IINTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS
STEVE ZELDITCH
Abstract.
This is a survey of results obtained jointly with Boris Hanin and Peng Zhou on interfaces inspectral asymptotics, both for Schr¨odinger operators on L ( R d ) and for Toeplitz Hamiltonians acting onholomorphic sections of ample line bundles L → M over K¨ahler manifolds ( M, ω ). By an interface is meanta hypersurface, either in physical space R d or in phase space, separating an allowed region where spectralasymptotics are standard and a forbidden region where they are non-standard. The main question is to givethe detailed transition between the two types of asymptotics across the hypersurface (i.e. interface). In thereal Schr¨odinger setting, the asymptotics are of Airy type; in the K¨ahler setting they are of Erf (Gaussianerror function) type.A principal purpose of this survey is to compare the results in the two settings. Each is apparentlyuniversal in its setting. This is now established for Toeplitz operators, but in the Schr¨odinger setting it isonly established for the simplest model operator, the isotropic harmonic oscillator. It is explained that thelatter result is most comparable to the behavior of the canonical degree operator on the Bargmann-Fockspace of a line bundle, a new construction introduced in these notes. Introduction
This is a mainly expository article on interfaces in spectral asymptotics. Interfaces are studied in manyfields of mathematics and physics but seem to be a novel area of spectral asymptotics. Spectral asymptoticsrefers to the behavior of spectral projections and nodal sets for a quantum Hamiltonian ˆ H (cid:126) , which might be aSchr¨odinger operator on L ( R d ) or on a Riemannian manifold ( M, g ), with or without boundary, or a ToeplitzHamiltonian acting on holomorphic sections H ( M, L k ) of line bundles over a K¨ahler manifold. Interfaceasymptotics refers to the change in behavior of the spectral projections or nodal sets as a hypersurface iscrossed, either in physical space (configuration space) or in phase space. Interfaces exist in diverse settingsand indeed the purpose of this article is to compare interface behavior in different settings and to considerpossible future settings that have yet to be explored.What is meant by an ‘interface’ in the sense of this article? The general idea is that there is a hypersurfacein the phase space separating two regions in which the asymptotic behavior of a spectral projections kernel hasdifferent types of behavior: In the first, that we will term the ‘allowed’ region, the asymptotics are constantand, after normalization, equal 1, so that one has a plateau over the region; in the second ‘forbidden’ regionthe asymptotics are rapidly decaying, so that one has a rather flat 0 region. The interface is the shape of thegraph of the spectral kernel connecting 1 and 0 in a thin region separating the allowed and forbidden region.One expects that when scaled properly, the limit shape is universal. More precisely, universality holds ineach type of model (e.g. Schr¨odinger or K¨ahler ) but is model-dependent: one expects ‘Airy interfaces’ inthe Schr¨odinger setting and Erf interfaces in the K¨ahler setting. The separation into different regions for thespectral projections kernel often coincides with the separation of other spectral behavior, such as nodal setsof the eigenfunctions.The terminology (classically) ‘allowed’ and (classically) ‘forbidden’ is standard in quantum mechanics forregions inside, resp. outside, of an energy surface in phase space, or more commonly, the projection of theseregions to configuration space. This will indeed be the meaning of ‘interface’ for most of this article. We willdescribe results of B. Hanin, P. Zhou and the author [HZZ15, HZZ16] on the different behavior of nodal setsof Schr¨odinger eigenfunctions in allowed resp. forbidden regions for the simplest Schr¨odinger Hamiltonianˆ H (cid:126) , namely the isotropic Harmonic oscillator on R d . We then consider phase space interfaces of Wignerdistributions for the same model, following [HZ19, HZ19b]. We then turn to phase space interfaces in the Date : August 11, 2020.Research partially supported by NSF grant DMS-1810747. a r X i v : . [ m a t h . SP ] A ug STEVE ZELDITCH
K¨ahler (complex holomorphic) setting, and discuss results of Pokorny-Singer [PS], Ross-Singer [RS], P. Zhouand the author [ZZ16, ZZ17] on interfaces for partial Bergman kernel asymptotics. In Section 8 we explainthat the exact analogue of the results on Wigner distributions for the isotropic harmonic oscillator in thecomplex setting is a series of results on interfaces for disc bundles in the Bargmann-Fock space of a linebundle. This Bargmann-Fock space and the interface results constitute the new results of the article.Roughly speaking, interfaces in spectral asymptotics involve two types of localization: (i) spectral, i.e.quantum, localization where the eigenvalues are constrained to lie in an interval I , (ii) classical, i.e. phasespace, localization where a phase space point is constrained to lie in an open set U of phase space. Ithas long been understood that spectral localization E j ( (cid:126) ) ∈ I implies phase space localization in the sensethat quantum objects decay in the complement of the allowed region H − ( I ). But the study of interfacesis devoted to the precise behavior of quantum objects as one crosses the interface between allowed andforbidden regions, and more generally, considers all possible combinations of spectral localization E j ( (cid:126) ) ∈ I and phase space localization ζ ∈ U , where U may have any position relative to H − ( I ).Often, the interface corresponds to a sharp cutoff in a spectral parameter and signals something dis-continuous. In fact, the earliest studies of interface asymptotics are classical analysis studies of Bernsteinpolynomials of discontinuous functions with jump discontinuities [Ch, L, Lev, Mir, O]. These studies wereintended to be analogues of Gibbs phenomena for Fourier series of discontinuous functions, which have beengeneralized to wave equations on Riemannian manifolds in [PT97].In this article we review the following results on interface asymptotics: • Interface behavior for spectral projections and for nodal sets of random eigenfunctions of energy E N ( (cid:126) ) = (cid:126) ( N + d ) = E of the isotropic harmonic oscillator on R d across the caustic set in physicalspace, where the potential V ( x ) = | x | / E . • Interface behavior for Wigner distributions of the same eigenspace projections, and more generallyfor various types of Wigner-Weyl sums across an energy surface in phase space; • Interface behavior for the holomorphic analogues of such Wigner distributions, namely for partialBergman kernels for general Berezin-Toeplitz Hamiltonians on general K¨ahler phase spaces. • Interface results for partial Bergman kernels corresponding to the canonical S action on the totalspace L ∗ of the dual line bundle of an ample line bundle L → M over a K¨ahler manifold.In the case of Schr¨odinger operators, the results are only proved in the special case of the isotropicharmonic oscillator. It is plausible that some of the results should be universal among Schr¨odinger operators,but at the present time the generalizations have not been formulated or proved. See Section 9.1 for furtherproblems. Among other gaps in the theory, Wigner distributions per se are only defined when the Riemannianmanifold is R d and are closely connected to the representation theory of the Heisenberg and metaplecticgroups. Wigner distributions of eigenfunctions are special types of “microlocal lifts” of eigenfunctions;there is no generally accepted canonical microlocal lift on a general Riemannian manifold. Despite therestrictive setting, Wigner distributions are important in mathematical physics, in particular in quantumoptics. The results in the complex holomorphic (K¨ahler ) setting are much more complete, due to the factthat the theory of Bergman kernels is technically simpler and more complete than the corresponding theoryof Wigner distributions for Schr¨odinger operators. The results are proved for any Toeplitz Hamiltonian onany projective K¨ahler manifold. In fact, the exact analogue of the Wigner result is proved in Section 8, wherea new construction is introduced in this article: the Bargmann-Fock space of a holomorphic line bundle. Itis a Gaussian space of holomorphic functions on the total space L ∗ of the dual of a holomorphic Hermitianline bundle L → M over a K¨ahler manifold. This total space carries a natural S action and this S actionplays the role of the propagator of the isotropic Harmonic Oscillator. Thus, the interfaces are the boundariesof the co-disc bundles D ∗ E ⊂ L ∗ of different energy levels (i.e. radii). The interface results in Section 8 are a‘new result’ of this article, but the proofs are similar to, and simpler than, those in [ZZ17, ZZ18] .This survey is organized as follows: S always denotes the unit circle NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 3 (1) In Section 2, we review the basic linear models: the Harmonic oscillator in the Schr¨odinger repre-sentation on L ( R d ) and in the Bargmann-Fock (holomorphic) representation on entire holomorphicfunctions on C d . We also present a list of analogies between the real Schr¨odinger setting and the com-plex holomorphic quantization. Section 3 is devoted to the Bergman kernel on Bargmann-Fock space,and the Bargmann-Fock representations of the Heisenberg and Symplectic groups on Bargmann-Fockspace.(2) In Section 4, we review the interface results in physical space for spectral projections for the isotropicHarmonic Oscillator. These imply interface results for nodal sets of random eigenfunctions in a fixedeigenspace.(3) In Section 5, we change the setting to phase space T ∗ R d and review the interface results in physicalspace for Wigner distributions of spectral projections for the isotropic Harmonic Oscillator.(4) In Section 6, we switch to the complex holomorphic setting and review interface results for partialBergman kernels on general compact K¨ahler manifolds.(5) In Section 7 we specialize to the isotropic harmonic oscillator on the standard Bargmann-Fock spaceand describe its interfaces;(6) In Section 8 we introduce a new model: the Bargmann-Fock space of a holomorphic line bundle. Wethen consider interfaces with respect to a natural S action on this space, generalalizing the previousresult on the Bargmann-Fock isotropic Harmonic oscillator.(7) In Section 9.1 we list some further problems on interfaces.(8) In Section 10 we give some background to the holomorphic setting.1.1. Results surveyed in this article.
The articles surveyed in this article are the following:
References [HZZ15] Boris Hanin, Steve Zelditch, Peng Zhou Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator,International Mathematics Research Notices, Vol. 2015, No. 13, pp. 4813-4839, (2015) (arXiv:1310.4532)[HZZ16] Boris Hanin, Steve Zelditch and Peng Zhou, Scaling of harmonic oscillator eigenfunctions and their nodal sets aroundthe caustic. Comm. Math. Phys. 350 (2017), no. 3, 1147-1183 (arXiv:1602.06848).[HZ19] B. Hanin and S. Zelditch, Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator,arXiv:1901.06438.[HZ19b] B. Hanin and S. Zelditch, Interface Asymptotics of Wigner-Weyl Distributions for the Harmonic Oscillator,arXiv:1903.12524.[ZZ16] S. Zelditch and P. Zhou, Interface asymptotics of partial Bergman kernels on S -symmetric Kaehler manifolds, toappear in J. Symp. Geom. (arXiv:1604.06655).[ZZ17] S. Zelditch and P. Zhou, Central Limit theorem for spectral Partial Bergman kernels, to appear in Geom. Topl.arXiv:1708.09267.[ZZ18] S. Zelditch and P. Zhou, Interface asymptotics of Partial Bergman kernels around a critical level (arXiv:1805.01804).[ZZ18b] S. Zelditch and P. Zhou, Pointwise Weyl law for Partial Bergman kernels, Algebraic and Analytic Microlocal Analysis pp. 589- 634. M. Hitrik, D. Tamarkin, B. Tsygan, S. Zelditch (eds). Springer Proceedings in Mathematics and Statistics,Springer-Verlag (2018). The basic linear models
As mentioned above, our aim in this survey is not only to describe interface results in various settingsbut to compare the results in the real Schr¨odinger setting and the complex holomorphic Bargmann-Fock orBerezin-Toeplitz setting. The real setting is self-explanatory to mathematical physicists but the complexholomorphic setting is probably less familiar. In this section, we give some background on the basic linearmodels (isotropic Harmonic Oscillator in both settings) to make the relations between the real and complexsettings more familiar. We then give a list of analogies between the two settings. In addition, we present alist of open problems on interfaces to amplify the scope of spectral interface problems. It would be laborious
STEVE ZELDITCH to present all of the background for the geometric setting before getting to the main results and phenomena,so we have put that background into an Appendix Section 10.A preliminary remark: Since the early days of quantum mechanics, it was understood that there are manyequivalent representations (or ‘pictures’) of quantum mechanics. In the case of R d they correspond to differentbut unitarily equivalent representations of the Heisenberg and metaplectic groups (see [F] for background).The most common are the Schr¨odinger representation on L ( R d ) and the Bargmann-Fock representation on H ( C d , e −| Z | dL ( Z )), the Bargmann-Fock space of entire holomorphic functions on C d which are in L withrespect to Gaussian measure; here dL is Lebesgue measure. One refers to R d as ‘configuration space’ or‘physical space’ and to T ∗ R d as phase space. Of course, T ∗ R d (cid:39) C d , so that Bargmann-Fock space employsa complex structure on phase space. A natural unitary intertwining operator is the Bargmann transform(see (26) below). We refer to [F] and to [HSj16] for background on Bargmann-Fock space and metaplecticoperators.The first item is to give background on the isotropic Harmonic oscillator in both the Schr¨odinger repre-sentation and the Bargmann-Fock representation.2.1. Schr¨odinger representation of the isotropic Harmonic oscillator.
The Schr¨odinger representa-tion of quantum mechanics is too familiar to need a detailed review here. The isotropic Harmonic Oscillatoron L ( R d , dx ). is the operator, (cid:98) H (cid:126) = d (cid:88) j =1 (cid:32) − (cid:126) ∂ ∂x j + x j (cid:33) . (1)It has a discerete spectrum of eigenvalues E N ( (cid:126) ) = (cid:126) ( N + d/ , ( N = 0 , , , . . . ) (2)with multiplicities given by the composition function p ( N, d ) of N and d (i.e. the number of ways to write N as an ordered sum of d non-negative integers). That is, the eigenspaces V (cid:126) ,E N ( (cid:126) ) := { ψ ∈ L ( R d ) : (cid:98) H (cid:126) ψ = E N ( (cid:126) ) ψ } , (3)have dimensions given by dim V (cid:126) N ,E = p ( N, d ) = 1( d − N d − (1 + O ( N − )) . (4)When E N ( (cid:126) ) = E we also write (cid:126) = (cid:126) N ( E ) := EN + d , (5)An orthonormal basis of its eigenfunctions is given by the product Hermite functions, φ α,h ( x ) = h − d/ p α (cid:16) x · h − / (cid:17) e − x / h , (6)where α = ( α , . . . , α d ) ≥ (0 , . . . ,
0) is a d − dimensional multi-index and p α ( x ) is the product (cid:81) dj =1 p α j ( x j )of the hermite polynomials p k (of degree k ) in one variable.The eigenspace projections are the orthogonal projectionsΠ (cid:126) ,E N ( (cid:126) ) : L ( R d ) → V (cid:126) ,E N ( (cid:126) ) . (7)When E N ( (cid:126) ) = E (5), their Schwartz kernels are given in terms of an orthonormal basis by,Π h N ,E ( x, y ) = (cid:88) | α | = N φ α,h N ( x ) φ α,h N ( y ) . (8)The high multiplicities are due to the U ( d )-invariance of the isotropic Harmonic Oscillator. Due to extremedegeneracy of the spectrum of (1) when d ≥
2, the eigenspace projections have very special semi-classicalasymptotic properties, reflecting the periodicity of the classical Hamiltonian flow and of the Schr¨odingerpropagator exp[ − it (cid:126) (cid:98) H (cid:126) ]. In particular, the eigenspace projections (7) are semi-classical Fourier integraloperators (see e.g. [GU12, GUW, HZ19]. We exploit this very rare property to obtain scaling asymptoticsacross the caustic. This explains why the results to date are only available for isotropic oscillators. Forgeneral Harmonic Oscillators with incommensurate frequencies the eigenvalues have multiplicity one and the NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 5 eigenspace projections are of a very different type. For general Schr¨odinger operator, one would need to takeappropriate combinations of eigenspace projections with eigenvalues in an interval.As with any 1-parameter metaplectic unitary group [F, HSj16], one has an explicit Mehler formula for theSchwartz kernel U h ( t, x, y ) of the propagator, e − ih tH h . The Mehler formula [F] reads U h ( t, x, y ) = e − ih tH h ( x, y ) = 1(2 πih sin t ) d/ exp (cid:32) ih (cid:32) | x | + | y | t sin t − x · y sin t (cid:33)(cid:33) , (9)where t ∈ R and x, y ∈ R d . The right hand side is singular at t = 0 . It is well-defined as a distribution,however, with t understood as t − i
0. Indeed, since H h has a positive spectrum the propagator U h isholomorphic in the lower half-plane and U h ( t, x, y ) is the boundary value of a holomorphic function in { Im t < } .One may express the N th spectral projection as a Fourier coefficient of the propagator. It is somewhatsimpler to work with the number operator N , i.e. the Schr¨odinger operator with the same eigenfunctionsas H h and eigenvalues h | α | . If we replace U h ( t ) by e − ith N then the spectral projections Π h,E are simply theFourier coefficients of e − ith N . In [HZZ15, HZZ16] it is shown thatΠ h N ,E ( x, y ) = (cid:90) π − π U h ( t − i(cid:15), x, y ) e ih ( t − i(cid:15) ) E dt π . (10)The integral is independent of (cid:15) . Combining (10) with the Mehler formula (9), one has an explicit integralrepresentation of (8).2.1.1. Wigner distributions.
