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A spherical conical metric g on a surface Σ is a metric of constant curvature 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2π. The eigenfunctions of the Friedrichs Laplacian Δg with eigenvalue λ = 2 play a special role in this problem, as they represent local obstructions to deformations of the metric g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2-eigenfunctions, given in terms of a certain meromorphic data on Σ. As an application we give a description of all 2-eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2-eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2-eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2-eigenfunctions. [ABSTRACT FROM AUTHOR]

In this paper, we derive the sharp lower and upper bounds of nodal lengths of Laplacian eigenfunctions in the disc. [ABSTRACT FROM AUTHOR]

In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on triangles. We prove that the L2 norm of the (semiclassical) Neumann data on each side is equal to the length of the side divided by the area of the triangle. The novel feature of this result is that it is not an asymptotic, but an exact formula. The proof is by simple integrations by parts. [ABSTRACT FROM AUTHOR]

We give a characterization of the extremal sequences for the Bellman function of three variables of the dyadic maximal operator in relation to Kolmogorov's inequality. In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue. For this approach we use the one introduced in a paper by A. D. Melas and the author, where the respective Bellman function has been precisely evaluated. [ABSTRACT FROM AUTHOR]

In this paper we consider diffusions on the half line (0,∞) such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity which takes finite values at positive times. We study the behavior of hitting times of large barriers and, in a dual way, the behavior of the process starting at infinity for small time. In particular, we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion. [ABSTRACT FROM AUTHOR]

The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a long-standing open problem and has been solved so far only in a certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in which the waves propagate with near sonic speed, have large wave length, and carry low energy. In this paper we derive a similar result in a complementary asymptotic regime related to fast and strongly localized waves with high energy. In particular, we show that the spectrum of the linearized FPUT operator contains asymptotically no unstable eigenvalues except for the neutral ones that stem from the shift symmetry and the spatial discreteness. This ensures that high-energy waves are linearly stable in some orbital sense, and the corresponding nonlinear stability is granted by the general, non-asymptotic part of the seminal Friesecke-Pego result and the extension by Mizumachi. Our analytical work splits into two principal parts. First we refine two-scale techniques that relate high-energy waves to a nonlinear asymptotic shape ODE and provide accurate approximation formulas. In this way we establish the existence, local uniqueness, smooth parameter dependence, and exponential localization of fast lattice waves for a wide class of interaction potentials with algebraic singularity. Afterwards we study the crucial eigenvalue problem in exponentially weighted spaces, so that there is no unstable essential spectrum. Our key argument is that all proper eigenfunctions can asymptotically be linked to the unique bounded and normalized solution of the linearized shape ODE, and this finally enables us to disprove the existence of unstable eigenfunctions in the symplectic complement of the neutral ones. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ RN be an open bounded connected set. We consider the eigenvalue problem -Δpu = λρ∣u∣p-2u in Ω with homogeneous Dirichlet boundary condition, where Δp is the p-Laplacian operator and ρ is an arbitrary function that takes only two given values 0 < α < β and that is subject to the constraint ∫Ω ρdx = αγ +β(∣Ω∣-γ) for a fixed 0 < γ < ∣Ω∣. The optimization of the map ρ ↔ λ1(ρ), where λ1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω and we show that the minimizers inherit the same symmetry. [ABSTRACT FROM AUTHOR]

Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let Ω be a C2 bounded region in Rn (n = 2, 3). The double layer potential K : L2(∂Ω) → L2(∂Ω) is defined by (Kψ)(x) ≡ ∫ ∂ Ω ψ(y)·vyE(x, y) dsy, where E(x, y) = ∫1/2π log1/∣x-y∣ , if n = 2, 1/π log1/∣x-y∣ , if n = 3, dsy is the line or surface element and vy is the outer normal derivative on ∂Ω. It is known that K is a compact operator on L2(∂Ω) and consists of at most a countable number of eigenvalues, with 0 as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of ∂Ω. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any Hs (Ω) solution u of the problem s (0.1) (-Δ)su = f(u) in Ω,u = 0 on ∂ύ, belongs to L∞ (Ω) for the nonlinearity of f(t) being subcritical and critical. This implies that the solution u is classical if f(t) is C1, γ for some 0 < γ < 1. [ABSTRACT FROM AUTHOR]

