Interface asymptotics of Partial Bergman kernels around a critical level

Zelditch, Steve; Zhou, Peng

doi:10.48550/arxiv.1805.01804

Cited by 3 publications

(15 citation statements)

References 3 publications

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“…for each t and in particular for t = 2πk for some k ∈ Z. Using the trace asymptotics of [12], we prove (in the notation of Theorem 1.1):…”

Section: Introductionmentioning

confidence: 96%

“…After generalizing the averaging technique of Weinstein [10] and Guillemin [5], we prove the result in two ways: (i) using special parametrices for the isotropic propagators U(t) on L 2 (R d ) developed in [3], and (ii) by using an eigespace decomposition for H 0 and conjugating the traces of U(t) on eigenspaces to the Bargmann-Fock representation and then to holomorphic sections of line bundles over the space CP d−1 of Hamilton orbits of H 0 . It turns out that the eigenspace traces of U(t) involve the central limit scaling of Toeplitz propagators studied in [11,12]. We then use the trace asymptotics of [12] to obtain a second proof.…”

Section: Introductionmentioning

confidence: 96%

“…It turns out that the eigenspace traces of U(t) involve the central limit scaling of Toeplitz propagators studied in [11,12]. We then use the trace asymptotics of [12] to obtain a second proof. The two approaches are in some sense dual, the first leading to Fourier integral representations, and the second to Fourier series representations, of Tr U(t).…”

Section: Introductionmentioning

confidence: 96%

“…The advantage of the conjugation is that the spectral projections for H 0 conjugate to the simpler Bergman kernel projections for holomorphic sections of the standard ample line bundles over projective space. In that setting, Theorem 1.1 can be reduced to the trace asymptotics calculated in [12] for semi-classial Toeplitz operators acting on holomorphic sections of line bundles over Kähler manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…The time-scaled Toeplitz Fourier integral operators U N ( t √ N ) are studied in [11,12] for unrelated reasons, and the pointwise and integrated asymptotics of those articles gives Theorem 1.3.…”

Section: Introductionmentioning

confidence: 99%

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Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Doll

Gannot²,

Wunsch³

2018

Commun. Math. Phys.

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Let H = H 0 + P denote the harmonic oscillator on R d perturbed by an isotropic pseudodifferential operator P of order 1 and let U (t) = exp(−itH). We prove a Gutzwiller-Duistermaat-Guillemin type trace formula for Tr U (t). The singularities occur at times t ∈ 2πZ and the coefficients involve the dynamics of the Hamilton flow of the symbol σ(P ) on the space CP d−1 of harmonic oscillator orbits of energy 1. This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of Weinstein and Guillemin to this order of perturbation, and then present two completely different calculations of Tr U (t). The first proof directly constructs a parametrix of U (t) in the isotropic calculus, following earlier work of Doll-Gannot-Wunsch. The second proof conjugates the trace to the Bargmann-Fock setting, the order 1 of the perturbation coincides with the 'central limit scaling' studied by Zelditch-Zhou for Toeplitz operators.

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“…for each t and in particular for t = 2πk for some k ∈ Z. Using the trace asymptotics of [12], we prove (in the notation of Theorem 1.1):…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Doll

Gannot²,

Wunsch³

2018

Commun. Math. Phys.

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Strictly regular and cscK metrics

Loi,

Zudda

2020

Preprint

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A Kähler metric g with integral Kähler form is said to be partially regular if the partial Bergman kernel associated to mg is a positive constant for all integer m sufficiently large. The aim of this paper is to prove that for all n ≥ 2 there exists an n-complex dimensional manifold equipped with strictly partially regular and cscK metric g. Further, for n ≥ 3, the (constant) scalar curvature of g can be chosen to be zero, positive or negative.

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Interfaces in spectral asymptotics and nodal sets

Zelditch¹

2020

Preprint

Self Cite

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This is a survey of results obtained jointly with Boris Hanin and Peng Zhou on interfaces in spectral asymptotics, both for Schrödinger operators on L 2 (R d ) and for Toeplitz Hamiltonians acting on holomorphic sections of ample line bundles L → M over Kähler manifolds (M, ω). By an interface is meant a hypersurface, either in physical space R d or in phase space, separating an allowed region where spectral asymptotics are standard and a forbidden region where they are non-standard. The main question is to give the detailed transition between the two types of asymptotics across the hypersurface (i.e. interface). In the real Schrödinger setting, the asymptotics are of Airy type; in the Kähler setting they are of Erf (Gaussian error function) type.A principal purpose of this survey is to compare the results in the two settings. Each is apparently universal in its setting. This is now established for Toeplitz operators, but in the Schrödinger setting it is only established for the simplest model operator, the isotropic harmonic oscillator. It is explained that the latter result is most comparable to the behavior of the canonical degree operator on the Bargmann-Fock space of a line bundle, a new construction introduced in these notes.

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Interface asymptotics of Partial Bergman kernels around a critical level

Cited by 3 publications

References 3 publications

Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Refined Weyl Law for Homogeneous Perturbations of the Harmonic Oscillator

Strictly regular and cscK metrics

Interfaces in spectral asymptotics and nodal sets

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