For any Schwartz kernel K (cid:126) ∈ L ( R d × R d ) one may define the Wigner distri-bution of K (cid:126) by W K, (cid:126) ( x, ξ ) := (cid:90) R d K (cid:126) (cid:16) x + v , x − v (cid:17) e − i (cid:126) vξ dv (2 πh ) d , (11)The map from K (cid:126) → W K, (cid:126) defines the unitary ‘Wigner transform’, W (cid:126) : L ( R d × R d ) → L ( T ∗ R d ) . The inverse Wigner transform is given by (see page 79 of [F]) f ⊗ g ∗ ( x, y ) = (cid:90) W f,g ( x + y , ξ ) e i (cid:104) x − y,ξ (cid:105) dξ. (12)Here, W f,g := W f ⊗ g ∗ is the Wigner transform of the rank one operator f ⊗ g ∗ .The unitary group U ( d ) acts on L ( R d × R d ) by conjugation, U ( g ) · K = gKg ∗ . where we identify K ( x, y ) ∈ L ( R d × R d ) with the associated Hilbert-Schmidt operator. Metaplectic covariance implies that, W (cid:126) U ( g ) = T g W (cid:126) . Definition
The Wigner distributions W (cid:126) ,E N ( (cid:126) ) ( x, p ) ∈ L ( T ∗ R d ) of the eigenspace projections Π (cid:126) ,E N ( (cid:126) ) are defined by, W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) = (cid:90) R d Π (cid:126) ,E N ( (cid:126) ) (cid:16) x + v , x − v (cid:17) e − i (cid:126) v · ξ dv (2 πh ) d . (13)When E N ( (cid:126) ) = E , the Wigner distribution W (cid:126) ,E N ( (cid:126) ) of a single eigenspace projection (13) is the ‘quan-tization’ of the energy surface of energy E and should therefore be localized at the classical energy level H ( x, ξ ) = E , where H ( x, ξ ) = (cid:80) dj =1 ( ξ j + x j ). We denote the (energy) level sets by,Σ E = { ( x, ξ ) ∈ T ∗ R d : H ( x, ξ ) := 12 ( || x || + || ξ || ) = E } . (14)The Hamiltonian flow of H is 2 π periodic, and its orbits form the complex projective space CP d − (cid:39) Σ E / ∼ where ∼ is the equivalence relation of belonging to the same Hamilton orbit. Due to this periodicity, theprojections (7) are semi-classical Fourier integral operators (see [GU12, GUW, HZZ15]). This is also true forthe Wigner distributions (13). Their properties are basically unique to the isotropic oscillator (1). Theseproperties are visible in Figure 1 depicting the graph of W (cid:126) , / . STEVE ZELDITCH
Weyl pseudo-differential operators, metaplectic covariance.
A semi-classical Weyl pseudo-differentialoperator is defined by the formula, Op wh ( a ) u ( x ) = (cid:90) R d (cid:90) R d a (cid:126) ( 12 ( x + y ) , ξ ) e i (cid:126) (cid:104) x − y,ξ (cid:105) u ( y ) dydξ. See [F, Zw] for background. By using the identity (cid:104) Op w ( a ) f, f (cid:105) = (cid:90) T ∗ R d a ( x, ξ ) W f,f ( x, ξ ) dxdξ, of [F, Proposition 2.5] for orthonormal basis elements f = φ α, (cid:126) N of V (cid:126) ,E N ( (cid:126) ) and summing over α , one obtainsthe (well-known) identity, Tr Op wh ( a )Π (cid:126) ,E N ( (cid:126) ) = (cid:90) T ∗ R d a ( x, ξ ) W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) dxdξ. (15)This formula is one of the key properties of Wigner distributions and Weyl quantization.The Wigner transform (40) taking kernels to Wigner functions is therefore an isometry from Hilbert-Schmidt kernels K ( x, y ) on R d × R d to their Wigner distributions on T ∗ R d [F]. From (15) and this isometry,it is straightforward to check that, ( i ) (cid:82) T ∗ R d W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) dxdξ = TrΠ (cid:126) ,E N ( (cid:126) ) = dim V (cid:126) ,E N ( (cid:126) ) = (cid:0) N + d − d − (cid:1) ( ii ) (cid:82) T ∗ R d (cid:12)(cid:12) W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) (cid:12)(cid:12) dxdξ = TrΠ (cid:126) ,E N ( (cid:126) ) = dim V (cid:126) ,E N ( (cid:126) ) = (cid:0) N + d − d − (cid:1) ( iii ) (cid:82) T ∗ R d W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) W (cid:126) ,E M ( (cid:126) ) ( x, ξ ) dxdξ = TrΠ (cid:126) ,E N ( (cid:126) ) Π (cid:126) ,E M ( (cid:126) ) = 0 , for M (cid:54) = N. , (16)In these equations, N = E (cid:126) − d , and (cid:0) N + d − d − (cid:1) is the composition function of ( N, d ) (i.e. the number of waysto write N as an ordered us of d non-negative integers). Thus, the sequence, { (cid:112) dim V (cid:126) ,E N ( (cid:126) ) W (cid:126) ,E N ( (cid:126) ) } ∞ N =1 ⊂ L ( R n )is orthonormal.In comparing (15), (16)(i)-(ii) one should keep in mind that W (cid:126) ,E N ( (cid:126) ) is rapidly oscillating in { H ≤ E } with slowly decaying tails in the interior of { H ≤ E } , with a large ‘bump’ near Σ E and with maximumgiven by Proposition 5.7. Integrals (e.g. of a ≡
1) against W (cid:126) ,E N ( (cid:126) ) involve a lot of cancellation due to theoscillations. The square integrals in (ii) enhance the ‘bump’ and decrease the tails and of course are positive.Another key property of Weyl quantization is its metaplectic covariance (see Section 3.2 for background).Let Sp (2 d, R ) = Sp ( T ∗ R d , σ ) denote the symplectic group and let µ ( g ) denote the metaplectic representationof its double cover. Then, µ ( g ) Op wh ( a ) µ ( g ) = Op wh ( a ◦ T g ) , where T g : T ∗ R d → T ∗ R d denotes translation by g . See [F] and Section 3.2 for background. In particular, U ∈ U ( d ) acts on L ( T ∗ R d ) by translation T U offunctions, using the identification T ∗ R d (cid:39) C d defined by the standard complex structure J . U ( d ) ⊂ Sp (2 d, R )is a subgroup of the symplectic group and the complete symbol H ( x, ξ ) of (1) is U ( d ) invariant, so bymetaplectic covariance, ˆ H (cid:126) commutes with the metaplectic represenation of U ( d ) . Bargmann-Fock space and the Toeplitz representation of the isotropic oscillator
Bargmann-Fock space of degree k on C m +1 is defined by H k = { f ( z ) holomorphic function on C m +1 , (cid:90) C m +1 | f | e − k | z | dV ol C m +1 < ∞} . The volume form on C m +1 is d Vol C m +1 = ω m +1 / ( m + 1)!, and dL ( z ) denotes Lebesgue measure. We notethat (cid:90) C m +1 e − k | z | dL ( z ) = ω m +1 (cid:90) ∞ e − kρ ρ m +1 dρ = ω m +1 (cid:90) ∞ e − kx x m dx and that (cid:90) ∞ e − kx x m dx = k − ( m +1) Γ( m + 1) = m ! k − ( m +1) , NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 7 where we use polar coordinates ( θ, ρ ) on C m +1 and where ω m +1 = | S m +1 | is the surface measure of the unitsphere in C m +1 . We normalize the Gaussian measure to have mass 1 and denote it by, d Γ m +1 ,k := k ( m +1) m ! ω m +1 e − k | z | dL ( z ) . (17)Let us fix k = 1. An orthonormal basis is given by the holomorphic monomials, { z α √ α ! }| α ∈ N m +1 , where α = ( α , . . . , α m +1 ) is a lattice point in the orthant α j ∈ N and z α = (cid:81) m +1 j =1 z α j j , α ! := (cid:81) m +1 j =1 α j !. Ifwe fix the degree | α | = (cid:80) m +1 j =1 α j we get the subspaces H N = Span { z α : | α | = N } , and one has the orthogonal decompositon, L ( C m +1 , d Γ m +1 ,k ) = ∞ (cid:77) N =0 H N . Further, there is a canonical isomorphism H N (cid:39) H ( CP m , O ( N ))between H N and the space of holomorphic sections of the N th power of the standard line bundle O (1) → CP m over projective space. The isomorphism is essentially by the liftˆ s ( z, λ ) = λ ⊗ N ( s ( z ))of a section s ∈ H ( M, O ( N )) to the total space O ( − → CP m of the line bundle dual to O (1), as anequivariant holomorphic function ˆ s of degree N . The lifted function vanishes at the zero section. If oneblows down the zero section to a point, then O ( − (cid:39) C m +1 and the lifted sections are, again, homoge-neous holomorphic polynomials of degree N . This implies that Bargmann-Fock space is, as a vector space,isomorphic to (cid:76) ∞ N =0 H ( CP m , O ( N )) . The direct sum is endowed with the Bargmann-Fock Hilbert spaceinner product and, up to a scalar, this inner product on H N is the same as the Fubini-Study inner producton H ( M, O ( N )).The degree k Bargmann-Fock Bergman kernel is the orthogonal projection from L ( C m +1 , d Γ m +1 ,k ) → H k .Its Schwartz kernel relative to Gaussian measure d Γ m +1 ,k is given byΠ k ( z, w ) = (cid:18) k π (cid:19) m +1 e kz ¯ w , i.e. for any function f ∈ L ( C m +1 , d Γ m +1 ,k ), its orthogonal projection to Bargmann-Fock space is given by(Π k f )( z ) = (cid:90) C m Π k ( z, w ) f ( w ) d Γ m +1 ,k ( dw )) . More generally, fix (
V, ω ) be a real 2 m dimensional symplectic vector space. Let J : V → V be a ω compatible linear complex structure, that is g ( v, w ) := ω ( v, Jw ) is a positive-definite bilinear form and ω ( v, w ) = ω ( Jv, Jw ). There exists a canonical identification of V ∼ = C m up to U ( m ) action, identifying ω and J . We denote the BF space for ( V, ω, J ) by H k,J .To put Bargmann-Fock space into the general framework of holomorphic line bundles over K¨ahler mani-folds, we let M = C m with coordinate z i = x i + √− y i , L → M be the trivial line bundle, let L ∼ = C m × C ,and let ω = i (cid:80) i dz i ∧ d ¯ z i be the K¨ahler form, whose potential is ϕ ( z ) = | z | := (cid:80) i | z i | . STEVE ZELDITCH
Lifting to the Heisenberg group.
It is useful to lift holomorphic sections of line bundles to equivari-ant functions on the dual L ∗ of the total space of the line bundle. Since they are equivariant with respectto the natural S action, one often restricts them to the unit circle bundle X = X h defined by a Hermitianmetric h on L ∗ .In the case of Bargmann-Fock space, X is the Heisenberg group H mred = C m × S , with group multiplication( z, θ ) ◦ ( z (cid:48) , θ (cid:48) ) = ( z + z (cid:48) , θ + θ (cid:48) + Im( z ¯ z (cid:48) )) . The circle bundle π : X → M can be trivialized as X ∼ = C m × S . The contact form on X is α = dθ + ( i/ (cid:88) j ( z j d ¯ z j − ¯ z j dz j ) . The contact form α = dθ + i (cid:80) j ( z j d ¯ z j − ¯ z j dz j ) on H mred is invariant under the left multiplication L ( z ,θ ) : ( z, θ ) (cid:55)→ ( z , θ ) ◦ ( z, θ ) = ( z + z , θ + θ + z ¯ z − ¯ z z i ) . The volume form on X = C m × S is d Vol X = ( dθ/ π ) ∧ ω m /m !.The action of the Heisenberg group is by Heisenberg translations on phase space. As seen in the nextLemma, Heisenberg translations are Euclidean translations in the C m component but also have a non-trivialchange in the angular component. The infinitesimal Heisenberg group action on X can be identified withthe contact vector field generated by a linear Hamiltonian function H : C m → R . Lemma [ZZ17, Section 3.2]
For any β ∈ C m , we define a linear Hamiltonian function on C m by H ( z ) = z ¯ β + β ¯ z. The Hamiltonian vector field on C m is ξ H = − iβ∂ z + i ¯ β∂ ¯ z , and its contact lift is ˆ ξ H = − iβ∂ z + i ¯ β∂ ¯ z −
12 ( z ¯ β + β ¯ z ) ∂ θ . The time t flow ˆ g t on X is given by left multiplication ˆ g t ( z, θ ) = ( − iβt, ◦ ( z, θ ) = ( z − iβt, θ − t Re ( β ¯ z )) . The lift of a holomorphic section of L k → C m is the CR-holomorphic function defined by,ˆ s ( z, θ ) = e k ( iθ − | z | ) s ( z ) . Indeed, the horizontal lift of ∂ ¯ z j is ∂ h ¯ z j = ∂ ¯ z j − i z j ∂ θ , and ∂ h ¯ z j ˆ s ( z, θ ) = 0.The corresponding lift of the degree k Bergman (or, Szeg¨o ) kernel ˆΠ k (ˆ z, ˆ w ) to X = C m × S is given byˆΠ k (ˆ z, ˆ w ) = (cid:18) k π (cid:19) m e k ˆ ψ (ˆ z, ˆ w ) , (18)where ˆ z = ( z, θ z ) , ˆ w = ( w, θ w ) and the phase function is ψ (ˆ z, ˆ w ) = i ( θ z − θ w ) + z ¯ w − | z | − | w | . (19)3.2. Metapletic Representation.
The Harmonic oscillator is a quadratic operator. Such operators formthe symplectic Lie algebra. Their representations on Bargmann-Fock space is a unitary representation ofthe Lie algebra. The integration this representation gives the metaplectic representation. There exist exactformulae for the Schwartz kernels of metaplectic propagators, generalizing the Mehler formula. We needthese formulae later on. A thorough treatment can be found in [F, HSj16].Let R m , ω = 2 (cid:80) mj =1 dx j ∧ dy j be a sympletic vector space. The space Sp ( m, R ) consists of lineartransformation S : R m → R m , such that S ∗ ω = ω . In coordinates, we write (cid:18) x (cid:48) y (cid:48) (cid:19) = S (cid:18) xy (cid:19) = (cid:18) A BC D (cid:19) (cid:18) xy (cid:19) . NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 9
The semi-direct product of the symplectic group and Heisenberg group (sometimes called the Jacobi group)thus consists of linear transformations fixing 0 together with Heisenberg translations moving 0 to any point.In complex coordinates z i = x i + iy i , we have then (cid:18) z (cid:48) ¯ z (cid:48) (cid:19) = (cid:18) P Q ¯ Q ¯ P (cid:19) (cid:18) z ¯ z (cid:19) =: A (cid:18) z ¯ z (cid:19) , where (cid:18) P Q ¯ Q ¯ P (cid:19) = W − (cid:18) A BC D (cid:19) W , W = 1 √ (cid:18) I I − iI iI (cid:19) . (20)The choice of normalization of W is such that W − = W ∗ .Thus, P = 12 ( A + D + i ( C − B )) . We say such
A ∈ Sp c ( m, R ) ⊂ M (2 n, C ). The following identities are often useful. Proposition ( [F] Prop 4.17) . Let A = (cid:18) P Q ¯ Q ¯ P (cid:19) ∈ Sp c , then(1) (cid:18) P Q ¯ Q ¯ P (cid:19) − = (cid:18) P ∗ − Q t − Q ∗ P t (cid:19) = K A ∗ K , where K = (cid:18) I − I (cid:19) . (2) P P ∗ − QQ ∗ = I and P Q t = QP t .(3) P ∗ P − Q t ¯ Q = I and P t ¯ Q = Q ∗ P . The (double cover) of Sp ( m, R ) acts on the Bargmann-Fock space H k of C m as an integral operator withthe following kernel: given M = (cid:18) P Q ¯ Q ¯ P (cid:19) ∈ Sp c , we define K k,M ( z, w ) = (cid:18) k π (cid:19) m (det P ) − / exp (cid:26) k (cid:0) z ¯ QP − z + 2 ¯ wP − z − ¯ wP − Q ¯ w (cid:1)(cid:27) where the ambiguity of the sign the square root (det P ) − / is determined by the lift to the double cover.When A = Id , then K k, A ( z, ¯ w ) = Π k ( z, ¯ w ). The lifted kernel upstairs on the reduced Heisenberg group X is given by, ˆ K k, A (ˆ z, ˆ w ) = K k,M ( z, ¯ w ) e k ( iθ z −| z | / k ( − iθ w −| w | / . (21)3.3. Toeplitz construction of the metaplectic representation.
The analogue of Weyl pseudo-differentialoperators on L ( R m ) is (Berezin-)Toeplitz operators on Bargmann-Fock space. Given the semi-classical pa-rameter k , the Berezin-Toeplitz quantization of a multiplication operator by a semi-classical symbol σ k ( Z, ¯ Z )on C m is defined by Π k σ k ( Z, ¯ Z )Π k . (22)It operators on Bargmann-Fock space by multiplying a holomorphic function by σ k and then projecting backonto Bargmann-Fock space. More generally, one could let σ k be a semi-classical pseudo-differential operator.The isotropic Harmonic oscillator is on represented on H k ( C d ) asˆ H k = Π k | Z | Π k . It is equally well representated by (cid:80) mj =1 a ∗ j a j + d = (cid:80) mj =1 z j ∂∂z j + d , where a j = ∂∂z j and a ∗ j = z j are theannihilation/creation operators. The operator (cid:80) mj =1 a ∗ j a j is called the degree or number operator since itsaction on a holomorphic polynomial is to give its degree. In a similar way, the infinitesimal metaplecticrepresentation of quadratic polynomials Q = Q ( z, ¯ z ) is by Toeplitz operators Π k Q Π k .The Toeplitz construction of the metaplectic representation is due to Daubechies [Dau80]. The integratedmetaplectic representation W J ( S ) of S ∈ M p ( n, R ) on H J is defined as follows: Let S ∈ Sp ( n, R ) and let U S be the unitary translation operator on L ( R n , dL ) defined by U S F ( x, ξ ) := F ( S − ( x, ξ )). The metaplecticrepresentation of S on H J is given by ([Dau80],(5.5) and (6.3 b)) W J ( S ) = η J,S Π J U S Π J , (23) where (see [Dau80] (6.1) and (6.3a)), η J,S = 2 − n det( I − iJ ) + S ( I + iJ ) (24)and Π J is the Bargmann-Fock Szeg¨o projector.In the notation of the previous section, a quadratic Hamiltonian function H : C m → R generates a one-parameter family of symplectic linear transformations A t = g t : C m → C m , which in general is only R -linearand not C -linear, i.e. M t does not preserve the complex structure of C m . Hence, one need to orthogonalproject back to holomorphic sections. To compensate for the loss of norm due to the projection, one needto multiply a factor η A t . Proposition
Let A : C m → C m be a linear symplectic map, A = (cid:18) P Q ¯ Q ¯ P (cid:19) , and let ˆ A : X → X be thecontact lift that fixes the fiber over , then ˆ K k, A (ˆ z, ˆ w ) = (det P ∗ ) / (cid:90) X ˆΠ k (ˆ z, ˆ A ˆ u ) ˆΠ k (ˆ u, ˆ w ) d Vol X (ˆ u ) Proof.