In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifolds by Seeley in 1969, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors (2014). [ABSTRACT FROM AUTHOR]

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus (ℝ/ℤ)³, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant- sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the second eigenfunctions of the p- Laplacian, p > 1, are not radial on the unit ball in RN, for any N ⩾ 2. Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs {τn,ψn} such that ψn is nonradial and has exactly 2n nodal domains. A few related open problems are also stated. [ABSTRACT FROM AUTHOR]

In this paper, we consider a blow-up solution for the complexvalued semilinear wave equation with power non-linearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter family. Then, we use a dynamical system formulation to show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic case. This gives the blow-up profile for the original equation in the non-characteristic case. Our analysis is not just a simple adaptation of the already handled real case. In particular, there is one more neutral-direction in our problem, which we control thanks to a modulation technique. [ABSTRACT FROM AUTHOR]

Let (M,g) be an ra-dimensional compact boundaryless Riemannian manifold with nonpositive sectional curvature. Then our conclusion is that we can give improved estimates for the Lp norms of the restrictions of eigenfunctions of the Laplace-Beltrami operator to smooth submanifolds of dimension k, for p > 2n/n - 1 when k = n -- 1 and p > 2 when k ≤ n -- 2, compared to the general results of Burq, Gérard and Tzvetkov. Earlier, Bérard gave the same improvement for the case when p = ∞, for compact Riemannian manifolds without conjugate points for n = 2, or with nonpositive sectional curvature for n ≥ 3 and k = n -- 1. In this paper, we give the improved estimates for n = 2, the Lp norms of the restrictions of eigenfunctions to geodesics. Our proof uses the fact that the exponential map from any point in x Є M is a universal covering map from ℝ² ≃ TxM to M, which allows us to lift the calculations up to the universal cover (ℝ²,g), where g is the pullback of g via the exponential map. Then we prove the main estimates by using the Hadamard parametrix for the wave equation on (ℝ²,g), the stationary phase estimates, and the fact that the principal coefficient of the Hadamard parametrix is bounded, by observations of Sogge and Zelditch. The improved estimates also work for n ≥ 3, with p > 4k/n -1. We can then get the full result by interpolation. [ABSTRACT FROM AUTHOR]

In this paper we provide a short proof of the fact that if X is a Banach space and f : X → ℝ υ {∞} is a proper function such that f -- x* attains its minimum for every x* ∈ X*, then all the sublevels of f are relatively weakly compact. This result has many applications [ABSTRACT FROM AUTHOR]

We study in one dimension the semi-classical limit of the exact eigenfunction $\Psi _{E(N,h)}^{h}$ of the Hamiltonian $H=-\frac{1}{2} h^{2} \Delta +V(x)$, for a potential $V$ being analytic, bounded below and $\lim _{|x|\to \infty }V(x)=+\infty $. The main result of this paper is that, for any given $E>\min _{x\in R^{1}} V(x)$ with two turning points, the exact $L^{2}$ normalized eigenfunction $|\Psi ^{h}_{E(N,h)}(q)|^{2}$ converges to the classical probability density, and the momentum distribution $|\hat \Psi ^{h}_{E(N,h)}(p)|^{2}$ converges to the classical momentum density in the sense of distribution, as $h\to 0$ and $N\to\infty$ with $(N+\frac{1}{2} )h =\frac{1}{\pi } \int _{V(x)

For 2\leq p<4, we study the L^p norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact 2-dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov [Duke Math. J. 138 (2007), pp. 445–486], and Hu [Forum Math. 21 (2009), pp. 1021–1052] found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For p=4, we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [Comm. Math. Phys. 350 (2017), pp. 1299–1325], Sogge [Math. Res. Lett. 24 (2017), pp. 549–570], and Bourgain [Geom. Funct. Anal. 1 (1991), pp. 147–187]. [ABSTRACT FROM AUTHOR]

We show that in two dimensional flat torus the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when the randomness is not gaussian. [ABSTRACT FROM AUTHOR]

