Superscars for arithmetic point scatters II

Pär Kurlberg; Stephen Lester; Lior Rosenzweig

doi:10.1017/fms.2023.33

Superscars for arithmetic point scatters II

Pär Kurlberg, Stephen Lester, Lior Rosenzweig

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Abstract

We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T2 = R2/Z2. Given any probability measure arising by placing delta masses, with equal weights, on Z2-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.

Original language	English
Article number	e37
Journal	Forum of Mathematics, Sigma
Volume	11
DOIs	https://doi.org/10.1017/fms.2023.33
Publication status	Published - 1 May 2023

Keywords

81Q50 58J51 37C83 81Q10

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10.1017/fms.2023.33

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title = "Superscars for arithmetic point scatters II",

abstract = "We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T2 = R2/Z2. Given any probability measure arising by placing delta masses, with equal weights, on Z2-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.",

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author = "P{\"a}r Kurlberg and Stephen Lester and Lior Rosenzweig",

note = "Funding Information: P.K. was partially supported by the Swedish Research Council (2016-03701,2020-04036). S.L. was partially supported by the Engineering and Physical Sciences Research Council (EP/T028343/1). We would also like to thank Dimitris Koukouloupolos, St{\'e}phane Nonnenmacher, Maksym Radziwi{\l}{\l}, Ze{\'e}v Rudnick and Steve Zelditch for very helpful discussions and suggestions. Publisher Copyright: {\textcopyright} The Author(s), 2023. Published by Cambridge University Press.",

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doi = "10.1017/fms.2023.33",

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T1 - Superscars for arithmetic point scatters II

AU - Kurlberg, Pär

AU - Lester, Stephen

AU - Rosenzweig, Lior

N1 - Funding Information: P.K. was partially supported by the Swedish Research Council (2016-03701,2020-04036). S.L. was partially supported by the Engineering and Physical Sciences Research Council (EP/T028343/1). We would also like to thank Dimitris Koukouloupolos, Stéphane Nonnenmacher, Maksym Radziwiłł, Zeév Rudnick and Steve Zelditch for very helpful discussions and suggestions. Publisher Copyright: © The Author(s), 2023. Published by Cambridge University Press.

PY - 2023/5/1

Y1 - 2023/5/1

N2 - We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T2 = R2/Z2. Given any probability measure arising by placing delta masses, with equal weights, on Z2-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.

AB - We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T2 = R2/Z2. Given any probability measure arising by placing delta masses, with equal weights, on Z2-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.

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Superscars for arithmetic point scatters II

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