The contact lift ˆ A : C m × S → C m × S is given by A acting on the first factor:ˆ A : ( z, θ ) (cid:55)→ ( P z + Q ¯ z, θ ) , one can check that ˆ A ∗ α = α . The integral over X is a standard complex Gaussian integral, analogous to [F,Prop 4.31], and with determinant Hessian 1 / | det P | , hence we have (det P ∗ ) / / | det P | = (det P ) − / . (cid:3) Toeplitz Quantization of Hamiltonian flows.
The Toeplitz construction of the metaplectic repre-sentation generalizes to the construction of a Toeplitz quantization of any symplectic map on any K¨ahlermanifold as a Toeplitz operator on the quantizing line bundles [Z97]. In this section we briefly review theconstruction of a Toeplitz parametrix for the propagtor U k ( t ) of the quantum Hamiltonian (57). We referto Section 10 and to [ZZ17, ZZ18] for the details.Let ( M, ω, L, h ) be a polarized K¨ahler manifold, and π : X → M the unit circle bundle in the dualbundle ( L ∗ , h ∗ ). X is a contact manifold, equipped with the Chern connection contact one-form α , whoseassociated Reeb flow R is the rotation ∂ θ in the fiber direction of X . Any Hamiltonian vector field ξ H on M generated by a a smooth function H : M → R can be lifted to a contact Hamiltonian vector field ˆ ξ H on X , which generates a contact flow ˆ g t . The following Proposition from [Z97] expresses the lift of (75) to H ( X ) = (cid:76) k ≥ H k ( X ). Proposition
There exists a semi-classical symbol σ k ( t ) so that the unitary group (75) has the form ˆ U k ( t ) = ˆΠ k (ˆ g − t ) ∗ σ k ( t ) ˆΠ k (25) modulo smooth kernels of order k −∞ . Bargmann intertwining operator between Schr¨odinger and Bargmann-Fock.
The standardunitary intertwining operator between the Schrodinger representation and the Bargmann-Fock representationis the (Segal-)Bargmann transform, Bf ( Z ) = (cid:90) R n exp (cid:16) − ( Z · Z − √ Z · X + X · X ) / (cid:17) f ( X ) dX. (26)Its inverse is its adjoint, B ∗ F ( x ) = (cid:90) C n exp (cid:16) − ( ¯ Z · ¯ Z − √ Z · X + X · X ) / (cid:17) F ( Z ) e −| Z | L ( dZ ) . Another inversion formula is f ( x ) = π − n/ (2 π ) − n/ e −| x | (cid:90) R n ( Bf )( x + iy ) e −| y | / dy. The Bargmann transform is obtained from the Euclidean heat kernel by analytic continuation in the firstvariable. It might be surprising that this transform is useful in studying the Harmonic oscillator. One couldjust as well analytically continue the propagator (9), which also defines a unitary intertwining operator.However, that operator would simply analytically continue Hermite functions, which does not simply the
NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 11 analysis. The Bargmann transform maps Hermite functions to holomorphic polynomials, and the Hermiteoperator to the degree operator (up to a constant) and this is a significant simplification.One may also use the Bargmann transform to convert Wigner distributions associated to spectral pro-jections of the Harmonic oscillator to the much simpler orthogonal projections onto spaces of holomorphicpolynomials of fixed degree. The density of states (diagonal of a Bergman kernel) is known as a Husimidistribution in physics. An interesting historical fact is that Cahill-Glauber studied the relation betweenWigner distributions W Π (cid:126) ,EM ( x, ξ ) and the Bargmann-conjugate Bergman Husimi distributions B Π (cid:126) ,E N B ∗ ( Z, ¯ Z )in [CG69I, CG69II]. The Bargmann transform is the same as the spectral projections of the Bargmann-Fockquantization Π BF,k | Z | Π BF,k of | Z | . They showed that B x ⊗ B y (cid:82) W Π (cid:126) ,EN ( x + y , ξ ) e i (cid:104) x − y,ξ (cid:105) dξ = (cid:82) R n (cid:82) R n (cid:82) R n B ( x, Z ) B ( y, Z ) W Π (cid:126) ,EN ( x + y , ξ ) e i (cid:104) x − y,ξ (cid:105) dξdxdy is convolution of W Π (cid:126) ,EM ( x, ξ ) with a complex Gaussian.3.6. Analogies and correspondences between the real and complex settings.
We now list someimportant analogies to help navigate the results of this article, and to compare the results in the real andcomplex settings. The undefined notation and terminology will be provided in the relevant section of thisarticle. The reader is encouraged to consult this list as the article proceeds; it is probably not possible tounderstand much of it from the start.Microlocal analysis provides a generalization of this equivalence to general manifolds. The generalizationof the Bargmann transform (see Section 26) is called an FBI transform. It is well-recognized that thesetting of holomorphic sections of high powers L k → M of ample line bundles over K¨ahler manifolds isquite analogous to the setting of Schr¨odinger operators on Riemannian manifolds, to the extent that onemay expect parallel results in both domains. The role of the Planck constant (cid:126) in semi-classical analysisis analogous to k − in the line bundle setting. In fact, the relation between Wigner distributions and“Husimi distributions” (or partial Bergman density of states) was first given by Cahill-Glauber in 1969[CG69I, CG69II] for applications in quantum optics. We refer to [R87, Zw] for background in semi-classicalanalysis and to [BG81] for background on Toeplitz operators.Here is a list of analogies which are relevant to the present survey. • The cotangent bundle ( T ∗ R d , σ ) equipped with its canonical symplectic structure is analogous to aK¨ahler manifold ( M, ω ). One may equip T ∗ R d with a complex structure J so that it becomes theK¨ahler manifold C d . • The total space of the dual line bundle L ∗ of a holomorphic line bundle L → M is analogous to C d . Indeed, if M = CP d − (complex projective space), then C d = L ∗ where L ∗ = O ( −
1) is thetautological line bundle over CP d − . (More precisely, C d = O ( −
1) with the zero section ‘blowndown’.) • When L is an ‘ample’ line bundle, sections s k ∈ H ( M, L k ) in the space of holomorphic sections ofthe k th power of L lift in a canonical way to equivariant holomorphic functions ˆ s k on L ∗ . In thecase ( M, L ) = ( CP d − , O ( − L k are the holomorphic homogeneous polynomialson C d of degree k . • The total space L carries an S (circle) action, namely rotation in the fibers L z of π : L → M .The generator D θ of this circle action is analogous to the isotropic harmonic oscillator and to thedegree operator. Namely if D θ ˆ s k = k ˆ s k . The isotropic harmonic oscillator ˆ H (cid:126) on L ( R d ) is unitarilyequivalent to the degree operator on C d under the Bargmann transform. • In the case (
M, L ) = ( CP d − , O ( − H ( CP d − , O ( k )) is canonically isomorphic to the eigenspaceof eigenvalue k + d of the isotropic harmonic oscillator. • Eigenspace spectral projection kernels Π (cid:126) ,E N ( (cid:126) ) ( x, y ) for eigenspaces V N of isotropic harmonic oscil-lators are analogous to Bergman kernels Π h k ( z, w ) for spaces H ( M, L k ) of holomorphic sections ofpowers of a positive Hermitian line bundle ( L, h ) over a K¨ahler manifold (
M, ω ). • The Wigner distribution W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) of an eigenspace projection is analogous to the density ofstates Π h k ( z, z ) where Π h k is the Bergman kernel for H ( M, L k ). The density of states is thecontraction of the diagonal of the Bergman kernel. • Airy scaling asymptotics of scaled Wigner distributions of eigenspace projections of the isotropicharmonic oscillator around an energy surface Σ E ⊂ T ∗ R d are analogous to Gaussian error functionasymptotics of scaled Bergman kernels around an energy surface. Both live on ‘phase space’. Theeigenspace projections of the oscillator live on configuration (or, physical) space and have no simpleanalogue in the K¨ahler setting. • The unitary Bargman transform B : L ( R d ) → H ( C d , e −| Z | dL ( Z )) intertwines the real Schr¨odingerand holomorphic Bargmann-Fock representations of quantum mechanics on R d . There is no sim-ple analogue for general K¨ahler manifolds. It would be a unitary intertwining operator betweenthe Bargmann-Fock spaces of L ∗ and L ( N ) where N ⊂ M would be a totally real Lagrangiansubmanifold. See Section 26 for background.There is an important difference between the results on Wigner distributions and the results on partialBergman kernels, which indicates that there is much more to be done on interfaces in spectral asymptotics.Namely, in the K¨ahler setting we have two Hamiltonians: (i) A Toeplitz Hamiltonian ˆ H k := Π h k H Π h k (where H : M → R is a smooth function), and (ii) the operator D θ on L ∗ defining the degree k of a liftedsection. The latter is analogous to the isotropic oscillator. The interfaces for D θ are interfaces across ‘discbundles’ D ∗ R ⊂ L ∗ defined by a Hermitian metric h on L . The analogue of Airy scaling asymptotics ofWigner distributions is Gaussian error function asymptotics for lifts of Bergman kernels to L ∗ . A ToeplitzHamiltonian ˆ H k lifts to a Hamiltonian on L ∗ which commutes with D θ , and our results on partial Bergmankernels pertain to the pair. So far, we have not considered the analogous problem on L ( R d ) defined by asecond Schr¨odinger operator which commutes with the isotropic harmonic oscillator. As this brief discussionindicates, there are many types of interface phenomena that remain to be explored.4. Interface problems for Schr¨odinger equations
In this section we consider the simplest Schr¨odinger operator, namely the isotropic Harmonic Oscillatoron R d . We review three types of interface scaling results: • Scaling of the spectral projections kernel for a single eigenspace around the caustic. At the sametime, we consider scaling of nodal sets of random eigenfunctions around the caustic. • Scaling asymptotics of the Wigner distributions of the spectral projections kernel around an energylevel in phase space. • Scaling asymptotics of the Wigner distributions of Weyl sums of spectral projections kernels over aninterval of energies at the boundary of the interval.4.1.
Allowed and forbidden regions and the caustic.
Consider a general Schr¨odinger operator ˆ H (cid:126) := − (cid:126) ∆ + V on L ( R d ) with V ( x ) → ∞ as | x | → ∞ . Then ˆ H (cid:126) has a discrete spectrum of eigenfunctions E j ( (cid:126) ), ˆ H (cid:126) ψ (cid:126) ,j = E j ( (cid:126) ) ψ (cid:126) ,j . (27)In the semi-classical limit (cid:126) → , j → ∞ , E j ( (cid:126) ) = E, (28)the eigenfunctions of ˆ H (cid:126) are rapidly oscillating in the classically allowed region A E := { V ( x ) ≤ E } , NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 13 and exponentially decaying in the classically forbidden region F E := A cE = { V ( x ) > E } . This reflects the fact that a classical particle of energy E is confined to A E = { V ( x ) ≤ E } . We define the caustic to be C E := ∂ A E = { V ( x ) = E } . (29)The exponential decay rate of eigenfunctions in the forbidden region as (cid:126) → A E , resp. the forbidden region F E are given respectively by, A E = { x : | x | < E } , F E = { x : | x | > E } . (30)Thus, A E is the projection to R d of the energy surface { H = E } ⊂ T ∗ R d , F E is its complement, and thecaustic set is given by, C E = {| x | = 2 E } . The semi-classical limit at the energy level
E > (cid:126) → , N → ∞ with fixed E , so that (cid:126) only takes the values (5).4.2. Scaling asymptotics around the caustic in physical space.
Due to the homogeneity of theisotropic oscillator, it suffices to consider one value of E . We fix E = and consider E N ( (cid:126) ) = . Forthis choice of E , (7) is Π (cid:126) , . When d = 1 , the eigenspaces V (cid:126) N ,E have dimension 1 and it is a classical fact (based on WKB or ODEtechniques) that Hermite functions and more general Schr¨odinger eigenfunctions exhibit Airy asympotics atthe caustic (turning points). See for instance [O,T,FW]. It is not true for d > d = 1 result is toconsider the scaling asymtptoics of the eigenspace projection kernels (7) with x, y in an (cid:126) / -tube around C E .The first result states that individual eigenspace projection kernels (7) exhibit Airy scaling asymtoticsaround a point x ∈ C E of the caustic. Let x be a point on the caustic | x | = 1 for E = 1 /
2. Points in an (cid:126) / neighborhood of x may be expressed as x + (cid:126) / u with u ∈ R d . The caustic is a ( d − x is x , so the normal component of u is u x when | x | = 1, where u := (cid:104) x , u (cid:105) . Wealso put u (cid:48) := u − u x for the tangential component, and identify T x C E ∼ = T ∗ x C E ∼ = R d − . By rotationalsymmetry, we may assume x = (1 , , · · · , u = ( u , u , · · · , u d ) =: ( u ; u (cid:48) ). Theorem
Let x be a point on the caustic | x | = 1 for E = 1 / . Then for u, v ∈ R d , Π (cid:126) , / ( x + (cid:126) / u, x + (cid:126) / v ) = (cid:126) − d/ / Π ( u, v )(1 + O ( (cid:126) / )) , (31) where Π ( u , u (cid:48) ; v , v (cid:48) ) := 2 / (2 π ) − d +1 (cid:90) R d − e i (cid:104) u (cid:48) − v (cid:48) ,p (cid:105) Ai(2 / ( u + p / / ( v + p / dp, (32) and u := (cid:104) x , u (cid:105) , u (cid:48) := u − u x (similarly for v . ) On the diagonal, let | x | = (cid:12)(cid:12) x + (cid:126) / u (cid:12)(cid:12) = 1 + (cid:126) / s + O ( (cid:126) / ) with s = 2 (cid:104) x , u (cid:105) ∈ R . Then, Π (cid:126) ( x, x ) = 2 − d +1 π − d/ (cid:126) (1 − d ) / Ai − d/ ( s )(1 + O ( (cid:126) / )) . (33) The error terms in (31) and (33) are uniform when u, v, s vary over a compact set.
Above, Ai is the Airy function, and Ai − d/ is a weighted Airy function , defined for k ∈ R byAi k ( s ) := (cid:90) C T k exp (cid:18) T − T s (cid:19) dT πi , u ∈ R (34)where C is the usual contour for Airy function, running from e − iπ/ ∞ to e iπ/ ∞ on the right half of thecomplex plane (see Section 11.1 for a brief review of the Airy function). Remark When d = 3 , the kernel (32) with u (cid:48) = v (cid:48) , i.e. Π ( u , u (cid:48) ; v , u (cid:48) ) , coincides modulo the factorof √ λ with the Airy kernel K ( x, y ) of the Tracy-Widom distribution. The “allowed region” of this article isanalogous to the ‘bulk’ in random matrix theory, and the “caustic” of this article is analogous to the “edgeof the spectrum”. Nodal sets of random Hermite eigenfunctions.
Theorem 4.1 can be used to determine the interfacebehavior of nodal (zero) sets of random eigenfunctions of the isotropic oscillator of a fixed eigenvalue. Inmany ways, the isotropic oscillator is the analogue among Schr¨odinger operators on L ( R d ) of the Laplacianon a standard sphere S d , and the study of random Hermite eigenfunctions is somewhat analogous to thestudy of random spherical harmonics. However, there are no forbidden regions in the case of S d , and theinterface behavior of random Hermite eigenfunctions has no parallel for random spherical harmonics. Definition
A Gaussian random eigenfunction for H h with eigenvalue E is the random series Φ N ( x ) := (cid:88) | α | = N a α φ α,h N ( x ) , for a α ∼ N (0 , R i.i.d. Equivalently, it is the Gaussian measure γ N on V N which is given by e − (cid:80) α | a α | / (cid:81) da α . We denote by Z Φ N = { x : Φ N ( x ) = 0 } the nodal set of Φ N and by | Z Φ (cid:126) ,E | the random measure of integration over Z Φ N with respect to the Euclideansurface measure (the Hausdorff measure) of the nodal set. Thus for any ball B ⊂ R d , | Z Φ (cid:126) ,E | ( B ) = H d − ( B ∩ Z Φ N ) . Thus E | Z Φ (cid:126) ,E | is a measure on R n given by E | Z Φ (cid:126) ,E | ( B ) = (cid:90) V N H d − ( B ∩ Z Φ N ) dγ N . The first result gives semi-classical asymptotics of the hypersurface volumes of the nodal sets of randomHermite eigenfunctions of fixed eigenvalue in the allowed, resp. forbidden region.