This paper estimates the size of the first Dirichlet eigenfunction of a convex planar domain. The eigenfunction is shown to be well-approximated, uniformly for all convex domains, by the first Dirichlet eigenfunction of a naturally associated ordinary differential (Schrödinger) operator. In particular, the place where the eigenfunction attains its maximum is located to within a distance comparable to the inradius. [ABSTRACT FROM AUTHOR]

Motivated by a problem on conditions for the existence of clines in genetics, we show that the positive zeros of a cross-product of the regular Coulomb wave function and the Tricomi hypergeometric function are increasing with respect to one of the parameters. In particular, this implies that the eigenvalues of a certain boundary value problem are increasing with the dimension. [ABSTRACT FROM AUTHOR]

Using symmetrization techniques, we show that, for every N \geq 2, any second eigenfunction of the fractional Laplacian in the N-dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball. [ABSTRACT FROM AUTHOR]

In this note we consider the stationary Kirchhoff equation { -M(‖u‖2)Δu = f(x, u) inΩ, u = 0 on ∂Ω, where M is a continuous positive function and ‖ . ‖ is the standard norm in H10 (Ω). We show that the equation does not enjoy the usual comparison principles (both weak or strong) nor the sub and supersolutions method. [ABSTRACT FROM AUTHOR]

We study the localization effect for the eigenfunctions of the Laplace-Dirichlet problem in a thin three-dimensional plate with curved non-smooth bases. We show that the eigenfunctions are localized at the thickest region, or the longest traverse axis, of the plate and that the magnitude of the eigenfunctions decays exponentially as a function of the distance to this axis. We consider some extensions like mixed boundary value problems in thin domains. The obtained asymptotic formulas for eigenfunctions prove the existence of gaps in the essential spectrum of the Dirichlet Laplacian in an unbounded double-periodic curved piecewise smooth thin layer. [ABSTRACT FROM AUTHOR]

Many data-science problems can be formulated as an inverse problem, where the parameters are estimated by minimizing a proper loss function. When complicated black-box models are involved, derivative-free optimization tools are often needed. The ensemble Kalman filter (EnKF) is a particle-based derivative-free Bayesian algorithm originally designed for data assimilation. Recently, it has been applied to inverse problems for computational efficiency. The resulting algorithm, known as ensemble Kalman inversion (EKI), involves running an ensemble of particles with EnKF update rules so they can converge to a minimizer. In this article, we investigate EKI convergence in general nonlinear settings. To improve convergence speed and stability, we consider applying EKI with non-constant step-sizes and covariance inflation. We prove that EKI can hit critical points with finite steps in non-convex settings. We further prove that EKI converges to the global minimizer polynomially fast if the loss function is strongly convex. We verify the analysis presented with numerical experiments on two inverse problems. [ABSTRACT FROM AUTHOR]

We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov's theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C^{1+} regularity of the stable/unstable foliations. [ABSTRACT FROM AUTHOR]

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix–Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart–Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart–Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions. [ABSTRACT FROM AUTHOR]

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips, and so forth. A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic 0, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders (0,π) × S1r where r ∈ {0.5,1} is the radius of the circle S1r, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues. [ABSTRACT FROM AUTHOR]

Hill's method is a means to numerically approximate spectra of linear differential operators with periodic coefficients. In this paper, we address different issues related to the convergence of Hill's method. We show the method does not produce any spurious approximations, and that for self-adjoint operators, the method converges in a restricted sense. Furthermore, assuming convergence of an eigenvalue, we prove convergence of the associated eigenfunction approximation in the $ L^2$ [ABSTRACT FROM AUTHOR]

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and eigenvalue $-mu $, then $$d_{TV}(f(X),Z)le frac {2}{mu }mathbb {E}big ||nabla f(X)|^2-mathbb {E}|nabla f(X) |^2big |.$$ This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori. [ABSTRACT FROM AUTHOR]

Let $lambda (n)$ be the $n$th normalized Fourier coefficient of a holomorphic Hecke eigenform $f(z)in S_{k}(Gamma )$. In this paper we are interested in the average behavior of $lambda ^2(n)$ over sparse sequences. By using the properties of symmetric power $L$-functions and their Rankin-Selberg $L$-functions, we are able to establish that for any $varepsilon >0$, $$sum _{n leq x}lambda ^2(n^j)=c_{j-1} x+Oleft (x^{1-frac {2}{(j+1)^2+2}+varepsilon }right ),$$ where $j=2,3,4.$ [ABSTRACT FROM AUTHOR]

The article presents a study which aims to demonstrate the spectral decomposition of real symmetric quadratic λ-matrices. It says that spectral decomposition provides a basic operator representation in a vector space in terms of its eigenvalues and eigenfunctions in which the Gohberg, Lancaster, and Rodman (GLR) theory generalizes eigendecomposition to matrix polynomials. The results revealed that the existence of real spectral decomposition can be shown through real standard pair.

