Abstract. Let (M, g) be a compact, boundaryless manifold of dimension n with the property that either (i) n = 2 and (M, g) has no conjugate points, or (ii) the sectional curvatures of (M, g) are nonpositive. Let ∆ be the positive Laplacian on M determined by g. We study the L 2 → L p mapping properties of a spectral cluster of √ ∆ of width 1/ log λ. Under the geometric assumptions above, [1] Bérard obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a (log λ)
Let P = P (h) be a semiclassical pseudodifferential operator on a Riemannian manifold M . Suppose that u(h) is a localised, L 2 normalised family of functions such that P (h)u(h) is O(h) in L 2 , as h → 0. Then, for any submanifold Y ⊂ M , we obtain estimates on the L p norm of u(h) restricted to Y , with exponents that are sharp for h → 0. These results generalise those of Burq, Gérard and Tzvetkov [4] on L p norms for restriction of Laplacian eigenfunctions. As part of the technical development we prove some extensions of the abstract Strichartz estimates of Keel and Tao [7].Let P = P (h) be a semiclassical pseudodifferential operator on a Riemannian manifold M . We will assume that P has a real principal symbol, and that its full symbol is smooth in the semiclassical parameter h. Other more technical assumptions on P are given in Definition 1.6. We prove estimates for approximate solutions u = u(h) to the equation P (h)u(h) = 0. As usual in semiclassical analysis we assume that u(h) is defined at least for a sequence h n tending to zero.Our precise definition of approximate solution, or quasimode, is thatThis definition is natural with respect to localisation: if, and χ is a pseudodifferential operator of order zero (with a symbol smooth in h), then P (χu) is also O L 2 (h). We will make the assumption that u(h) can be localised, see Definition 1.3, and therefore will be able to reduce the problem to one of local analysis.Given a submanifold Y of M , we estimate the L p norm of the restriction of u to Y , assuming the normalisation condition u L 2 (M ) = 1. These estimates are of the form u L p (Y ) ≤ Ch −δ where δ depends on the dimension n of M , the dimension k of Y and p (except for one case where there is a logarithmic divergence) -see Theorem 1.7. In every case the exponent δ(n, k, p) given by Theorem 1.7 is optimal. Figure 1 shows the exponent δ for a hypersurface and, for comparison, the L p estimates over the whole manifold (Sogge [11] for spectral clusters and Koch-Tataru-Zworski [9] for semiclassical operators). The potential growth/concentration of the quasimodes of a semiclassical operator is of great interest due to the connection to Quantum Mechanics. It is from Quantum Mechanics that we get the important set of motivating examples,here ∆ g is the (positive) Laplace-Beltrami operator associated with the metric g. We can transition between this picture and the usual eigenfunction picture of Quantum Mechanics by dividing the eigenfunction equationwhere P is as in (1) with a potential term of h 2 V 0 (x) − 1. Therefore the higher eigenvalue asymptotics of eigenfunctions of Quantum Mechanical systems corresponds to the h → 0 limit in semiclassical analysis. When V 0 (x) = 0 this problem reduces to estimating the size of Laplacian eigenfunctions restricted to a submanifold. A complete set of estimates for Laplacian eigenfunctions on compact manifolds is given by Burq, Gérard and Tzvetkov [4].A
Abstract. Let M be a compact manifold of dimension n, P = P (h) a semiclassical pseudodifferential operator on M , andIn a previous article, the second-named author proved estimates on the L p norms, p ≥ 2, of u restricted to H, under the assumption that the u are semiclassically localised and under some natural structural assumptions about the principal symbol of P . These estimates are of the form Ch −δ(n,k,p) where k = dim H (except for a logarithmic divergence in the case k = n − 2, p = 2). When H is a hypersurface, i.e. k = n − 1, we have δ(n, n − 1, 2) = 1/4, which is sharp when M is the round n-sphere and H is an equator.In this article, we assume that H is a hypersurface, and make the additional geometric assumption that H is curved (in the sense of Definition 2.4 below) with respect to the bicharacteristic flow of P . Under this assumption we improve the estimate from δ = 1/4 to 1/6, generalising work of Burq-Gérard-Tzvetkov and Hu for Laplace eigenfunctions. To do this we apply the MelroseTaylor theorem, as adapted by Pan and Sogge, for Fourier integral operators with folding canonical relations.
For smooth bounded domains in R n , we prove upper and lower L 2 bounds on the boundary data of Neumann eigenfunctions, and prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of eigenvalue; this is achieved by working with an appropriate norm for boundary functions, which includes a 'spectral weight', that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for 'whispering gallery' type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru.Using this, we bound the distance from an arbitrary Helmholtz parameter E > 0 to the nearest Neumann eigenvalue, in terms of boundary normalderivative data of a trial function u solving the Helmholtz equation (∆−E)u = 0. This 'inclusion bound' improves over previously known bounds by a factor of E 5/6 . It is analogous to a recently improved inclusion bound in the Dirichlet case, due to the first two authors.Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the 42000th) from 9 digits to 14 digits, with little extra effort.
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