Theorem
Let x ∈ R d such that < | x | (cid:54) = √ E. Then the measure E | Z Φ (cid:126) ,E | has a density F N ( x ) withrespect to Lebesgue measure given by If x ∈ A E \{ } , F N ( x ) (cid:39) h − · c d (cid:113) E − | x | (1 + O ( h )) If x ∈ F E , F N ( x ) (cid:39) h − / · C d E / | x | / ( | x | − E ) / (1 + O ( h )) , where the implied constants in the ‘ O ’ symbols are uniform on compact subsets of the interiors of A E \{ } and F E , and where c d = Γ (cid:0) d +12 (cid:1) √ dπ Γ (cid:0) d (cid:1) and C d = Γ (cid:0) d +12 (cid:1) √ π Γ (cid:0) d (cid:1) . The key point is the different growth rates in h for the density of zeros in the allowed and forbiddenregion. In dimension one, eigenfunctions have no zeros in the forbidden region, but in dimensions d ≥ NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 15
The next result on nodal sets (Theorem 4.4) gives scaling asymptotics for the average nodal density that‘interpolate’ between (4.3) and (4.3). Fix x ∈ C E , where E = 1 /
2, and study the rescaled ensembleΦ x,α (cid:126) ,E ( u ) := Φ (cid:126) ,E ( x + (cid:126) α u )and the associated hypersurface measure (cid:12)(cid:12)(cid:12) Z x,α (cid:126) ,E (cid:12)(cid:12)(cid:12) ( B ) = H d − (cid:16) { Φ x,α (cid:126) ,E ( v ) = 0 } ∩ B (cid:17) , B ⊂ R d . The next result gives the asymptotics of E (cid:12)(cid:12)(cid:12) Z x,α (cid:126) ,E (cid:12)(cid:12)(cid:12) when α = 2 / k (see (34)). Theorem (Nodal set in a shrinking ball around a caustic point) . Fix E = 1 / and x ∈ C E , i.e. | x | = 1 .For any bounded measurable B ⊆ R d , E (cid:12)(cid:12)(cid:12) Z x, / (cid:126) ,E (cid:12)(cid:12)(cid:12) ( B ) = (cid:90) B F ( u ) du, where F ( u ) = (2 π ) − d +12 (cid:90) R d | Ω( u ) / ξ | e −| ξ | / dξ (1 + O ( (cid:126) / )) (35) and Ω = (Ω ij ) ≤ i,j ≤ n is the symmetric matrix Ω ij ( u ) = x i x j (cid:32) Ai − d/ ( s )Ai − d/ ( s ) − Ai − d/ ( s )Ai − d/ ( s ) (cid:33) + δ ij − − d/ ( s )Ai − d/ ( s ) . (36) where s = 2 (cid:104) u, x (cid:105) . The implied constant in the error estimate from (35) is uniform when u varies in compactsubsets of R d . Remark The leading term in F is (cid:126) -independent and positive everywhere since the matrix Ω ij ( u ) as alinear operator has nontrivial range. The matrix ( x i x j ) i,j in (36) is a rank projection onto the x − direction;since the dimension d ≥ , it cannot cancel out the second term. We refer to [HZZ15, HZZ16] for details. Remark Theorem 4.4 says that if x ∈ C E and (cid:101) B (cid:126) = x + (cid:126) / B for some bounded measurable B, then E (cid:12)(cid:12) Z Φ (cid:126) ,E (cid:12)(cid:12) ( (cid:101) B (cid:126) ) = (cid:126) / d − E (cid:12)(cid:12)(cid:12) Z x,α (cid:126) ,E (cid:12)(cid:12)(cid:12) ( B ) = (cid:126) − / (cid:90) (cid:101) B (cid:126) F ( (cid:126) − / ( y − x )) dy, which shows that the average (unscaled) density of zeros in a (cid:126) / − tube around C E grows like (cid:126) − / as (cid:126) → . Remark 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 thespectrum.
Discussion of the nodal results.
Computer graphics of Bies-Heller [BH] (reprinted as Figure 4.3in [HZZ15]) and the displayed graphics of Peng Zhou show that the nodal set in A E near the caustic ∂ A E consists of a large number of highly curved nodal components apparently touching the caustic while thenodal set in F E near ∂ A E consists of fewer and less curved nodal components all of which touch the caustic.This is because, if ψ ∈ V (cid:126) ,E is non-zero, ∆ ψ = ( V − E ) ψ forces ψ and ∆ ψ to have the same sign in F E . Ina nodal domain D we may assume ψ >
0, but then ψ is a positive subharmonic function in D and cannotbe zero on ∂ D without vanishing identically. Hence, every nodal component which intersects F E must alsointersect A E and therefore C E .The scaling limit of the density of zeros in a shrinking neighborhood of the caustic, or in annular subdo-mains of A E and F E at shrinking distances from the caustic is given in Theorem 4.4. The nodal set is very dense and busy in A E and rather sparse and ‘non-oscillating’ in F E .4.5. The Kac-Rice Formula.
The proof of Theorem 4.4 is based on the Kac-Rice formula for the averagedensity of zeros.
Lemma (Kac-Rice for Gaussian Fields) . Let Φ (cid:126) ,E be the random Hermite eigenfunction of (cid:98) H (cid:126) witheigenvalue E . Then the density of zeros of Φ (cid:126) ,E is given by F (cid:126) ,E ( x ) = (2 π ) − d +12 (cid:90) R d | Ω / ( x ) ξ | e −| ξ | / dξ, (37) where Ω( x ) is the d × d matrix Ω ij ( x ) = ( ∂ x i ∂ y j log Π (cid:126) ,E )( x, x )= (Π (cid:126) ,E · ∂ x i ∂ y j Π (cid:126) ,E )( x, x ) − ( ∂ x i Π (cid:126) ,E · ∂ y j Π (cid:126) ,E )( x, x )Π (cid:126) ,E ( x, x ) (38) and Π (cid:126) ,E ( x, y ) is the kernel of eigenspace projection (8) . We refer to [HZZ15, HZZ16] for background. The main task in proving results on zeros near the causticis therefore to work out the asymptotics of Π (cid:126) ,E ( x, x ) and its derivatives there.5. Interfaces in phase space for Schr¨odinger operators: Wigner distributions
We now turn to phase space interfaces. Instead of studying the scaling asymptotics of the spectralprojections (7) Π (cid:126) ,E N ( (cid:126) ) : L ( R d ) → V (cid:126) ,E N ( (cid:126) ) (39)we study the scaling asymptotics of their semi-classical Wigner distributions W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) := (cid:90) R d Π (cid:126) ,E N ( (cid:126) ) (cid:16) x + v , x − v (cid:17) e − i (cid:126) v · ξ dv (2 πh ) d (40)across the phase space energy surface (14).When E N ( (cid:126) ) = E + o (1) as (cid:126) → W (cid:126) ,E N ( (cid:126) ) is thought of as the ‘quantization’ of the energy surface, and(40) is thought of as an approximate δ -function on (14). This is true in the weak* sense, but the pointwisebehavior is quite a bit more complicated and is studied in [HZ19]. NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 17 - - - - - - Figure 1.
The Wigner function W (cid:126) ,E N ( (cid:126) ) of the eigenspace projection Π (cid:126) ,E N ( (cid:126) ) is alwaysradial (see Proposition 5.1). Displayed above is the graph of the Airy function (orange) andof W (cid:126) ,E N ( (cid:126) ) with N = 500 (blue) as a function of the rescaled radial variable ρ in a (cid:126) / tube around the energy surface H ( x, ξ ) = E N ( (cid:126) ) = 1 / . Theorem 5.3 predicts that, whenproperly scaled, W (cid:126) ,E N ( (cid:126) ) should converge to the Airy function (with the rate of convergencebeing slower farther from the energy surface, which is defined here by ρ = 0).Wigner distributions were introduced in [W32] as phase space densities. Heuristically, the Wigner distri-bution (7) is a kind of probability density in phase space of finding a particle of energy E N ( (cid:126) ) at the point( x, ξ ) ∈ T ∗ R d . This is not literally true, since W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) is not positive: it oscillates with heavy tailsinside the energy surface (14), has a kind of transition across Σ E and then decays rapidly outside the energysurface. The purpose of this paper is to give detailed results on the concentration and oscillation propertiesof these Wigner distributions in three phase space regimes, depending on the position of ( x, ξ ) with respectto Σ E .There is an exact formula for the Wigner distributions (13) of the eigenspace projections for the isotropicHarmonic oscillator in terms of Laguerre functions (see Appendix 11.2 and [T] for background on Laguerrefunctions). Proposition
The Wigner distribution of Definition 2.1 is given by, W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) = ( − N ( π (cid:126) ) d e − H/ (cid:126) L ( d − N (4 H/ (cid:126) ) , H = H ( x, ξ ) = | x | + | ξ | , (41) where L ( d − N is the associated Laguerre polynomial of degree N and type d − . See [O, JZ] for d = 1 and [T, Theorem 1.3.5] and [HZ19] for general dimensions. The second result is aweak* limit result for normalized Wigner distributions. Proposition
Let a be a semi-classical symbol of order zero and let Op wh ( a ) be its Weyl quantization.Then, as (cid:126) → , with E N ( (cid:126) ) → E , V (cid:126) ,E N ( (cid:126) ) (cid:90) T ∗ R d a ( x, ξ ) W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) dxdξ → − (cid:90) Σ E a dµ E , where dµ E is Liouville measure on Σ E and − (cid:82) Σ E a dµ E = µ E (Σ E ) (cid:82) Σ E a dµ E . Thus, W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) → δ Σ E in the sense of weak* convergence. But this limit is due to the oscillationsinside the energy ball; the pointwise asymptotics are far more complicated.5.1. Interface asymptotics for Wigner distributions of individual eigenspace projections.
Ourfirst main result gives the scaling asymptotics for the Wigner function W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) of the projection ontothe E -eigenspace of (cid:98) H (cid:126) when ( x, ξ ) lies in an (cid:126) / neighborhood of the energy surface Σ E . Theorem
Fix
E > , d ≥ . Assume E N ( (cid:126) ) = E and let (cid:126) = (cid:126) N ( E ) (5) . Suppose ( x, ξ ) ∈ T ∗ R d satisfies H ( x, ξ ) = E + u (cid:18) (cid:126) E (cid:19) / , u ∈ R , H ( x, ξ ) = (cid:107) x (cid:107) + (cid:107) ξ (cid:107) with | u | < (cid:126) − / . Then, W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) = π (cid:126) ) d (cid:0) (cid:126) E (cid:1) / (cid:0) Ai( u/E ) + O (cid:0) (1 + | u | ) / u (cid:126) / (cid:1)(cid:1) , u < π (cid:126) ) d (cid:0) (cid:126) E (cid:1) / Ai( u/E ) (cid:0) O (cid:0) (1 + | u | ) / u (cid:126) / (cid:1)(cid:1) , u > x ) is the Airy function. The Airy scaling of W (cid:126) ,E N ( (cid:126) ) is illustrated in Figure 2. The assumption(42) may be stated more invariantly that ( x, ξ ) lies in the tube of radius O ( (cid:126) / ) around Σ E defined by thegradient flow of H with respect to the Euclidean metric on T ∗ R d . The asymptotics are illustrated in figure1. Due to the behavior of the Airy function Ai( s ), these formulae show that in the semi-classical limit (cid:126) → E N ( (cid:126) ) → E , W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) concentrates on the energy surface surface Σ E , is oscillatory inside the energyball { H ≤ E } and is exponentially decaying outside the ball.5.2. Interior Bessel asymptotics.
In addition to the Airy asymptotics in an (cid:126) / -tube around Σ E , W (cid:126) ,E N ( (cid:126) ) exhibits Bessel asymptotics in the interior of Σ E . There are two (or three, depending on taste)uniform asymptotic regimes for the Laguerre polynomial L ( α ) n ( x ): Bessel, Trigonometric, Airy.For t ∈ [0 , A ( t ) = 12 [ (cid:112) t − t + sin − √ t ] , t ∈ [0 , . For t < − is replaced by sinh − and the by i/ J d − be the Besselfunction (of the first kind) of index d − Theorem
Fix
E > and suppose E N ( (cid:126) ) = E. For each ( x, ξ ) ∈ T ∗ R d write H E := H ( x, ξ ) E = (cid:107) x (cid:107) + (cid:107) ξ (cid:107) E , ν E := 4 E (cid:126) . Fix < a < / . Uniformly over a ≤ H E ≤ − a , there is an asymptotic expansion, W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) = 2(2 π (cid:126) ) d (cid:20) J d − ( ν E A ( H E )) A ( H E ) d − α ( H E ) + O (cid:18) ν − E (cid:12)(cid:12)(cid:12)(cid:12) J d ( ν E A ( H E )) A ( H E ) d (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:21) . In particular, uniformly over H E in a compact subset of (0 , , we find W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) = (2 π (cid:126) ) − d +1 / P H,E cos ( ξ (cid:126) ,E,H ) + O (cid:16) (cid:126) − d +3 / (cid:17) , (44) where we’ve set ξ (cid:126) ,E,H = − π − H (cid:126) (cid:0) H − E − (cid:1) / + 2 E (cid:126) cos − (cid:16) H / E (cid:17) and P E,H := (cid:16) πE / (cid:0) H − E − (cid:1) / ( H E ) d/ (cid:17) − . Small ball integrals.
The interior Bessel asymptotics do not encompass the behavior of W (cid:126) ,E N ( (cid:126) ) inshrinking balls around ρ = 0. In that case, we have, Proposition
For (cid:15) > sufficiently small and for any a ( x, ξ ) ∈ C b ( T ∗ R d ) , (cid:90) T ∗ R d a ( x, ξ ) W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) ψ (cid:15), (cid:126) ( x, ξ ) dxdξ = O ( (cid:126) − d − d(cid:15) (cid:107) a (cid:107) L ∞ ( B ( (cid:126) / − (cid:15) )) ) . (45) where ψ (cid:15), (cid:126) is a smooth radial cut-off that is identically on the ball of radius (cid:126) / − (cid:15) and is identically outside the ball of radius (cid:126) / − (cid:15) . The errors blow up when u = (cid:126) − / . NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 19
Figure 2.
The Wigner function W (cid:126) ,E N ( (cid:126) ) of the eigenspace projection Π (cid:126) ,E N ( (cid:126) ) is alwaysradial (see Proposition 5.1). Displayed above is the blow-up of the Wigner function at (0 , Exterior asymptotics. If E N ( (cid:126) ) → E , then W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) concentrates on Σ E and is exponentiallydecaying in the complement H = H ( x, ξ ) > E . The precise statement is, Proposition
Suppose that H E = H ( x, ξ ) /E > and let E N ( (cid:126) ) = E . Then, there exists C > sothat | W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) | ≤ C (cid:126) − d + e − E (cid:126) [ √ H E − H E − cosh − √ H E ] . Moreover, as H ( x, ξ ) → ∞ , there exists C > so that | W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) | ≤ C (cid:126) − d + e − H ( x,ξ ) (cid:126) . Supremum at ρ = 0 . The reader may notice the ‘spike’ at the origin ρ = 0; it is the point at which W (cid:126) ,E N ( (cid:126) ) has its global maximum (see Figure 2). The height is given by W (cid:126) ,E N ( (cid:126) ) (0 ,
0) = ( − N ( π (cid:126) ) d L d − N (0) = ( − N ( π (cid:126) ) d Γ( N + d )Γ( N +1)Γ( d ) (cid:39) ( − N π d C d (cid:126) − d N d − . (46)The last statement follows from the explicit formula L ( d − N (0) = Γ( N + d )Γ( N +1)Γ( d ) = ( N + d − N !( d − (see e.g. [T,(1.1.39)]).On the complement of the ball B (0 , (cid:126) − (cid:15) ), the Wigner distribution is much smaller than at its maximum.The following is proved by combining the estimates of Theorem 5.3 , Theorem 5.4 and Proposition 5.6. Proposition
For any (cid:15) > , sup ( x,ξ ): H ( x,ξ ) ≥ (cid:15) | W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) | ≤ C (cid:126) − d + . The supremum in this region is achieved in { H ≤ E } at ( x, ξ ) satisfying (42) where u is the global maximumof Ai(x) . Why the spike at ρ = 0? It is observed in [HZ19] that W (cid:126) ,E N ( (cid:126) ) is an eigenfunction of the (essentiallyisotropic) Schr¨odinger operator (cid:18) − (cid:126) ξ + ∆ x ) + H ( x, ξ ) (cid:19) W (cid:126) ,E N ( (cid:126) ) = E N ( (cid:126) ) W (cid:126) ,E N ( (cid:126) ) . (47)on T ∗ R d . By [HZZ15, Lemma 10], the eigenspace spectral projections for the isotropic harmonic oscillatorin dimension d satisfies, Π h,E ( x, x ) = (2 πh ) − ( d − (cid:16) E − | x | (cid:17) d − ω d − (1 + O ( h )) , for a dimensional constant ω d . We apply this result to the eigenspace projections for (47) in dimension2 d and find that at the point (0 ,
0) its diagonal value is of order (cid:126) − d +1 . We then express this eigenspace projection in terms of an orthonormal basis for the eigenspace. From the inner product formulae (16), it isseen that one of the orthonormal basis elements is √ dim V (cid:126) ,EN ( (cid:126) ) W (cid:126) ,E N ( (cid:126) ) . Note that dim V (cid:126) ,E N ( (cid:126) ) (cid:39) (cid:126) − d +1 in dimension 2 d . Due to the normalization and (46),1 (cid:112) dim V (cid:126) ,E N ( (cid:126) ) W (cid:126) ,E N ( (cid:126) ) (0 , (cid:39) (cid:126) − d +1+ d − = (cid:126) − d + . There exists a simple spectral geometric explanation for the order of magnitude at the origin: All eigen-functions of (47) with the exception of the radial eigenfunction W (cid:126) ,E N ( (cid:126) ) (0 ,
0) vanish at the origin (0 ,
0) sincethey transform by non-trivial characters of U ( d ) and (0 ,
0) is a fixed point of the action. Consequently, thevalue of the eigenspace projection on the diagonal at (0 ,
0) is the square of W (cid:126) ,E N ( (cid:126) ) (0 ,
0) and that accountsprecisely for the order of growth.5.6.