There exists a generalized Hankel transform of order $\alpha\geq -1/2$ on $\mathbb{R}$, which is based on the eigenfunctions of the Dunkl operator \[\qquad\quad\quad T_\alpha f(x) = f^\prime(x) +\bigl(\alpha+\frac{1}{2}\bigr) \frac{f(x)-f(-x)}{x},\quad f\in C^1(\mathbb{R}).\qquad\qquad\qquad] For $\alpha=-1/2$ this transform coincides with the usual Fourier transform on $\mathbb{R}$. In this paper the operator $T_\alpha$ replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on $\mathbb{R}$. It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on $L^2(\mathbb{R})$; moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order $\alpha\geq -1/2$ on $[0,\infty[.$ [ABSTRACT FROM AUTHOR]

In the present paper, deformations of harmonic 2-spheres in the unit $n$-sphere $S^{n}$ respecting the degree are studied. The limit maps of such deformations are characterized as harmonic maps with extra eigenfunctions. The space $Harm_{d}(S^{2},S^{n})$ of harmonic 2-spheres in $S^{n}$ with fixed degree $d$ is described in terms of such deformations and the limit maps. [ABSTRACT FROM AUTHOR]

We present an iterative method based on repeatedly inverting the Monge-Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain Ω ⊂ Rn. We prove that the iterates uk generated by this method converge as k → ∞ to a solution of the Monge-Ampère eigenvalue problem {detD2u = λMA (−u)n in Ω, u = 0 on ∂ Ω. Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates uk := uk/||uk||L∞(Ω) converge to the eigenfunction of unit height. In addition, we show that limk → ∞ R(uk) = limk → ∞ R(uk) = λMA, where the Rayleigh quotient R(u) is defined as R(u) := ∫Ω (−u) det D2u/∫Ω (−u)n+1. Our method converges for a wide class of initial choices u0 that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Ampère eigenvalue λMA. [ABSTRACT FROM AUTHOR]

We prove a multiplier theorem of Mihlin-Hörmander-type for operators of the form −Δx − V(x) Δy on Rd1x × Rd2y, where V(x) = ∑j=1d1 Vj(xj), the Vj are perturbations of the power law t → ||t||2σ, and σ ∈ (1/2,∞). The result is sharp whenever d1 ≥ σ d2. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schrödinger operators, which are stable under perturbations of the potential. [ABSTRACT FROM AUTHOR]

We consider Riemannian symmetric spaces X of noncompact-type with rank one, which accommodates all hyperbolic spaces. We characterize the eigenfunctions of the Laplace-Beltrami operator on X with arbitrary complex eigenvalues through an asymptotic version of the ball mean value property as the radius of the ball tends to infinity. [ABSTRACT FROM AUTHOR]

We consider a compact Riemannian manifold M (possibly with boundary) and Σ ⊂ M \ ∂M an interior hypersurface (possibly with boundary). We study observation and control from Σ for both the wave and heat equations. For the wave equation, we prove controllability from Σ in time T under the assumption (TGCC) that all generalized bicharacteristics intersect Σ transversally in the time interval (0,T). For the heat equation we prove unconditional controllability from Σ. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on Σ under TGCC and unconditional exponential lower bounds on such Cauchy data. [ABSTRACT FROM AUTHOR]

We prove that if Ω is an open bounded domain with smooth and connected boundary, for every p ∈ (1, + ∞) the first Dirichlet eigenvalue of the normalized p-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (nonoptimal) lower bound for the eigenvalue in terms of the measure of Ω, and we address the open problem of proving a Faber-Krahn-type inequality with balls as optimal domains. [ABSTRACT FROM AUTHOR]

