivalent to,

 12(∑j(xj+iℏ2∂ξj)2+(ξj+iℏ2∂xj)2)Wℏ,EN(ℏ)=EN(ℏ)Wℏ,EN(ℏ). (29)

This equation is related to that of [T, (1.3.14)]. The imaginary part vanishes, , since is radial. We then simplify (29) to

 (−ℏ28(Δξ+Δx)+H(x,ξ))Wℏ,EN(ℏ)=EN(ℏ)Wℏ,EN(ℏ). (30)

### 2.7. The special point (x,ξ)=(0,0)

In this section we briefly return to the observation in Section 1.6 that