Sums of eigenspace projections.
Let us begin by introducing the three types of spectral localizationwe are studying and the interfaces in each type. • (i) (cid:126) -localized Weyl sums over eigenvalues in an (cid:126) -window E N ( (cid:126) ) ∈ [ E − a (cid:126) , E + b (cid:126) ] of width O ( (cid:126) ).More generally we consider smoothed Weyl sums W (cid:126) ,E,f with weights f ( (cid:126) − ( E N ( (cid:126) ) − E )); see (49)for such (cid:126) -energy localization. This is the scale of individual spectral projections but is substantiallymore general than the results of [HZ19]. The scaling and asymptotics are in Theorem 5.9. For generalSchr¨odinger operators, (cid:126) - localization around a single energy level leads to expansions in terms ofperiodic orbits. Since all orbits of the classical isotropic oscillator are periodic, the asymptotics maybe stated without reference to them. The generalization to all Schr¨odinger operators will be studiedin a future article. • (ii) Airy-type (cid:126) / -spectrally localized Weyl sums W (cid:126) ,f, / ( x, ξ ) over eigenvalues in a window [ E − a (cid:126) / , E + a (cid:126) / ] of width O ( (cid:126) / ). See Definition 5.10 for the precise definition. The levelset Σ E isviewed as the interface. The scaling asympotics of its Wigner distribution across the interface aregiven in Theorems 5.11 and 5.12. To our knowledge, this scaling has not previously been consideredin spectral asymptotics. • (iii) Bulk Weyl sums (cid:80) N : (cid:126) ( N + d ) ∈ [ E ,E ] W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) over energies in an (cid:126) -independent ‘window’[ E , E ] of eigenvalues; this ‘bulk’ Weyl sum runs over (cid:39) (cid:126) − distinct eigenvalues; See Definition5.13. We are mainly interested in its scaling asymptotics around the interface Σ E (see Theorem5.16). However, we also prove that the Wigner distribution approximates the indicator function ofthe shell { E ≤ H ≤ E } ⊂ T ∗ R d (see Proposition 5.15). As far as we know, this is also a newresult and many details are rather subtle because of oscillations inside the energy shell. Indeed, theresults of [HZ19] show that the indvidual terms in the sum grow like W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) (cid:39) (cid:126) − d +1 / when H ( x, ξ ) ∈ ( E , E ) . Proposition 5.15, in constrast, shows although the bulk Weyl sums have (cid:39) (cid:126) − such terms, their sum has size (cid:126) − d , implying significant cancellation.We are particularly interested in ‘interface asymptotics’ of the Bulk Wigner-Weyl distributions W (cid:126) ,f,δ ( (cid:126) ) around the edge (i.e. boundary) of the spectral interval when ( x, ξ ) is near the corresponding classical energysurface Σ E . Such edges occur when f is discontinuous, e.g. the indicator function of an interval. In otherwords, we integrate the empirical measures (48) below over an interval rather than against a Schwartz testfunction. At the interface, there is an abrupt change in the asymptotics with a conjecturally universal shape.Theorem 5.9 gives the shape of the interface for (cid:126) -localized sums, Theorem 5.11 gives the shape for (cid:126) / localized sums and Theorem 5.16 gives results on the bulk sums.Our results concern asymptotics of integrals of various types of test functions against the weighted em-pirical measures, dµ ( x,ξ ) (cid:126) ( τ ) := ∞ (cid:88) N =0 W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) δ E N ( (cid:126) ) ( τ ) , (48) NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 21 - - - - - Figure 3.
Plot with (cid:126) ≈ . , E = 1 / π (cid:126) ) d W (cid:126) , [0 ,E ] ( x, ξ ) when H ( x, ξ ) = E + u ( (cid:126) / E ) / as a function of u (blue) against itsintegrated Airy limit (cid:82) ∞ Ai( λ + u/E ) dλ (red) from Theorem 5.16.and of recentered and rescaled versions of these measures (see (52) below). A key property of Wignerdistributions of eigenspace projections (40) is that the measures (48) are signed, reflecting the fact thatWigner distributions take both positive and negative values, and are of infinite mass: Proposition
The signed measures (48) are of infinite mass (total variation norm). On the other hand,the mass of (48) is finite on any one-sided interval of the form, [ −∞ , τ ] . Also, (cid:82) R dµ ( x,ξ ) (cid:126) = 1 for all ( x, ξ ) . Moreover, the L norms of the terms W (cid:126) ,E N ( (cid:126) ) grows in N like N d − . Hence, the measures (48) are highlyoscillatory and the summands can be very large.5.7. Interior asymptotics for (cid:126) -localized Weyl sums.
The first result we present pertains to the (cid:126) -spectrally localized Weyl sums of type (i), defined by W (cid:126) ,E,f ( x, ξ ) := (cid:88) N f ( (cid:126) − ( E − E N ( (cid:126) ))) W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) , f ∈ S ( R ) . (49) Theorem
Fix
E > , and let W (cid:126) ,E,f be the Wigner distribution as in (49) with f an even Schwartzfunction. If H ( x, ξ ) > E, then W (cid:126) ,E,f ( x, ξ ) = O ( (cid:126) ∞ ) . In contrast, when < H ( x, ξ ) < E , set H E := H ( x, ξ ) /E and define t + , ± ,k := 4 πk ± − (cid:16) H / E (cid:17) , t − , ± ,k := 4 π (cid:18) k + 12 (cid:19) ± − (cid:16) H / E (cid:17) , k ∈ Z . Fix any δ > . Then W (cid:126) ,f,E ( x, ξ ) = (cid:126) − d +1 (cid:0) O δ ( (cid:126) − δ ) (cid:1) (2 E ) / (2 π ) d H d/ E ( H − E − / (cid:88) ± , ± ∈{ + , −} e ± i ( π − E (cid:126) )( ± ) d (cid:88) k ∈ Z (cid:98) f ( t ± , ± ,k ) e iE (cid:126) t ± , ± ,k , where the notation O δ means the implicit constant depends on δ. Note that there are potentially an infinite number of ‘critical points’ in the support of ˆ f .5.8. Interface asymptotics for smooth (cid:126) / -localized Weyl sums. We now consider spectrally local-ized Wigner distributions that are both spectrally localized and phase-space localized on the scale δ ( (cid:126) ) = (cid:126) / .They are mainly relevant when we study interface behavior around Σ E of Weyl sums. Definition
Let H ( x, ξ ) = ( (cid:107) x (cid:107) + (cid:107) ξ (cid:107) ) / , and assume that ( x, ξ ) satisfiy H ( x, ξ ) = E + u ( (cid:126) / E ) / . (50) Let δ ( h ) = (cid:126) / and define the interface-localized Wigner distributions by W (cid:126) ,f, / ( x, ξ ) : = (cid:80) N f ( h − / ( E − E N ( (cid:126) ))) W (cid:126) ,E N ( x, ξ ) Theorem
Assume that ( x, ξ ) satisfies (50) with | u | < (cid:126) − / . Fix a Schwartz function f ∈ S ( R ) withcompactly supported Fourier transform. Then W (cid:126) ,f, / ( x, ξ ) = (2 π (cid:126) ) − d I ( u ; f, E ) + O ((1 + | u | ) (cid:126) − d +2 / ) , where I ( u ; f, E ) = (cid:90) R f ( − λ/C E )Ai (cid:16) λ + uE (cid:17) dλ, C E = ( E/ / . More generally, there is an asymptotic expansion W (cid:126) ,f, / ( x, ξ ) (cid:39) (2 π (cid:126) ) d (cid:88) m ≥ (cid:126) m/ I m ( u ; f, E ) in ascending powers of (cid:126) / where I m ( u ; f, E ) are uniformly bounded when u stays in a compact subset of R . The calculations show that the results are valid with far less stringent conditions on f than f ∈ S ( R ) and (cid:98) f ∈ C ∞ . To obtain a finite expansion and remainder it is sufficient that (cid:82) R | (cid:98) f ( t ) || t | k dt < ∞ for all k. It isnot necessary that ˆ f ∈ C k for any k > Sharp (cid:126) / -localized Weyl sums. Next we consider the sums of Definition 5.10 when f is the indicatorfunction of a spectral interval, f = [ λ − ,λ + ] . Equivalently, we fix integers 0 < n ± such that λ ± = (cid:126) / n ± are bounded , and consider the corresponding Wigner-Weyl sums W (cid:126) ,f, / ( x, ξ ) of Definition 5.10: W / ,E,λ ± ( x, ξ ) := (cid:88) N : λ − (cid:126) / ≤ E N ( (cid:126) ) − E<λ + (cid:126) / W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) = N ( E, (cid:126) )+ n + − (cid:88) N = N ( E, (cid:126) )+ n − W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) , (51)where N ( E, (cid:126) ) = E/ (cid:126) − d/
2. Thus, the sums run over spectral intervals of size (cid:39) (cid:126) / centered at a fix E > (cid:39) (cid:126) − / Wigner functions for spectral projections of individual eigenspaces.The following extends Theorem 5.11 to sharp Weyl sums at the cost of only giving a 1-term expansion plusremainder.
Theorem
Assume that ( x, ξ ) satisfies (cid:16) (cid:107) x (cid:107) + (cid:107) ξ (cid:107) (cid:17) / E + u (cid:0) (cid:126) E (cid:1) / with | u | < (cid:126) − / . Then, W / ,E,λ ± ( x, ξ ) = (2 π (cid:126) ) − d C E (cid:90) − λ − − λ + Ai (cid:16) uE + λC E (cid:17) dλ + O (cid:16) (cid:126) − d +1 / − δ + (1 + | u | ) (cid:126) − d +2 / − δ (cid:17) , where C E = ( E/ / . Theorem 5.12 can be rephrased in terms of weighted empirical measures dµ u,E, (cid:126) := (cid:126) d (cid:88) N W (cid:126) ,E N ( (cid:126) ) (cid:16) E + u ( (cid:126) / E ) / (cid:17) δ [ (cid:126) − / ( E − E N ( (cid:126) ))] . (52)obtained by centering and scaling the family (48). Thus, for ( x, ξ ) satisfying (cid:16) (cid:107) x (cid:107) + (cid:107) ξ (cid:107) (cid:17) / E + u (cid:0) (cid:126) E (cid:1) / , and for f ∈ S ( R ), W (cid:126) ,f, / ( x, ξ ) := (cid:126) − d (cid:90) R f ( τ ) dµ u,E, (cid:126) ( τ ) , W / ,E,λ ± ( x, ξ ) = (cid:126) − d (cid:90) λ + λ − dµ u,E, (cid:126) ( τ ) . NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 23
Bulk sums.
We next consider Weyl sums of eigenspace projections corresponding to an energy shell(or window) [ E , E ]. We consider both sharp and smoothed sums. Definition
Define the ‘bulk’ Wigner distributions for an (cid:126) -independent energy window [ E , E ] by W (cid:126) , [ E ,E ] ( x, ξ ) : (cid:88) N : E N ( (cid:126) ) ∈ [ E ,E ] W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) . (53) More generally for f ∈ C b ( R ) define W (cid:126) ,f ( x, ξ ) := ∞ (cid:88) N =1 f ( (cid:126) ( N + d/ W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) . (54)Our first result about the bulk Weyl sums concerns the smoothed Weyl sums W (cid:126) ,f . Proposition
For f ∈ S ( R ) with ˆ f ∈ C ∞ , W (cid:126) ,f ( x, ξ ) admits a complete asymptotic expansion as (cid:126) → of the form, W (cid:126) ,f ( x, ξ ) (cid:39) ( π (cid:126) ) − d (cid:80) ∞ j =0 c j,f,H ( x, ξ ) (cid:126) j , with c ,f,H ( x, ξ ) = f ( H ( x, ξ )) = (cid:82) R ˆ f ( t ) e itH ( x,ξ ) dt. In general c k,f,H ( x, ξ ) is a distribution of finite order on f supported at the point ( x, ξ ) . The proof merely involves Taylor expansion of the phase.5.11.
Interior/exterior asymptotics for bulk Weyl sums of Definition 5.13.
From Proposition 5.14,it is evident that the behavior of W (cid:126) , [ E ,E ] ( x, ξ ) depends on whether H ( x, ξ ) ∈ ( E , E ) or H ( x, ξ ) / ∈ [ E , E ].Some of this dependence is captured in the following result. Proposition
We have, W (cid:126) , [ E ,E ] ( x, ξ ) = ( i ) (2 π (cid:126) ) − d (1 + O ( (cid:126) / )) , H ( x, ξ ) ∈ ( E , E ) , ( ii ) O ( (cid:126) − d +1 / ) , H ( x, ξ ) < E , ( iii ) O ( (cid:126) ∞ ) , H ( x, ξ ) > E The two ‘sides’ 0 < H ( x, ξ ) < E and H ( x, ξ ) > E also behave differently because the Wigner dis-tributions have slowly decaying tails inside an energy ball but are exponentially decaying outside of it. Ifwe write W (cid:126) , [ E ,E ] ( x, ξ ) = W (cid:126) , [0 ,E ] ( x, ξ ) − W (cid:126) , [0 ,E ] ( x, ξ ), we see that the two cases with H ( x, ξ ) > E are covered by results for W (cid:126) , [0 ,E ] with E = E or E = E . When H ( x, ξ ) < E , then both terms of W (cid:126) , [0 ,E ] ( x, ξ ) − W (cid:126) , [0 ,E ] ( x, ξ ) have the order of magnitude (cid:126) − d and the asymptotics reflect the cancellationbetween the terms. The boundary case where H ( x, ξ ) = E , or H ( x, ξ ) = E is special and is given inTheorem 5.11.5.12. Interface asymptotics for bulk Weyl sums of Definition 5.13.
Our final result concerns theasymptotics of W (cid:126) , [ E ,E ] ( x, ξ ) in (cid:126) -tubes around the ‘interface’ H ( x, ξ ) = E . Again, it is suffcient toconsider intervals [0 , E ]. It is at least intuitively clear that the interface asymptotics will depend only on theindividual eigenspace projections with eigenvalues in an (cid:126) / -interval around the energy level E , and sincethey add to 1 away from the boundary point, one may expect the asymptotics to be similar to the interfaceasymptotics for individual eigenspace projections in [HZ19]. Theorem
Assume that ( x, ξ ) satisfies | x | + | ξ | − E = u (cid:0) (cid:126) E (cid:1) / with | u | < (cid:126) − / . Then, for any (cid:15) > W (cid:126) , [0 ,E ] ( x, ξ ) = (2 π (cid:126) ) − d (cid:20)(cid:90) ∞ Ai (cid:16) uE + τ (cid:17) dτ + O ( (cid:126) / − (cid:15) | u | / ) + O ( | u | / (cid:126) / − (cid:15) ) (cid:21) , where the implicit constant depends only on d, (cid:15). The Airy scaling the Wigner function is illustrated in Figure 4.
Figure 4.
Scaling at energy surface of Wigner function of projection onto energy interval [0 , / Heuristics.
Wigner distributions are normalized so that the Wigner distribution of an L normalizedeigenfunction has L norm 1 in T ∗ R d . Due to the multiplicity N d − of eigenspaces (3), the L norm of W (cid:126) ,E N ( (cid:126) ) is of order N d − .In the main results, we sum over windows of eigenvalues, e.g. λ − (cid:126) / ≤ E − E N ( (cid:126) ) < λ + (cid:126) / (51),resp. E N ( (cid:126) ) ∈ [0 , E ] in (5.13). Inevitably, the asymptotics are joint in ( (cid:126) , N ). As (cid:126) ↓
0, the number of N contributing to the sum grows at the rate (cid:126) − , resp. (cid:126) − . Due to the N -dependence of the L norm,terms with higher N have norms of higher weight in N than those of small N but the precise size of thecontribution depends on the position of ( x, ξ ) relative to the interface { H = E } and of course the relation(2). W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) peaks when H ( x, ξ ) = E N ( (cid:126) ), exponentially decays in (cid:126) when H ( x, ξ ) > E N ( (cid:126) ) andhas slowly decaying tails inside the energy ball { H < E N ( (cid:126) ) } , which fall into three regimes: (i) Besselnear 0, (ii) oscillatory or trigonometric in the bulk, and (iii) Airy near { H = E } . In terms of N , when(2) holds, and H ( x, ξ ) < E N ( (cid:126) ), then W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) (cid:39) (cid:126) − d +1 / (cid:39) N d − / . Near the peak point, when H ( x, ξ ) − E N ( (cid:126) ) ≈ (cid:126) / , we have in contrast W (cid:126) ,E N ( (cid:126) ) ( x, ξ ) (cid:39) (cid:126) − d +1 / (cid:39) E N d − / . It follows that the terms with a high value of N and with E N ( (cid:126) ) ≥ H ( x, ξ ) in (48) contribute high weights.There are an infinite number of such terms, and so (48) is a signed measure of infinite mass (as stated inProposition 5.8.) This is why we mainly consider the restriction of the measures (48) to compact intervals.5.14. Remark on nodal sets in phase space.