Let X be an irreducible m-symmetric Markov process on E with strong Feller property. In addition, suppose X has a tightness property. We then show that the semigroup of X is a compact operator on L²(E;m) and every eigenfunction has a bounded continuous version. [ABSTRACT FROM AUTHOR]

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on T³ = R³/Z³ (three-dimensional “arithmetic random waves”). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to the two-dimensional case addressed by Marinucci et al., [Geom. Funct. Anal. 26 (2016), pp. 926–960] the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results from Benatar and Maffiucci [Int. Math. Res. Not. IMRN (to appear)] that establish an upper bound for the number of nondegenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result. [ABSTRACT FROM AUTHOR]

We prove a stability estimate for the functions that are almost extremals for the Bellman function related to the Lp norm of the dyadic maximal operator in the case p ≥ 2. This estimate gives that such almost extremals are also almost "eigenfunctions" for the dyadic maximal operator, in the sense that the Lp distance between the maximal operator applied to the function and a certain multiple of the function is small. [ABSTRACT FROM AUTHOR]

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequalities and restricted Sogge-type Lp moment estimates of eigenfunctions. [ABSTRACT FROM AUTHOR]

We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree k ≥ 0. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as h2t and the eigenfunctions as ht in the H1-seminorm, where h is the mesh-size, t ∈ [s, k + 1] depends on the smoothness of the eigenfunctions, and s > 1/2 results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus h2k+2 for the eigenvalues and hk+1 for the eigenfunctions. Our theoretical findings, which improve recent error estimates for Hybridizable Discontinuous Galerkin (HDG) methods, are verified on various numerical examples including smooth and non-smooth eigenfunctions. Moreover, we observe numerically in one dimension for smooth eigenfunctions that the eigenvalues superconverge as h2k+4 for a specific value of the stabilization parameter. [ABSTRACT FROM AUTHOR]

We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point, respectively, continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier-Bohr coefficients of weakly almost periodic measures. [ABSTRACT FROM AUTHOR]

In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results [Anal. PDE 11 (2018), 855-871], which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by Nonlinearity 28 (2015), 3263-3288, Adv. Math. 290 (2016), 938-966 small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get logarithmic improvements on the inner radius of eigenfunctions on such manifolds. We also get improvements for manifolds with ergodic geodesic flows. In addition using the small scale equidistribution results of Comm. Math. Phys. 350 (2017), 279-300, one gets polynomial betterments of Comm. Partial Differential Equations 33 (2008), 1611-1621 for toral eigenfunctions in dimensions n ≥ 3. The results work only for a full density subsequence of eigenfunctions. [ABSTRACT FROM AUTHOR]

We prove the Plancherel formula for a four-parameter family of discrete Fourier transforms and their multivariate generalizations stemming from corresponding generalized Schur polynomials. For special choices of the parameters, this recovers the sixteen classic discrete sine- and cosine transforms DST-1,. . .,DST-8 and DCT-1,. . .,DCT-8, as well as recently studied (anti)symmetric multivariate generalizations thereof. [ABSTRACT FROM AUTHOR]

In this note we analyse Lp estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular, we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition, we produce a family of flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check Lp growth estimates against the simple flat model examples. Finally, we present a heuristic that for any subset determines which quasimode in the family is expected to produce sharp examples. [ABSTRACT FROM AUTHOR]

Let G be a smooth connected linear algebraic group and let X be a G-torsor. Totaro asked: if X admits a zero-cycle of degree d ≥ 1, then does X have a closed étale point of degree dividing d? This question is entirely unexplored in the literature for algebraic tori. We settle Totaro’s question affirmatively for algebraic tori of rank ≤ 2. [ABSTRACT FROM AUTHOR]

We study the first Dirichlet eigenfunction of a class of Schrödinger operators with a convex, non-negative, potential V on a convex, planar domain Ω. In the case where the diameter of Ω is large and the potential V varies on different length scales in orthogonal directions, we find two length scales L1 and L2 and an orientation of the domain Ω which determine the shape of the level sets of the eigenfunction. As an intermediate step, we also establish bounds on the first eigenvalue in terms of the first eigenvalue of an associated ordinary differential operator. [ABSTRACT FROM AUTHOR]