PKPär Kurlberg

Pär Kurlberg

KTH Royal Institute of Technology
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Pär Kurlberg has a total of 65 co-authors. Here are the top co-authors by number of shared publications:

Latest Publications

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    • 14 July 2025
    • Published by HAL in HAL
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    Superscars in the Šeba billiard

    by Pär Kurlberg, Henrik Ueberschär

    We consider the Laplacian with a delta potential (also known as a "point scatterer", or "Fermi pseudopotential") on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes -"old" eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and "new" eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential.Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that they are localized along 4 wave vectors in momentum space -we therefore prove the existence of so-called "superscars" as predicted by Bogomolny and Schmit [6]. This result contrasts the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus [15]. Further, in the strong coupling limit we show that a weaker form of localization holds for an essentially full density subsequence of the new eigenvalues; in particular quantum ergodicity does not hold. We also explain how our results can be modified for rectangles with Dirichlet boundary conditions with a point scatterer in the interior. In this case our results extend previous work of Keating, Marklof and Winn who proved the existence of localized semiclassical measures under a clustering condition on the spectrum of the Laplacian.

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    On Quantum Ergodicity for Higher Dimensional Cat Maps

    by Pär Kurlberg, Alina Ostafe, Zeev Rudnick, Igor E. Shparlinski

    We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in $$\operatorname {Sp}(2g,{{\mathbb {Z}}})$$ Sp ( 2 g , Z ) , which we take to be ergodic. Under some natural assumptions, we show that there is a density one sequence of integers N so that as N tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant N are uniformly distributed. For the two-dimensional case ( $$g=1$$ g = 1 ), this was proved by Kurlberg and Rudnick (Duke Math J 103:47–78, 2000). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, such as a bound of Bourgain (J Am Math Soc 18:477–499, 2005) for Mordell sums, and a study of tensor product structures for the cat map, which has never been exploited in this context.

    Cited by 4
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    Poisson spacing statistics for lattice points on circles

    by Pär Kurlberg, Stephen Lester

    We show that along a density one subsequence of admissible radii, the nearest neighbor spacing between lattice points on circles is Poissonian.

    Cited by 5
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    Prime and Möbius correlations for very short intervals in $\fq[x]$

    by Pär Kurlberg, Lior Rosenzweig

    We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):=\{f(x) + a : a \in\fp\}$ for $f(x)\in\fp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\"{o}bius $\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M\"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\"{o}bius sums is {\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\"{o}bius $\mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)\in\fq[x]$ and intervals of the form $f(x)+a$ for $a\in\fq$, where $p$ is fixed and $q=p^{l}$ grows.

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    Superscars for arithmetic point scatters II

    by Pär Kurlberg, Stephen Lester, Lior Rosenzweig

    We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus ${\mathbb T}^2 = {\mathbb R}^2/{\mathbb Z}^{2}$ . Given any probability measure arising by placing delta masses, with equal weights, on ${\mathbb Z}^2$ -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.

    Cited by 3
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    Level Repulsion for Arithmetic Toral Point Scatterers in Dimension 3

    by Pär Kurlberg

    We show that arithmetic toral point scatterers in dimension three (“Šeba billiards on $${\mathbb R}^3/{\mathbb Z}^3$$ R 3 / Z 3 ”) exhibit strong level repulsion between the set of “new” eigenvalues. More precisely, let $$\Lambda := \{ \lambda _{1}< \lambda _{2} < \ldots \}$$ Λ : = { λ 1 < λ 2 < … } denote the unfolded set of new eigenvalues. Then, given any $$\gamma >0$$ γ > 0 , $$\begin{aligned} \frac{|\{ i \le N : \lambda _{i+1}-\lambda _{i} \le \epsilon \}|}{N} = O_{\gamma }(\epsilon ^{4-\gamma }) \end{aligned}$$ | { i ≤ N : λ i + 1 - λ i ≤ ϵ } | N = O γ ( ϵ 4 - γ ) as $$N \rightarrow \infty $$ N → ∞ (and $$\epsilon >0$$ ϵ > 0 small.) To the best of our knowledge, this is the first mathematically rigorous demonstration of a level repulsion phenomena for the quantization of a deterministic system.

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    The defect of toral Laplace eigenfunctions and arithmetic random waves

    by Pär Kurlberg, Igor Wigman, Nadav Yesha

    We study the defect (or ‘signed area’) distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or in a random Gaussian scenario (‘arithmetic random waves’). In either case we exploit the associated symmetry of the eigenfunctions to show that the expectation (spatial or Gaussian) vanishes. In the deterministic setting, we prove that the variance of the defect of flat eigenfunctions, restricted to balls shrinking above the Planck scale, vanishes for ‘most’ energies. Hence the defect of eigenfunctions restricted to most of the said balls is small. We also construct ‘esoteric’ eigenfunctions with large defect variance, by choosing our eigenfunctions so that to mimic the situation on the hexagonal torus, thus breaking the symmetries associated to the standard torus. In the random Gaussian setting, we establish various upper and lower bounds for the defect variance w.r.t. the Gaussian probability measure. A crucial ingredient in the proof of the lower bound is the use of Schmidt’s subspace theorem.

    Cited by 8
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    The Chebotarev density theorem for function fields — Incomplete intervals

    by Pär Kurlberg, Lior Rosenzweig

    Cited by 2
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    Superscars for Arithmetic Point Scatterers II

    by Pär Kurlberg, Stephen Lester, Lior Rosenzweig

    We consider momentum push-forwards of measures arising as quantum limits (semi-classical measures) of eigenfunctions of a point scatterer on the standard flat torus $\mathbb T^2 = \mathbb R^2/\mathbb Z^{2}$. Given any probability measure arising by placing delta masses, with equal weights, on $\mathbb Z^2$-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 is 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, e.g. singular continous 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.

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    The Chebotarev density theorem for function fields -- incomplete intervals

    by Pär Kurlberg, Lior Rosenzweig

    We prove a Polya-Vinogradov type variation of the the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" $I \subset \mathbb{F}_p$, provided $(p^{1/2}\log p)/|I| = o(1)$. Applications include density results for irreducible trinomials in $\mathbb{F}_p[x]$, i.e. the number of irreducible polynomials in the set $\{ f(x) = x^{d} + a_{1} x + a_{0} \in \mathbb{F}_p[x] \}_{a_{0} \in I_{0}, a_{1}\in I_{1}}$ is $\sim |I_{0}|\cdot |I_{1}|/d$ provided $|I_{0}| > p^{1/2+\epsilon}$, $|I_{1}| > p^{\epsilon}$, or $|I_{1}| > p^{1/2+\epsilon}$, $|I_{0}| > p^{\epsilon}$, and similarly when $x^{d}$ is replaced by any monic degree $d$ polynomial in $\mathbb{F}_p[x]$. Under the above assumptions we can also determine the distribution of factorization types, and find it to be consistent with the distribution of cycle types of permutations in the symmetric group $S_{d}$.

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