In Section 4 we discussed nodal sets of random eigenfunc-tions of the isotropic Harmonic oscillator. It would also make sense to consider nodal sets in phase space T ∗ R d for Wigner distributions W Φ (cid:126) ,E of random eigenfunctions of the isotropic Harmonic oscillator. Thisis of interest because Wigner distributions are signed, i.e. not positive, and their nodal sets and domainssignal the extent of this ‘defect’ in their interpretation as phase space densities. But so far, this has not beendone. However, the covariance function is simply the Wigner distribution of the spectral projection kernels,so the analysis of Wigner distributions and of their interfaces across energy surfaces provides the necessarytechniques.In the next section we consider interfaces for partial Bergman kernels. The analogue in the complexdomain of random nodal sets of isotropic oscillator eigenfunctions is zero sets of random homogeneousholomorphic polynomials of fixed degree in C d . This is essentially the same as studying such zero sets oncomplex projective space CP d − , and to that extent the theory has already been developed. But interfacephenomena for complex zero sets has not so far been studied. NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 25 Interfaces in phase space: partial Bergman kernels
In this section, we continue to study phase space distributions of orthogonal projections, but change fromthe Schr¨odinger quantization to the holomorphic quantization. The holomorphic setting consists of Berezin-Toeplitz operators acting on holomorphic sections of line bundles over K¨ahler manifolds, and is analyticallysimpler than the real Schr¨odinger setting. Hence we are able to present much more general results. Instead offixing a model Schr¨odinger operator like the isotropic Harmonic Oscillator, we consider all possible ToeplitzHamiltonians on holomorphic sections of Hermitian line bundles (
L, h ) → ( M, ω ) over all possible projectiveK¨ahler manifolds. Here it is assumed that i∂∂ log h = ω , i.e. that ( L, h ) is a positive, ample line bundle.For background on Bargmann-Fock space, and on line bundles over general K¨ahler manifolds, we refer toSection 10.Motivation to study partial Bergman kernels comes from two sources. On the one hand, they arise in manyproblems of complex geometry (see [Ber1, HW17, HW18, PS, RS] besides the articles surveyed here). On theother hand, they arise in the IQHE (integer quantum Hall effect). The author’s interest was stimulated byconversations with A. Abanov, S. Klevtsov and P. Wiegmann during a Simons’ Center program on complexgeometry and the IQHE. We refer to [W, Wieg, CFTW] for some physics articles where interfaces in thedensity of states of the IQHE are studied. It should be emphasized that there are many types of partialBergman kernels, and the ones most interesting in physics are still out of reach of the rigorous techniquesdescribed here. What we study here are spectral partial Bergman kernels , i.e. orthogonal projection kernelsonto spectral subspaces for Toeplitz Hamiltonians. By no means do all pBK’s (partial Bergman kernels)arise from spectral problems, but the spectral pBK’s are the only types for which there exist general results(or almost any results) and sometimes the pBK’s of interest in the IQHE are spectral pBK’s.We do not review the basic definitions here (see Section 10) but head straight for the interface results. Inplace of the spectral projections of the previous sections, we consider partial Bergman kernels on “polarized”K¨ahler manifolds (
L, h ) → ( M m , ω, J ), i.e. K¨ahler manifolds of (complex) dimension m equipped with aHermitian holomorphic line bundle whose curvature form F ∇ for the Chern connection ∇ satisfies ω = iF ∇ .Partial Bergman kernels Π k, S k : L ( M, L k ) → S k ⊂ H ( M, L k ) (55)are Schwarz kernels for orthogonal projections onto proper subspaces S k of the holomorphic sections of L k .For general subspaces, there is little one can say about the asymptotics of the partial density of statesΠ k, S k ( z ), i.e. the contraction of the diagonal of the kernel. But for certain sequences S k of subspaces, thepartial density of states Π k, S k ( z ) has an asymptotic expansion as k → ∞ which roughly gives the probabilitydensity that a quantum state from S k is at the point z . More concretely, in terms of an orthonormal basis { s i } N k i =1 of S k , the partial Bergman densities defined byΠ k, S k ( z ) = N k (cid:88) i =1 (cid:107) s i ( z ) (cid:107) h k . (56)When S k = H ( M, L k ), Π k, S k = Π k : L ( M, L k ) → H ( M, L k ) is the orthogonal (Szeg¨o or Bergman)projection. We also call the ratio Π k, S k ( z )Π k ( z ) the partial density of states.Corresponding to S k there is an allowed region A where the relative partial density of states Π k, S k ( z ) / Π k ( z )is one, indicating that the states in S k “fill up” A , and a forbidden region F where the relative density ofstates is O ( k −∞ ), indicating that the states in S k are almost zero in F . On the boundary C := ∂ A betweenthe two regions there is a shell of thickness O ( k − ) in which the density of states decays from 1 to 0. The √ k -scaled relative partial density of states is asymptotically Gaussian along this interface, in a way reminis-cent of the central limit theorem. This was proved in [RS] for certain Hamiltonian holomorphic S actions,then in greater generality in [ZZ17]. In fact, it is a universal property of partial Bergman kernels defined by C ∞ Hamiltonians.To begin with, we define the subspaces S k . They are defined as spectral subspaces for the quantizationof a smooth function H : M → R . By the standard (Kostant) method of geometric quantization, one canquantize H as the self-adjoint zeroth order Toeplitz operator H k := Π k ( ik ∇ ξ H + H )Π k : H ( M, L k ) → H ( M, L k ) (57) acting on the space H ( M, L k ) of holomorphic sections. Here, ξ H is the Hamiltonian vector field of H , ∇ ξ H is the Chern covariant deriative on sections, and H acts by multiplication. We denote the eigenvalues(repeated with multiplicity) of ˆ H k (57) by µ k, ≤ µ k, ≤ · · · ≤ µ k,N k , (58)where N k = dim H ( M, L k ), and the corresponding orthonormal eigensections in H ( M, L k ) by s k,j .Let E be a regular value of H . We denote the partial Bergman kernels for the corresponding spectralsubspaces by Π k,E : H ( M, L k ) → H k,E , (59)where S k := H k,E := (cid:77) µ k,j
0, the mapΦ :
C × ( − δ, δ ) → M, ( z, t ) (cid:55)→ γ z ( t ) (66)is a diffeomorphism onto its image. NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 27
Main Theorem.
Let ( L, h ) → ( M, ω, J ) be a polarized K¨ahler manifold. Let H : M → R be a smoothfunction and E a regular value of H . Let S k ⊂ H ( X, L k ) be defined as in (60) . Then we have the followingasymptotics on partial Bergman densities Π k, S k ( z ) : (cid:18) Π k, S k Π k (cid:19) ( z ) = (cid:40) if z ∈ A if z ∈ F mod O ( k −∞ ) . For small enough δ > , let Φ :
C × ( − δ, δ ) → M be given by (66) . Then for any z ∈ C and t ∈ R , we have (cid:18) Π k, S k Π k (cid:19) (Φ( z, t/ √ k )) = Erf(2 √ πt ) + O ( k − / ) , (67) where Erf( x ) = (cid:82) x −∞ e − s / ds √ π is the cumulative distribution function of the Gaussian, i.e., P X ∼ N (0 , ( X As a quick illustration, holomorphic sections of the trivial line bundle over C are holomorphicfunctions on C . We equip the bundle with the Hermitian metric where has the norm-square e −| z | . The k th power has metric e − k | z | Fix (cid:15) > and define the subspaces S k = ⊕ j ≤ (cid:15)k z j of sections vanishing to orderat most (cid:15)k at , or sections with eigenvalues µ < (cid:15) for operator H k = ik ∂ θ quantizing H = | z | . The fulland partial Bergman densities are Π k ( z ) = k π , Π k,(cid:15) ( z ) = (cid:18) k π (cid:19) (cid:88) j ≤ (cid:15)k k j j ! | z j | e − k | z | , As k → ∞ , we have lim k →∞ k − Π k,(cid:15) ( z ) = (cid:40) | z | < (cid:15) | z | > (cid:15). For the boundary behavior, one can consider sequence z k , such that | z k | = (cid:15) (1 + k − / u ) , lim k →∞ k − Π k,(cid:15) ( z k ) = Erf( u ) . This example is often used to illustrate the notion of ‘filling domains’ in the IQH (integer Quantum Hall)effect. In IQH, one considers a free electron gas confined in plane R (cid:39) C , with a uniform magnetic field inthe perpendicular direction. A one-particle electron state is said to be in the lowest Landau level (LLL) if ithas the form Ψ( z ) = e −| z | / f ( z ), where f ( z ) is holomorphic as in Example 6.2. The following image of thedensity profile is copied from [W], where the picture on the right illustrates how the states ( √ k z ) j √ j ! e − k | z | / with j ≤ (cid:15)k fill the disc of radius √ (cid:15) , so that the density profile drops from 1 to 0.The example is S symmetric and therefore the simpler results of [ZZ16] apply. For more general domains D ⊂ C , it is not obvious how to fill D with LLL states. The Main Theorem answers the question when D = { H ≤ E } for some H . For a physics discussion of Erf asymptotics and their (as yet unknown)generalization to the fractional QH effect, see [Wieg, CFTW]. The usual Gaussian error function erf(x) = (2 π ) − / (cid:82) x − x e − s / ds is related to Erf by Erf(x) = (1 + erf( x √ )) . It does not matter whether the endpoints are included in the interval, since contribution from the eigenspaces V k,µ with µ = E i are of lower order than k m . What does a cylinder mean? It is vector space with an invertible operator. What does the gluingmean? Inclusion of open subset of skeleton, to some bigger piece. Open piece with stop boundary,included into the bigger one, then apply a stop removal, which is a localization. Like reading GHK, or KS. r m ⇢ r` B ⌫ = 1 Figure 5. “The density profile of the ν = 1 droplet, where the first m levels (representedby the thick lines) are filled.” From Fig 7.11 in [W].6.1. Three families of measures at different scales. The rationale for viewing the Erf asymptotics ofscaled partial Bergman kernels along the interface C is explained by considering three different scalings ofthe spectral problem. ( i ) dµ zk ( x ) = (cid:80) j Π k,j ( z ) δ µ k,j ( x ) , ( ii ) dµ z, k ( x ) = (cid:80) j Π k,j ( z ) δ √ k ( µ k,j − H ( z )) ( x ) , ( iii ) dµ z, ,τk ( x ) = (cid:80) j Π k,j ( z ) δ k ( µ k,j − H ( z ))+ √ kτ ( x ) , (68)where as usual, δ y is the Dirac point mass at y ∈ R . We use µ ( x ) = (cid:82) x −∞ dµ ( y ) to denote the cumulativedistribution function.We view these scalings as analogous to three scalings of the convolution powers µ ∗ k of a probability measure µ supported on [ − , 1] (say). The third scaling (iii) corresponds to µ ∗ k , which is supported on [ − k, k ]. Thefirst scaling (i) corresponds to the Law of Large Numbers, which rescales µ ∗ k back to [ − , −√ k, √ k ].Our main results give asymptotic formulae for integrals of test functions and characteristic functionsagainst these measures. To obtain the remainder estimate (67), we need to apply semi-classical Tauberiantheorems to µ z, k and that forces us to find asymptotics for µ z, ,τk .6.2. Unrescaled bulk results on dµ zk . The first result is that the behavior of the partial density of statesin the allowed region { z : H ( z ) < E } is essentially the same as for the full density of states, while it is rapidlydecaying outside this region.We begin with a simple and general result about partial Bergman kernels for smooth metrics and Hamil-tonians. Theorem 1. Let ω be a C ∞ metric on M and let H ∈ C ∞ ( M ) . Fix a regular value E of H and let A , F , C be given by (62) . Then for any f ∈ C ∞ ( R ) , we have Π k ( z ) − (cid:90) E −∞ f ( λ ) dµ zk ( λ ) → (cid:40) f ( H ( z )) ifz ∈ A ∈ F . (69) In particular, the density of states of the partial Bergman kernel is given by the asymptotic formula: Π k ( z ) − Π k,E ( z ) ∼ (cid:40) O ( k −∞ ) ifz ∈ A O ( k −∞ ) ifz ∈ F . (70) where the asymptotics are uniform on compact sets of A or F . NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 29 In effect, the leading order asymptotics says that the normalized measure Π k ( z ) − dµ zk → δ H ( z ) . Thisis a kind of Law of Large Numbers for the sequence dµ zk . The theorem does not specify the behavior of µ zk ( −∞ , E ) when H ( z ) = E . The next result pertains to the edge behavior.6.3. √ k -scaling results on dµ z, / k . The most interesting behavior occurs in k − -tubes around the interface C between the allowed region A and the forbidden region F . For any T > 0, the tube of ‘radius’ T k − around C = { H = E } is the flowout of C under the gradient flow of HF t := exp( t ∇ H ) : M → M, for | t | < T k − / . Thus it suffices to study the partial density of states Π k,E ( z k ) at points z k = F β/ √ k ( z )with z ∈ H − ( E ) . The interface result for any smooth Hamiltonian is the same as if the Hamiltonianflow generate a holomorphic S -actions, and thus our result shows that it is a universal scaling asymptoticsaround C . Theorem 2. Let ω be a C ∞ metric on M and let H ∈ C ∞ ( M ) . Fix a regular value E of H and let A , F , C be given by (62) . Let F t : M → M denote the gradient flow of H by time t . We have the following results: (1) For any point z ∈ C , any β ∈ R , and any smooth function f ∈ C ∞ ( R ) , there exists a completeasymptotic expansion, (cid:88) j f ( √ k ( µ k,j − E ))Π k,j ( F β/ √ k ( z )) (cid:39) (cid:18) k π (cid:19) m ( I + k − I + · · · ) , (71) in descending powers of k , with the leading coefficient as I ( f, z, β ) = (cid:90) ∞−∞ f ( x ) e − ( x |∇ H | ( z ) | − β |∇ H ( z ) | ) dx √ π |∇ H ( z ) | . (2) For any point z ∈ C , and any α ∈ R , the cumulative distribution function µ z, / k ( α ) = (cid:82) α −∞ dµ z, / k is given by µ z, / k ( α ) = (cid:88) µ k,j Let E be a regular value of H and z ∈ H − ( E ) . If (cid:15) is small enough, such that the Hamiltonianflow trajectory starting at z does not loop back to z for time | t | < π(cid:15) , then for any Schwarz function f ∈ S ( R ) with ˆ f supported in ( − (cid:15), (cid:15) ) and ˆ f (0) = (cid:82) f ( x ) dx = 1 , and for any α ∈ R we have (cid:90) R f ( x ) dµ z, ,αk ( x ) = (cid:18) k π (cid:19) m − / e − α (cid:107) ξH ( z ) (cid:107) √ π (cid:107) ξ H ( z ) (cid:107) (1 + O ( k − / )) . Critical levels. In this section we consider interfaces at critical levels. Let H : M → R be a smoothfunction with Morse critical points.Henceforth, to simplify notation, we use K¨ahler local coordinates u centered at z to write points in the k − (cid:15) tube around C by z = z + k − (cid:15) u := exp z ( k − (cid:15) u ) , u ∈ T z . C The abuse of notation in dropping the higher order terms of the normal exponential map is harmless sincewe are working so close to C . At regular points z we may use the exponential map along N z C but we alsowant to consider critical points. More generally we write z + u for the point with K¨ahler normal coordinate u . In these coordinates, ω ( z + u ) = i m (cid:88) j =1 du j ∧ d ¯ u j + O ( | u | ) . We also choose a local frame e L of L near z , such that the corresponding ϕ = − log h ( e L , e L ) is given by ϕ ( z + u ) = | u | + O ( | u | ) . See [ZZ17] for more on such adapted frames and Heisenberg coordinates.Clearly, the formula (71) breaks down at critical points and near such points on critical levels. Our maingoal in this paper is to generalize the interface asymptotics to the case when the Hamiltonian is a Morsefunction