### Abstract

In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels Π_{k,[E1},E_{2} ](z, w) and the associated DOS (density of states) Π_{k,[E1},E_{2} ](z) across the interface C between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions H:M →ℝ on a Kähler manifold. The allowed region is H^{−1}([E_{1},E_{2} ]) and the interface C is its boundary. In prior articles it was assumed that the endpoints E_{j} were regular values of H. This article completes the series by giving parallel results when an endpoint is a critical value of H. In place of the Erf scaling asymptotics in a k^{−1}2 tube around C for regular interfaces, one obtains δ-asymptotics in k^{−1}4-tubes around singular points of a critical interface. In k^{−1}2 tubes, the transition law is given by the osculating metaplectic propagator.

Original language | English (US) |
---|---|

Pages (from-to) | 471-492 |

Number of pages | 22 |

Journal | Arkiv for Matematik |

Volume | 57 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Arkiv for Matematik*,

*57*(2), 471-492. https://doi.org/10.4310/ARKIV.2019.v57.n2.a12

}

*Arkiv for Matematik*, vol. 57, no. 2, pp. 471-492. https://doi.org/10.4310/ARKIV.2019.v57.n2.a12

**Interface asymptotics of Partial Bergman kernels around a critical level.** / Zelditch, Steve; Zhou, Peng.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Interface asymptotics of Partial Bergman kernels around a critical level

AU - Zelditch, Steve

AU - Zhou, Peng

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels Πk,[E1,E2 ](z, w) and the associated DOS (density of states) Πk,[E1,E2 ](z) across the interface C between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions H:M →ℝ on a Kähler manifold. The allowed region is H−1([E1,E2 ]) and the interface C is its boundary. In prior articles it was assumed that the endpoints Ej were regular values of H. This article completes the series by giving parallel results when an endpoint is a critical value of H. In place of the Erf scaling asymptotics in a k−12 tube around C for regular interfaces, one obtains δ-asymptotics in k−14-tubes around singular points of a critical interface. In k−12 tubes, the transition law is given by the osculating metaplectic propagator.

AB - In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels Πk,[E1,E2 ](z, w) and the associated DOS (density of states) Πk,[E1,E2 ](z) across the interface C between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions H:M →ℝ on a Kähler manifold. The allowed region is H−1([E1,E2 ]) and the interface C is its boundary. In prior articles it was assumed that the endpoints Ej were regular values of H. This article completes the series by giving parallel results when an endpoint is a critical value of H. In place of the Erf scaling asymptotics in a k−12 tube around C for regular interfaces, one obtains δ-asymptotics in k−14-tubes around singular points of a critical interface. In k−12 tubes, the transition law is given by the osculating metaplectic propagator.

UR - http://www.scopus.com/inward/record.url?scp=85074101574&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85074101574&partnerID=8YFLogxK

U2 - 10.4310/ARKIV.2019.v57.n2.a12

DO - 10.4310/ARKIV.2019.v57.n2.a12

M3 - Article

AN - SCOPUS:85074101574

VL - 57

SP - 471

EP - 492

JO - Arkiv for Matematik

JF - Arkiv for Matematik

SN - 0004-2080

IS - 2

ER -