Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

Wolpert, Scott A.

doi:10.1090/s0002-9947-03-03154-4

Cited by 8 publications

(7 citation statements)

References 24 publications

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“…Systematic expositions in this context may be found in [Br81,G83,I02]. It is 'folklore', and proved also in this article, that |a n (λ k )| = O k,ǫ (n −N ) for all N ≥ 0 for n ≥ λ k + ǫ, and for any ǫ > 0; see also [Wo04,Xi19] for some estimates of this type. If one thinks of λ k as the 'energy' and n as the angular momentum, then rapid decay occurs in the "forbidden region" where the angular momentum exceeds the energy.…”

Section: Introductionmentioning

confidence: 60%

“…The case of c = 1 and H has non-degenerate second fundamental form requires quite different techniques from the totally geodesic case, and is under current investigation. Prior results in this case are the estimates on Fourier coefficients of cuspidal eigenfunctions restricted to closed horocycles are given in [Wo04]; see Section 2.6. We make a number of remarks on these cases but they are not studied in this article.…”

Section: Introductionmentioning

confidence: 99%

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Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

Wyman¹,

Xi²,

Zelditch³

2021

Preprint

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This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ϕ j of a compact Riemannian manifold to a submanifold H ⊂ M . We fix a number c ∈ (0, 1) and study the asymptotics of the thin sums,λ) := j,λj ≤λ k:|µ k −cλj |<ǫ H ϕ j ψ k dV H 2 where {λ j } are the eigenvalues of √ −∆ M , and {(µ k , ψ k )} are the eigenvalues, resp. eigenfunctions, of √ −∆ H . The inner sums represent the 'jumps' of N c ǫ,H (λ) and reflect the geometry of geodesic c-bi-angles with one leg on H and a second leg on M with the same endpoints and compatible initial tangent vectors ξ ∈ S c H M, π H ξ ∈ B * H, where π H ξ is the orthogonal projection of ξ to H. A c-bi-angle occurs when |πH ξ| |ξ| = c. Smoothed sums in µ k are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as ǫ varies, at certain values of ǫ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article [WXZ20] where the inner sums run over k : | µ k λj − c| ≤ ǫ and where geodesic bi-angles do not play a role.

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

Wyman¹,

Xi²,

Zelditch³

2021

Preprint

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“…It is known that the length-functions l c , l c ′ associated to non-intersecting curves c and c ′ Poisson-commute w.r.t. the Weil-Petersson symplectic form [21].…”

Section: Length-twist Coordinates -mentioning

confidence: 99%

“…. , l 3g−3+s } as the set of "action"-variables, it is amusing to note that the corresponding "angle"-variables are nothing but the twist-angles corresponding to the deformation of cutting Σ s g along c i and twisting by some angle ϕ i before gluing back along c i [21]. The set {l 1 , .…”

Section: Length-twist Coordinates -mentioning

confidence: 99%

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On the Relation Between Quantum Liouville Theory and the Quantized Teichmüller Spaces

Teschner

2004

Int. J. Mod. Phys. A

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We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture. Dedicated to A.A. Belavin on his 60 th birthday

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Semiclassical limits for the hyperbolic plane

Wolpert¹

2001

Duke Math. J.

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Research supported in part by NSF Grants DMS-9504176 and DMS-9800701 σ r Q (u, v)dV for σ r the symbol evaluated at τ = r and dV Haar measure [48,50]. We generalize a result of S. Zelditch [50, Prop.2.1] and show that the microlocal lift satisfies a partial differential equation; the equation is basic to our approach. A second microlocal lift is also considered.the assignment u to Q M (u) is equivariant for the left action on SL(2; R). An integration by parts argument provides a bound for the difference of Q (u, v) and Q M (u) for M large of order O(r −1 ) given uniform bounds for u 2m . The Fejér sum construction provides an alternative to introducing Friedrichs symmetrization, [9,48]. Our initial interest is the microlocal lift of the Macdonald-Bessel functions.The Macdonald-Bessel functions were introduced one-hundred years ago in a paper presented to the London Mathematical Society by H. M. Macdonald [33]. The paper includes a formula for the product of two Macdonald-Bessel functions in terms of the integral of a third Bessel function. We will develop formulas for the microlocal lift from such an identity [31,33].To present our main result we first describe a family of measures on SL(2; R). The square root (the double-cover) of the unit cotangent bundle of H is equivalent to SL(2; R). A geodesic has two unit cotangent fields and four square root unit cotangent fields. The geodesic-indicator measure ∆ αβ on SL(2; R) is the sum over the four lifts of the geodesic αβ of the lifted infinitesimal arc-length element. We further write ∆for the sum over Γ ∞ , the discrete group of integer translations. We consider for t = 2πnr −1 the microlocal lift of K(z, t) = (ry sinh πr) 1/2 K ir (2π|n|y)e 2πinx , and define the distributionfor ∆t = 2πr −1 . We show in Theorem 4.9 that for αβ the geodesic on H with Euclidean center the origin and radius |t| −1 and dV Haar measure then in the sense of tempered distributions Q symm (t)dV is close to π 2 8 ∆ Γ∞( αβ) uniformly for r large and |t| restricted to a compact subset of R + . At high-energy the microlocal lift of a Macdonald-Bessel function is concentrated along a single geodesic. Accordingly, at high-energy the behavior of the microlocal lift of a sum of Macdonald-Bessel functions is explicitly a matter of the space of geodesics and sums of coefficients. We introduce in Chapter 3 a coefficient summation scheme for studying quantities quadratic in the eigenfunction. The scheme provides a positive measure. In particular for (x, t) ∈ R × R + define the distributionfor d t denoting the Lebesgue-Stieljes derivative in t and for convolution inx with the Fejér kernel F N . We use a slight improvement of the J. M. Deshouillers-H. Iwaniec coefficient sum bound [11] and show for a unit-norm automorphic eigenfunction that Ω ϕ,N is a uniformly bounded tempered distributions. In fact for {ϕ j } a sequence of unit-norm automorphic eigenfunctions the sequences {Q(ϕ j , ϕ j )} and {Ω ϕ j ,N } are relatively compact. We consider a weak * convergent sequence and write Q limit = lim j Q...

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Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

Cited by 8 publications

References 24 publications

Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

On the Relation Between Quantum Liouville Theory and the Quantized Teichmüller Spaces

Semiclassical limits for the hyperbolic plane

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