and the interface C = { H = E } is a critical level, so that C contains a non-degenerate critical point z c of H . To allow for non-standard scaling asymptotics, we study the smoothed partial Bergman densitynear the critical value E = H ( z c ),Π k,E,f,δ ( z ) := (cid:88) j (cid:107) s k,j ( z ) (cid:107) · f ( k δ ( µ k,j − E ))where f ∈ S ( R ) with Fourier transform ˆ f ∈ C ∞ c ( R ), and 0 ≤ δ ≤ 1. This is the smooth analog of summingover eigenvalues within [ E − k − δ , E + k − δ ].The behavior of the scaled density of states is encoded in the following measures, dµ zk ( x ) = (cid:80) j (cid:107) s k,j ( z ) (cid:107) δ µ k,j ( x ) ,dµ z,δk ( x ) = (cid:80) j (cid:107) s k,j ( z ) (cid:107) δ k δ ( µ k,j − H ( z )) ( x ) ,dµ ( z,u,(cid:15) ) ,δk ( x ) = (cid:80) j (cid:107) s k,j ( z + k − (cid:15) u ) (cid:107) δ k δ ( µ k,j − H ( z )) ( x ) . (74)For each measure µ we denote by d ˆ µ the normalized probability measure d ˆ µ ( x ) = µ ( R ) − dµ ( x ) . For all z ∈ M , we have the following weak limit, reminiscent of the law of large numbers;ˆ µ zk ( x ) (cid:42) δ H ( z ) ( x ) . For z ∈ M with dH ( z ) (cid:54) = 0, (71) shows thatˆ µ z, / k ( x ) (cid:42) e − x | dH ( z ) | dx √ π | dH ( z ) | . Interface asymptotics at critical levels. The next result generalizes the ERF scaling asymptoticsto the critical point case. We use the following setup: Let z c be a non-degenerate Morse critical point of H ,then for small enough u ∈ C m , we denote the Taylor expansion components by H ( z c + u ) = E + H ( u ) + O ( | u | ) . where E = H ( z c ) , H ( u ) = 12 Hess z c H ( u, u ) . NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 31 Theorem For any f ∈ S ( R ) with ˆ f ∈ C ∞ c ( R ) , we have Π k,E,f, / ( z c + k − / u ) := (cid:88) j (cid:107) s k,j ( z c + k − / u ) (cid:107) · f ( k / ( µ k,j − E )) = (cid:18) k π (cid:19) m f ( H ( u )) + O f ( k m − / ) . More over, the normalized rescaled pointwise spectral measure d ˆ µ ( z c ,u, / , / k ( x ) := (cid:80) j (cid:107) s k,j ( z c + k − / u ) (cid:107) δ k / ( µ k,j − E ) ( x ) (cid:80) j (cid:107) s k,j ( z c + k − / u ) (cid:107) converges weakly ˆ µ ( z c ,u, / , / k ( x ) (cid:42) δ H ( u ) ( x ) . We notice that the scaling width has changed from k − to k − / due to the critical point. The differencein scalings raises the question of what happens if we scale by k − around a critical point. The result isstated in terms of the metaplectic representation on the osculating Bargmann-Fock space at z c . Theorem Let (cid:29) T > be small enough, such that there is no non-constant periodic orbit withperiods less than T . Then for any f ∈ S ( R ) with ˆ f ∈ C ∞ c (( − T, T )) , we have Π k,E,f, ( z c + k − / u ) = (cid:18) k π (cid:19) m (cid:90) R ˆ f ( t ) U ( t, u ) dt π + O ( k m − / ) where U ( t, u ) is the metaplectic quantization of the Hamiltonian flow of H ( u ) defined as U ( t, u ) = (det P ) − / exp(¯ u ( P − − u + u ¯ QP − u/ − ¯ uP − Q ¯ u/ . Here P = P ( t ) , Q = Q ( t ) be complex m × m matrices such that if u ( t ) = exp( tξ H ) u , then (cid:18) u ( t )¯ u ( t ) (cid:19) = (cid:18) P ( t ) Q ( t )¯ Q ( t ) ¯ P ( t ) (cid:19) (cid:18) u ¯ u (cid:19) . rem Unlike the universal Erf decay profile in the / √ k -tube around the smooth part of C , we cannotgive the decay profile of Π k,I ( z ) near the critical point z c . The reason is that there are eigensections thathighly peak near z c and with eigenvalues clustering around H ( z c ) . Hence it even matters whether we use [ E , E ] or ( E , E ) . See the following case where the Hamiltonian action is holomorphic, where the peaksection at z c is an eigensection, and all other eigensections vanishes at z c . The next result pertains to Hamiltonians generating R actions, as studied in [RS, ZZ16]. The Hamiltonianflow always extends to a holomorphic C action. Proposition Assume H generate a holomorphic Hamiltonian R action. The pointwise spectral measure dµ z c k ( x ) is always a delta-function µ z c k = δ H ( z c ) ( x ) , ∀ k = 1 , · · · Equivalently, for any spectral interval I , lim k →∞ Π k,I ( z c ) = (cid:40) E ∈ I E / ∈ I . The above result follows immediately from: Proposition Let z c be a Morse critical point of H , E = H ( z c ) . Then (1) The L -normalized peak section s k,z c ( z ) = C ( z c )Π k ( z, z c ) is an eigensection of ˆ H k with eigenvalue H ( z c ) . And all other eigensections orthogonal to s k,z c vanishes at z c . (2) If s k,j ∈ H ( M, L k ) is an eigensection of ˆ H k with eigenvalue µ k,j < E , then s k,j vanishes on W + ( z c ) . (3) If s k,j ∈ H ( M, L k ) is an eigensection of ˆ H k with eigenvalue µ k,j > E , then s k,j vanishes on W − ( z c ) . In particular, this shows the concentration of eigensection near z c . Depending on whether the spectralinteval I includes boundary point H ( z c ) or not, the partial Bergman density will differ by a large Gaussianbump of height ∼ k m . Sketch of Proof. As in [ZZ17, ZZ18] the proofs involve rescaling parametrices for the propagator U k ( t ) = exp itk ˆ H k (75)of the Hamiltonian (57). The parametrix construction is reviewed in Section 3.4. We begin by observingthat for all z ∈ M , the time-scaled propagator has pointwise scaling asymptotics with the k − scaling: Proposition ([ZZ17] Proposition 5.3) . If z ∈ M , then for any τ ∈ R , ˆ U k ( t/ √ k, ˆ z, ˆ z ) = (cid:18) k π (cid:19) m e it √ kH ( z ) e − t (cid:107) dH ( z ) (cid:107) (1 + O ( | t | k − / )) , where the constant in the error term is uniform as t varies over compact subset of R . The condition dH ( z ) (cid:54) = 0 in the original statement in [ZZ17] is never used in the proof, hence both statementand proof carry over to the critical point case. We therefore omit the proof of this Proposition.We also give asymptotics for the trace of the scaled propagator U k ( t/ √ k ). It is based on stationary phaseasymptotics and therefore also reflects the structure of the critical points. Theorem If t (cid:54) = 0 , the trace of the scaled propagator U k ( t/ √ k ) = e i √ kt ˆ H k admits the followingaymptotic expansion (cid:82) z ∈ M U k ( t/ √ k, z ) d Vol M ( z ) = (cid:0) k π (cid:1) m ( t √ k π ) − m (cid:80) z c ∈ crit ( H ) e it √ kH ( zc ) e ( iπ/ sgn (Hess zc ( H )) √ | det(Hess zc ( H )) | · (1 + O ( | t | k − / )) where sgn (Hess z c ( H )) is the signature of the Hessian, i.e. the number of its positive eigenvalues minus thenumber of its negative eigenvalues. Interfaces for the Bargmann-Fock isotropic Harmonic oscillator We continue the discussion of Bargmann-Fock space from Section 3 by considering partial Bargmann-FockBergman kernels. In this section, we tie together the results on Wigner distributions of spectral projectionsfor the isotropic Harmonic oscillator, and on density of states for partial Bergman kernels associated to thenatural S action on Bargmann-Fock space. This is the most direct analogue of the Schr¨odinger results.The classical Bargmann-Fock isotropic Harmonic oscillator corresponds to the degree operator on H ( CP m , O ( N )).The total space of the associated line bundle is C m +1 . The harmonic operator generates the standard diag-onal S action on C m +1 , e iθ · ( z , . . . , z m +1 ) = ( e in θ z , . . . , e in m θ z m +1 ) . Its Hamiltonian is H (cid:126)n ( Z ) = (cid:80) m +1 j =1 n j | z j | . The critical point set of H (cid:126)n is its minimum set.The eigenspaces H k,m,N consist of monomials z α with | α | = N . Given the Planck constant k , theeigenspace projection is given by Π h kBF ,N ( Z, W ) = (cid:88) | α | = N ( kZ ) α )( k ¯ W ) α α ! , (76)as a kernel relative to the Bargmann-Fock Gaussian volume form. The partial Bergman kernels arising fromspectral projections of the isotropic oscillator thus have the form,Π h kBF ,E = (cid:88) N : Nk ≥ E Π h kBF ,N ( Z, W ) . We claim that the eigenspace projector (76) satisfies,Π h kBF ,N ( Z, Z ) = C N,k,m || Z || N , (77)where C N,k,m = p ( N, m + 1) ω m k N Γ( N + m + 1) . Here, ω m = Vol(S ) is the surface are of the unit sphere in C m +1 . Also, dim H k,m,N = p ( m + 1 , N ) , the partition function which counts the number of ways to express N as a sum of m + 1 positive integers. NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 33 To prove this, we first observe that the U ( m + 1)-invariance of the Harmonic oscillator Hamiltonian H = || Z || implies that U ∗ Π h kBF ,N U = Π h kBF ,N and therefore Π h kBF ,N ( U Z, U Z ) = Π h kBF ,N ( Z, Z ) . It follows thatΠ h kBF ,N ( Z, Z ) = F ( || Z || ) is radial. It is also homogeneous of degree 2 N , hence is a constant multiple C N,k,m || Z || N as claimed in (77). The constant is calculated from the fact that p ( m, N ) = dim H k,m,N = k m +1 ( m +1)! (cid:82) C m +1 Π h kBF ,N ( Z, Z ) e − k || Z || dL ( Z )= ω m C m k m +1 (cid:82) ∞ e − kρ ρ N ρ m +1 dρ = ω m C m k m +1 (cid:82) ∞ e − kρ ρ N ρ m dρ = k m +1 ( m +1)! k − ( N + m +1) ω m C m,k,N Γ( N + m + 1) . Solving for C m,k,N establishes the formula. It also follows that the density of states is given by, (cid:80) N ≥ (cid:15)k Π h kBF ,N ( Z, Z ) = k m +1 ( m − ω m e − k || Z || (cid:80) N ≥ (cid:15)k ( k || Z || ) N p ( m,N )Γ( N + m +1) (cid:39) k m +1 ( m − ω m e − k || Z || (cid:80) N ≥ (cid:15)k ( k || Z || ) N N ! , (78)since p ( m + 1 , N ) (cid:39) m +1)! N m (1 + O ( N − )) (4); also, Γ( N + m + 1) = ( N + m )! (cid:39) ( N + m ) · · · ( N + 1) N ! (cid:39) N m N !. 8. Bargmann-Fock space of a line bundle and interface asymptotics In this section, we introduce a new model, the Bargmann-Fock space of an ample line bundle π : L → M over a K¨ahler manifold, and generalize the results of the preceding section to density of states for partialBergman kernels associated to the natural S action on the total space L ∗ of the dual line bundle. We let X h = ∂D ∗ h ⊂ L ∗ be the unit S -bundle given by the boundary of the unit codisc bundle, D ∗ h = { ( z, λ ) ∈ L ∗ : | λ | z < } . We sketch the proof that ‘interfaces’ for the Hamiltonian generating the standard S actionon the Bargmann-Fock space of L satisfy the central limit theorem or cumulative Gaussian Erf interfaces asin the compact case of [ZZ16]. The Hamiltonian is simply the norm-square function N ( z, λ ) := | λ | h z , so theenergy balls are simply the co-disc bundles D ∗ E = { ( z, λ ) ∈ L ∗ : | λ | h z ≤ E } . As usual, we equivariantly lift sections s k ∈ H ( M, L k ) to ˆ s k ∈ H k ( L ∗ ), which are homogeneous of degree k in the sense that ˆ s k ( rx ) = r k ˆ s k ( x ) . Volume forms. X h is a contact manifold with contact volume form dV = α ∧ ( π ∗ ω ) m . This contactvolume form induces a volume form dV ol L ∗ on L ∗ , generalizing the Lebesgue volume form dV ol C m in thestandard Bargmann-Fock space. Namely, the K¨ahler metric ω h of the Hermitian metric h on L lifts to thepartial K¨ahler metric π ∗ ω h . Then, ω L ∗ = π ∗ ω h + dλ ∧ d ¯ λ is a K¨ahler metric on L ∗ with potential | λ | e − φ where φ = log | e L | h z is the local K¨ahler potential on M . Since L ∗ (cid:39) X h × R + we may use polar coordinates ( x, ρ ) on L ∗ , which correspond to coordinates ( z, λ ) ∈ M × C ina local trivialization by ρ = | λ | h z and x = ( z, e iθ ). Since dim R X = 2 m + 1 when dim C M = m , the volumeform on L ∗ is given by dV ol L ∗ ( x, ρ ) = ρ m +1 dV ( x ) dρ. We then endow L ∗ with the (normalized) Gaussian measure analogous to (17), d Γ m +1 , (cid:126) := (cid:126) − ( m +1) Vol(X h )Γ(m + 1) e −|| Z || / (cid:126) dV ol L ∗ ( Z ) (79)To check that the measure has mass 1, we note that (cid:90) L ∗ e −|| Z || / (cid:126) dV ol L ∗ ( Z ) = Vol(X h ) (cid:90) ∞ e − ρ / (cid:126) ρ d ρ = Vol(X h ) (cid:126) m+1 Γ(m + 1) . Here, we denote a general point of L ∗ by Z = ρx with ρ ∈ R + , x ∈ X h . In the future we put C m ( h ) = 1Vol(X h )Γ(m + 1) , so that we do not have to keep track of this constant. Definition The Bargmann-Fock space of ( L, h ) is the Hilbert space H BF, (cid:126) ( L ∗ ) := ∞ (cid:77) N =0 H N ( L ∗ ) of entire square integrable holomorphic functions on L ∗ with respect to the inner product || f || (cid:126) ,BF =: (cid:126) − ( m +1) Vol(X h )Γ(m + 1) (cid:90) L ∗ | f ( Z ) | e −|| Z || / (cid:126) dV ol L ∗ ( Z ) . (80)8.2. Orthonormal basis. If s ∈ H ( M, L k ) then || ˆ s k || L ( X h ) = 1 m ! (cid:90) X h | ˆ s ( x ) | dV ( x ) = (cid:90) M || s ( z ) || h k dV ω , (81)where the right side is the inner product on H ( M, L k ), where dV ω = ω m /m !. Let N k = dim H ( M, L k )and let { ˆ s k,j } N k j =1 be any orthonormal basis of H k ( L ∗ ), corresponding to an orthonormal basis { s k,j } of H ( M, L k ). We let (cid:126) = k − . We also change the notation for powers of a bundle k → N to agree with thenotation for the real Harmonic oscillator but retain the notation (cid:126) = k − . Thus, in effect, there are twosemi-classical parameters: N and k , parallel to the parameters N and (cid:126) − for the Schr¨odinger representationof the harmonic oscillator. The lifts ˆ s N,j of an orthonormal basis s N,j of H ( M, L N ) are orthogonal but nolonger normalized. Lemma There exists a constant c m = (Vol(X h )Γ(m + 1)) − so that { c m (cid:126) − N/ s N,j ( Z ) √ ( N + m +1)! } is an or-thonormal basis of H BF .Proof. We have, || ˆ s N || BF, (cid:126) = || ˆ s || L ( X h ) C m (cid:126) − ( m +1) (cid:82) ∞ e − ρ / (cid:126) ρ N +2 m +1 dρ, = C m || ˆ s || L ( X h ) (cid:126) N Γ( N + m + 1) = C m (cid:126) N ( N + m )! || ˆ s || L ( X h ) , since (cid:126) − ( m +1) (cid:82) ∞ e − ρ / (cid:126) ρ N +2 m +1 dρ = (cid:126) N Γ( N + m + 1). Putting c m = C − m completes the proof. (cid:3) Corollary In the notation above, an orthonormal basis of H BF, (cid:126) ( L ∗ ) is given by { c m (cid:126) − N ˆ s N,j √ ( N + m )! } . Bargmann-Fock Bergman kernel of a line bundle. We now define the Bargmann-Fock Bergmankernel: Definition The Bargmann-Fock Bergman kernel is the kernel of the orthogonal projection, ˆΠ BF, (cid:126) : L ( L ∗ ) → H BF ( L ∗ ) , with respect to the Gaussian measure Γ m +1 , (cid:126) of the inner product (80) . The density of states is the positivemeasure, ˆΠ BF, (cid:126) ( Z, Z ) d Γ m +1 , (cid:126) ( Z )Let Π h N : L ( M, L N ) → H ( M, L N ) be the orthogonal projection with respect to the inner product(81). It lifts to the orthogonal projection ˆΠ N : L ( X h ) → H N ( X h ) with respect to the inner product on L ( X h ) defined by (81). Again by (80), ˆΠ N is equal up to the constant C N to the orthogonal projection H BF ( L ∗ ) → H N . The next Lemma is an immediate consequence of Corollary 8.3. NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 35 Lemma The Bargmann-Fock Bergman kernel on H BF ( L ∗ ) is given for Z = ( z, λ ) , W = ( w, µ ) ∈ L ∗ by ˆΠ BF, (cid:126) ( Z, W ) := c m (cid:80) ∞ N =0 (cid:126) − N ( N + m )! ˆΠ N ( Z, W ) = C m (cid:80) ∞ N =0 (cid:126) − N ( λµ ) N ( N + m )! ˆΠ N ( z, , w, , where the equivariant kernel ˆΠ N on X h is extended by homogeneity to L ∗ . The density of states is given by ˆΠ BF, (cid:126) ( Z, Z ) e −|| Z || / (cid:126) : = c m (cid:126) − ( m +1) e −|| Z || / (cid:126) (cid:80) ∞ N =0 (cid:126) − N ( N + m )! ˆΠ N ( Z, Z )= c m (cid:126) − ( m +1) e −|| Z || / (cid:126) (cid:80) ∞ N =0 (cid:126) − N | λ | N ( N + m )! Π h N ( z ) , where Π h N ( z ) is the metric contraction of Π N ( z, z ) on M . The following is the main result of this section: Proposition Let (cid:126) = k − . For Z = ( z, λ ) , the density of states equlas ˆΠ BF,k ( Z ) := c m k m +1 e − k || Z || ∞ (cid:88) N =0 | λ | N ( N + m )! k N N m [1 + O ( 1 N )] dV ol L ∗ ( Z ) . Proof. We recall that the density of states admits an asymptotic expansion,Π h N ( z ) (cid:39) N m m ! [1 + a ( z ) N + · · · ] , so by Lemma 8.5, the density of states equalsˆΠ BF, (cid:126) ( Z, Z ) d Γ m +1 , (cid:126) := c m (cid:126) − ( m +1) e −|| Z || / (cid:126) ∞ (cid:88) N =1 (cid:126) − N | λ | N ( N + m )! N m [1 + a ( z ) N + · · · ] dV ol L ∗ ( Z ) , where C m is a dimensional constant. Substituting (cid:126) = k − completes the proof. (cid:3) We note that N m ( N + m )! (cid:39) N ! , so that the asymptotics of Proposition 8.6 agree with the Bargmann-Fockcase (78).8.4. Interface asymptotics. The Hamiltonian is the norm square of the Hermitian metric itself, i.e. H ( z, λ ) = | λ | h z . The sublevel set { H ≤ E } is the disc bundle of radius E . We denote its boundary by Σ E . The normaldirection to Σ E is the gradient ∇ H direction, is given by the radial vector on L ∗ generated by the natural R + action in the fibers dual to the S action generated by H . Together, the R + and S actions define thestandard C ∗ action on L ∗ and ∇ H = Jξ H where ξ H = ∂∂θ is the Hamilton vector field of H . Thus, theasymptotics of such partial Bergman kernels falls into the C ∗ equivariant setting of [ZZ16].We fix E and consider the partial Bargmann-Fock Bergman kernel of L ∗ with the energy interval [0 , E ].Then as in the standard case, the exterior interface asymptotics pertain to the sums, (cid:88) N ≥ (cid:15)k Π h kBF ,N ( Z, Z ) = k m +1 ω m m ! e − k || Z || (cid:88) N ≥ (cid:15)k ( k || Z || ) N N m ( N + m )! [1 + a ( z ) N + · · · ] , (82)or to the complementary sums. Comparison with the standard Bargmann-Fock case of (78) shows thatthe agree to leading order, due to the Bergman kernel asymptotics of the summands Π N ( z, , z, H ( z, λ ) = | λ | and |∇ H ( z, λ ) | = | ∂∂θ | = λ .We refer to orbits of the R + action as radial orbits. Theorem Let Π h kBF , ( E, ∞ ] ( Z, Z ) = (cid:80) N ≥ Ek Π h kBF ,N ( Z, Z ) . Let Z = ( z, λ ) ∈ L ∗ and let Z E = ( z, λ E ) ∈ Σ E with | λ E | h z = E . Let Z k = e β √ k · Z E = ( z, e β √ k λ E ) be sequence of points approaching ( z, λ E ) along aradial R + orbit, where β ∈ R . Then, as k → ∞ , Π h kBF , ( E, ∞ ] ( Z k ) = k m Erf (cid:32) √ k E − e β √ k EE (cid:33) (1 + O ( k − / )) = k m Erf ( − β ) (1 + O ( k − / )) . (83) The proof of Theorem 8.7 is essentially the same as for Theorem 6, or better the same as in [ZZ16] for the C ∗ equivariant case. The only difference is that L ∗ is of infinite volume, but this does not affect pointwiseasymptotics. However, there is a more elementary proof in this case.Let x = | Z k | = | λ | h z = e β √ k Z E with | Z E | = E . It is well known that, as k → ∞ , e − kx (cid:88) N ≤ kE ( kx ) N N m ( N + m )! ∼ √ πx (cid:90) √ k E − x √ x −∞ e − t x dt. Indeed, Lemma 1 of [XX2] asserts that e − kx xk + y √ k (cid:88) N =1 ( kx ) N N ! ∼ √ π (cid:90) y √ x −∞ e − t dt + O ( Ax √ x + 1 √ k (( √ x + y ) ) . (84)We have, √ x = e β √ k E (cid:39) E + β √ k = ⇒ E − x √ x = E − e β √ k E e β √ k E = − E β √ k (1 + O ( 1 √ k )) . Then let kx + y √ k = kE , i.e. y √ k = E − x (cid:39) E β √ k , thus y = 2 βE , and use N m ( N + m )! (cid:39) N ! to obtain thedesired asymptotic.To see this asymptotic implies Theorem 8.7, we let √ k E − x √ x = β or E − x √ x = β √ k . Then we getΠ h kBF , ( E, ∞ ] ( Z k ) (cid:39) k m e − ke β √ k E (cid:88) N ≤ kE ( ke β √ k E ) N N m ( N + m )! ∼ k m √ πx (cid:90) β −∞ e − t E dt (1 + O ( 1 √ k ) Remark In [Sz50] , Szasz introduces the “Szasz operator” P f ( u, x ) := e − xu ∞ (cid:88) n =1 ( ux ) n n ! f ( nu ) , and shows that, for f ∈ C b ( R ) , lim u →∞ P f ( u, x ) = f ( x ) . If we let f ( v ) = [ E, ∞ ] ( v ) , then f ( nu ) = u ≤ nE .Szasz’s asymptotic does not apply at the point of discontinuity. Later, Mirjakan introduced the “Szasz-Mirjakan operator” [Mir] P f,N ( u, x ) := e − xu N (cid:88) n =1 ( ux ) n n ! f ( nu ) , and Omey [O] proved that if N = N ( n, x ) with lim n →∞ N − nx √ n = C < ∞ then lim n →∞ P f,N ( n, x ) = f ( x ) √ π (cid:82) C −∞ e − u du. [XX2, Lemma 1] is a refinement of this limit formula. This asymptotic formula arises in the analysis of Bernstein polynomials of discontinuous functions with ajump, and we refer to [Ch, Lev, O, Sz50, XX2] for the analysis.9. Further types of interface problems Further types of interface problems. Here are some further types of interface asymptotics: • Entanglement entropy: Sharp spectral cutoffs involve indicator functions E ,E ( ˆ H (cid:126) ) of a quantumHamiltonian. On the other hand, one might quantize the indicator function E ,E ( H ) of a classicalHamiltonian. This is obviously related but different, since the first is a projection and the secondis not. Entanglement entropy is a measure of how the second fails to be a projection and has beenstudied by Charles-Estienne [ChE18] and by the author (unpublished). • On a manifold M with boundary ∂M one may study the spectral projections kernel E D [0 ,λ ] ( x, x )of the Laplacian with Dirichlet boundary conditions. Away from ∂M , λ − n E D [0 ,λ ] ( x, x ) (cid:39) n = dim M . Yet E D [0 ,λ ] ( x, x ) = 0 on ∂M . What is the shape of the drop-off from 1 to 0 n a boundaryzone of width λ − ? NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 37 • For the hydrogen atom Hamiltonian ˆ H (cid:126) , there is a phase space interface Σ ⊂ T ∗ R d separating thebound states from the scattering states. The Hamiltonian flow is periodic on one side of Σ andunbounded on the other side and parabolic on Σ . The quantization of the bound state region is thediscrete spectral projection Π disc , (cid:126) ( x, y ). How does its Wigner distribution behave along Σ ? • Interfaces arise in the quantum Hall effect, a point process defined by a weight φ and a Laughlinstate which gives probabilities of N electrons to occur in a given configuration. The Laughlinstates concentrates as N → ∞ inside a ‘droplet’. The interface asymptotics across the droplet indimension one have been studied in [CFTW,Wieg] and others and from a mathematical point of viewby Hedenmalm and Wennmann [HW17, HW18]. In the next section, we discuss higher dimensionaldroplets. • Interfaces are studied for nonlinear equations such as the Allen-Cahn equation, and are related tophase transition problems; see e.g. [GG18] for references to the literature.9.2. Droplets in phase space. Let us describe droplets in more detail. Droplets in phase space arise ascoincidence sets in envelope problems for plurisubharmonic functions. The boundary of such coincidencesets is the interface. In special cases, it is the same interface that we have described for spectral interfaces.But in general, the interface is a free boundary that must be determined from the envelope, and even itsregularity is a problem. We refer to [Ber1] for the origins of the theory of dimensions > N ( h, ν ) induced by the data ( h, ν ) on the spaces H ( M, L N )of holomorphic sections of powers L N → M by || s || N ( h,ν ) := (cid:90) M | s ( z ) | h N dν ( z ) . (85)We let h be a general C Hermitian metric on L , and denote its positivity set by M (0) = { x ∈ M : ω φ | T x M has only positive eigenvalues } , (86)i.e. the set where ω φ is a positive (1 , 1) form. For a compact set K ⊂ M , also define the equilibrium potential φ eq = V ∗ h,K V ∗ h,K ( z ) = φ eq ( z ) := sup { u ( z ) : u ∈ P SH ( M, ω ) , u ≤ φ on K } , (87)where ω is a reference K¨ahler metric on M and P SH ( M, ω ) are the psh functions u relative to ω , P SH ( M, ω ) = { u ∈ L ( M, R ∪ ∞ ) : dd c u + ω ≥ , and u is ω − u.s.c. } . (88)Further define the coincidence set, D := { z ∈ M : φ ( z ) = φ e ( z ) } . (89)The boundary ∂D is the ‘interface’ and the problem is to determine its regularity and other properties. Itcarries an equilibrium measure defined by dµ φ = ( dd c φ eq ) m /m ! = D ∩ M (0) ( dd c φ ) m /m ! . (90)Here, d c = i ( ∂ − ¯ d ).Some droplets are classically forbidden regions for spectrally defined subpaces. The extent to which onemay construct a spectral problem with this property is unknown. Since the interface is usually only C , , itcannot be the level set (even a critical level) for a smooth (Morse-Bott) Hamiltonian in general.10. Appendix on K¨ahler analysis In this Appendix, we give a quick review of the basic notations of K¨ahler analysis. First we introduceco-circle bundle X ⊂ L ∗ for a positive Hermitian line bundle ( L, h ), so that holomorphic sections of L k fordifferent k can all be represented in the same space of CR-holomorphic functions on X , H ( X ) = ⊕ k H k ( X ).The Hamiltonian flow g t generated by ξ H on ( M, ω ) lifts to a contact flow ˆ g t generated by ˆ ξ H on X . Both notations φ eq and V ∗ h,K , and also P K ( φ ), are standard and we use them interchangeably. V ∗ h,K is called the pluri-complex Green’s function. Holomorphic sections in L k and CR-holomorphic functions on X . Let ( L, h ) → ( M, ω ) be apositive Hermitian line bundle, L ∗ the dual line bundle. Let X := { p ∈ L ∗ | (cid:107) p (cid:107) h = 1 } , π : X → M be the unit circle bundle over M .Let e L ∈ Γ( U, L ) be a non-vanishing holomorphic section of L over U , ϕ = − log (cid:107) e L (cid:107) and ω = i∂ ¯ dϕ .We also have the following trivialization of X : U × S ∼ = X | U , ( z ; θ ) (cid:55)→ e iθ e ∗ L | z (cid:107) e ∗ L | z (cid:107) . (91) X has a structure of a contact manifold. Let ρ be a smooth function in a neighborhood of X in L ∗ , suchthat ρ > ρ | X = 0 and dρ | X (cid:54) = 0. Then we have a contact one-form on Xα = − Re( i ¯ dρ ) | X , (92)well defined up to multiplication by a positive smooth function. We fix a choice of ρ by ρ ( x ) = − log (cid:107) x (cid:107) h , x ∈ L ∗ , then in local trivialization of X (91), we have α = dθ − d c ϕ ( z ) . (93) X is also a strictly pseudoconvex CR manifold. The CR structure on X is defined as follows: The kernelof α defines a horizontal hyperplane bundle HX := ker α ⊂ T X, (94)invariant under J since ker α = ker dρ ∩ ker d c ρ . Thus we have a splitting T X ⊗ C ∼ = H , X ⊕ H , X ⊕ C R. A function f : X → C is CR-holomorphic, if df | H , X = 0.A holomorphic section s k of L k determines a CR-function ˆ s k on X byˆ s k ( x ) := (cid:104) x ⊗ k , s k (cid:105) , x ∈ X ⊂ L ∗ . Furthermore ˆ s k is of degree k under the canonical S action r θ on X , ˆ s k ( r θ x ) = e ikθ ˆ s k ( x ). The inner producton L ( M, L k ) is given by (cid:104) s , s (cid:105) := (cid:90) M h k ( s ( z ) , s ( z )) d Vol M ( z ) , d Vol M = ω m m ! , and inner product on L ( X ) is given by (cid:104) f , f (cid:105) := (cid:90) X f ( x ) f ( x ) d Vol X ( x ) , d Vol X = α π ∧ ( dα ) m m ! . Thus, sending s k (cid:55)→ ˆ s k is an isometry.10.2. Szeg¨o kernel on X . On the circle bundle X over M , we define the orthogonal projection from L ( X )to the CR-holomorphic subspace H ( X ) = ˆ ⊕ k ≥ H k ( X ), and degree- k subspace H k ( X ):ˆΠ : L ( X ) → H ( X ) , ˆΠ k : L ( X ) → H k ( X ) , ˆΠ = (cid:88) k ≥ ˆΠ k . The Schwarz kernels ˆΠ k ( x, y ) of ˆΠ k is called the degree- k Szeg¨o kernel, i.e.( ˆΠ k F )( x ) = (cid:90) X ˆΠ k ( x, y ) F ( y ) d Vol X ( y ) , ∀ F ∈ L ( X ) . If we have an orthonormal basis { ˆ s k,j } j of H k ( X ), thenˆΠ k ( x, y ) = (cid:88) j ˆ s k,j ( x )ˆ s k,j ( y ) . NTERFACES IN SPECTRAL ASYMPTOTICS AND NODAL SETS 39 The degree- k kernel can be extracted as the Fourier coefficient of ˆΠ( x, y )ˆΠ k ( x, y ) = 12 π (cid:90) π ˆΠ( r θ x, y ) e − ikθ dθ. (95)We refer to (95) as the semi-classical Bergman kernels .10.3. Boutet de Monvel-Sj¨ostrand parametrix for the Szeg¨o kernel. Near the diagonal in X × X ,there exists a parametrix due to Boutet de Monvel-Sj¨ostrand [BSj] for the Szeg¨o kernel of the form,ˆΠ( x, y ) = (cid:90) R + e σ ˆ ψ ( x,y ) s ( x, y, σ ) dσ + ˆ R ( x, y ) . (96)where ˆ ψ ( x, y ) is the almost-CR-analytic extension of ˆ ψ ( x, x ) = − ρ ( x ) = log (cid:107) x (cid:107) , and s ( x, y, σ ) = σ m s m ( x, y )+ σ m − s m − ( x, y ) + · · · has a complete asymptotic expansion. In local trivialization (91),ˆ ψ ( x, y ) = i ( θ x − θ y ) + ψ ( z, w ) − ϕ ( z ) − ϕ ( w ) , where ψ ( z, w ) is the almost analytic extension of ϕ ( z ).10.4. Lifting the Hamiltonian flow to a contact flow on X h . In this seection we review the definitionof the lifting of a Hamiltonian flow to a contact flow, following [ZZ17, Section 3.1]. Let H : M → R be a Hamiltonian function on ( M, ω ). Let ξ H be the Hamiltonian vector field associated to H , such that dH = ι ξ H ω . The purpose of this section is to lift ξ H to a contact vector field ˆ ξ H on X . Let α denotethe contact 1-form (93) on X , and R the corresponding Reeb vector field determined by (cid:104) α, R (cid:105) = 1 and ι R dα = 0. One can check that R = ∂ θ . Definition (1) The horizontal lift of ξ H is a vector field on X denoted by ξ hH . It is determined by π ∗ ξ hH = ξ H , (cid:104) α, ξ hH (cid:105) = 0 . (2) The contact lift of ξ H is a vector field on X denoted by ˆ ξ H . It is determined by π ∗ ˆ ξ H = ξ H , L ˆ ξ H α = 0 . Lemma The contact lift ˆ ξ H is given by ˆ ξ H = ξ hH − HR. The Hamiltonian flow on M generated by ξ H is denoted by g t g t : M → M, g t = exp( tξ H ) . The contact flow on X generated by ˆ ξ H is denoted by ˆ g t ˆ g t : X → X, ˆ g t = exp( t ˆ ξ H ) . Lemma In local trivialization (91) , we have a useful formula for the flow, ˆ g t has the form (see [ZZ17, Lemma 3.2] ): ˆ g t ( z, θ ) = ( g t ( z ) , θ + (cid:90) t (cid:104) d c ϕ, ξ H (cid:105) ( g s ( z )) ds − tH ( z )) . Since ˆ g t preserves α it preserves the horizontal distribution H ( X h ) = ker α , i.e. D ˆ g t : H ( X ) x → H ( X ) ˆ g t ( x ) . (97)It also preserves the vertical (fiber) direction and therefore preserves the splitting V ⊕ H of T X . Its action inthe vertical direction is determined by Lemma 10.3. When g t is non-holomorphic, ˆ g t is not CR holomorphic,i.e. does not preserve the horizontal complex structure J or the splitting of H ( X ) ⊗ C into its ± i eigenspaces. Appendix Appendix on the Airy function. The Airy function is defined by, Ai ( z ) = 12 πi (cid:90) L e v / − zv dv, where L is any contour that beings at a point at infinity in the sector − π/ ≤ arg( v ) ≤ − π/ π/ ≤ arg( v ) ≤ π/ 2. In the region | arg z | ≤ (1 − δ ) π in C − { R − } write v = z + it on the upper half of L and v = z − it in the lower half. ThenAi( z ) = Ψ( z ) e − z / , with Ψ( z ) ∼ z − / ∞ (cid:88) j =0 a j z − j/ , a = 14 π − / . (98)11.2. Appendix on Laguerre functions . The Laguerre polynomials L αk ( x ) of degree k and of type α on[0 , ∞ ) are defined by e − x x α L αk ( x ) = 1 k ! d k dx k ( e − x x k + α ) . (99)They are solutions of the Laguerre equation(s), xy (cid:48)(cid:48) + ( α + 1 − x ) y ( x ) (cid:48) + ky ( x ) = 0 . For fixed α they are orthogonal polyomials of L ( R + , e − x x α dx ). An othonormal basis is given by L αk ( x ) = (cid:18) Γ( k + 1)Γ( k + α + 1) (cid:19) L αk ( x ) . We will have occasion to use the following generating function: ∞ (cid:88) k =0 L αk ( x ) w k = (1 − w ) − α − e − w − w x The most useful integral representation for the Laguerre functions is e − x/ L ( α ) n ( x ) = ( − n (cid:73) e − x · − z z z n (1 + z ) α +1 dz πiz , (100)where the contour encircles the origin once counterclockwise. 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