arXiv subjects

Suresh Eswarathasan

Publications and source records attributed to Suresh Eswarathasan.

18 recordsLinked to original sources

arXiv2024

Pointwise Weyl Laws for Quantum Completely Integrable Systems

The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of Hörmander, following important prior contributions by Gärding, Levitan, Avakumović, and Agmon-Kannai (to name only some), obtained pointwise asymptotics (or a "pointwise Weyl law") for a single elliptic, self-adjoint operator. Here, we establish a microlocalized pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, $\overline{P}=(P_1,P_2,\dots, P_n)$, where $P_i$ are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold $M^n$, with $\sum P_i^2$ elliptic and $[P_i,P_j]=0$ for $1\leq i,j\leq n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-Δ)^\frac12$. We illustrate our result with several examples, including surfaces of revolution.

math.AP↗

arXiv2024

Entropy of logarithmic modes

Let $(M,g)$ be a compact, boundaryless, Riemannian manifold whose geodesic flow on its unit sphere bundle is Anosov. Consider the (semiclassical) Laplace-Beltrami operator on $M$. Let $ε>0$. We study the semiclassical measures $μ_{sc}$ of quasimodes spectrally supported in intervals of width $ε\frac{h}{|\log h|}$, a critical-type regime when considering ``delocalization". We derive a lower bound for the Kolmogorov-Sinai entropy of $μ_{sc}$ that depends explicitly on $ε$, in the spirit of that given by Ananthamaran-Koch-Nonnenmacher.

math.SP↗

arXiv2023

Nodal sets of eigenfunctions of sub-Laplacians

Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In this note, we initiate the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians. A standard example is the sum of squares of bracket-generating vector fields on compact quotients of the Heisenberg group. Our results show that nodal sets behave in an anisotropic way which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation and desingularization at singular points. Furthermore, we provide a simple example demonstrating that for sub-Laplacians, the Hausdorff measure of nodal sets of eigenfunctions cannot be bounded above by $\sqrtλ$, which is the bound conjectured by Yau for Laplace-Beltrami operators on smooth manifolds.

math.AP↗

arXiv2021

Fractal uncertainty principle for discrete Cantor sets with random alphabets

In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the corresponding Cantor sets have a fixed dimension 0<log A/log M<2/3. We prove that the FUP with an improved exponent over Dyatlov-Jin (arXiv:1608.02238) holds for almost all alphabets, asymptotically as M tends to infinity. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additive energy assumption. The proof is based on a concentration of measure phenomenon in the space of alphabets.

math.CA↗

arXiv2021

Laplace-Beltrami spectrum of ellipsoids that are close to spheres and analytic perturbation theory

We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $N$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteen eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to our results from the analytic perturbation theory, and approximations for the eigenvalues of ellipsoids that degenerate into infinite cylinders or two-dimensional disks. We propose a conjecture on the exact number of nodal domains of near-sphere ellipsoids.

math.SP↗

arXiv2020

Restriction of Laplace-Beltrami eigenfunctions to arbitrary sets on manifolds

Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets $Γ$ of $M$. The sets $Γ$ that we consider are Borel measurable, Lebesgue-null but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $Γ$. For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in \cite{Ho68, Ho71, Sog88, BGT07}. Under an additional measure-theoretic assumption on $Γ$, the estimates are shown to be sharp in this range. As evidence of the genericity of the sharp estimates, we provide a large family of random, Cantor-type sets that are not submanifolds, where the above-mentioned sharp bounds hold almost surely.

math.AP↗

arXiv2020

Effective bounds for monochromatic connectivity measures in two dimensions

We establish numerical lower bounds for the monochromatic connectivity measure in two dimensions introduced by Sarnak and Wigman. This measure dictates among the nodal domains of a random plane wave what proportion have any given number of holes, and how they are nested. Our bounds provide the first effective estimate for the number of simply connected domains and for those that contain a single hole. The deterministic aspect of the proof is to find a single function with a prescribed zero set and, using a quantitative form of the implicit function theorem, to argue that the same configuration occurs in the zero set of any sufficiently close approximation to this function. The probabilistic aspect is to quantify the likelihood of a random wave being close enough to this function.

math.PR↗

arXiv2019

Direction distribution for nodal components of random band-limited functions on surfaces

Let $(M,g)$ be a smooth compact Riemannian surface with no boundary. Given a smooth vector field $V$ with finitely many zeroes on $M$, we study the distribution of the number of tangencies to $V$ of the nodal components of random band-limited functions. It is determined that in the high-energy limit, these obey a universal deterministic law, independent of the surface $M$ and the vector field $V$, that is supported precisely on the even integers $2 \mathbb{Z}_{> 0}$.

math.PR↗

arXiv2018

Tangent nodal sets for random spherical harmonics

In this note, we consider a fixed vector field $V$ on $S^2$ and study the distribution of points which lie on the nodal set (of a random spherical harmonic) where $V$ is also tangent. We show that the expected value of the corresponding counting function is asymptotic to the eigenvalue with a leading coefficient that is independent of the vector field $V$. This demonstrates, in some form, a universality for vector fields up to lower order terms.

math.SP↗

arXiv2017

Scarring of quasimodes on hyperbolic manifolds

Let $N$ be a compact hyperbolic manifold, $M\subset N$ an embedded totally geodesic submanifold, and let $-\hbar^2Δ_{N}$ be the semiclassical Laplace--Beltrami operator. For any $\varepsilon>0$, we explicitly construct families of \emph{quasimodes} of spectral width at most $\varepsilon\frac{\hbar}{|\log\hbar|}$ which exhibit a "strong scar" on $M$ in that their microlocal lifts converge weakly to a probability measure which places positive weight on $S^*M$ ($\hookrightarrow S^*N$). An immediate corollary is that \emph{any} invariant measure on $S^*N$ occurs in the ergodic decomposition of the semiclassical limit of certain quasimodes of width $\varepsilon \frac{\hbar}{|\log\hbar|}$

math.AP↗

arXiv2015

Intersections of sets, diophantine equations and Fourier analysis

A classical theorem due to Mattila (see \cite{Mat84}; see also \cite{M95}, Chapter 13) says that if $A,B \subset {\Bbb R}^d$ of Hausdorff dimension $s_A, s_B$, respectively, with $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$ and $dim_{\mathcal H}(A \times B)=s_A+s_B\ge d$, then $$ dim_{\mathcal H}(A \cap (z+B)) \leq s_A+s_B-d$$ for almost every $z \in {\Bbb R}^d$, in the sense of Lebesgue measure. In this paper, we replace the Hausdorff dimension on the left hand side of the first inequality above by the upper Minkowski dimension, replace the Lebesgue measure of the set of translates by a Hausdorff measure on a set of sufficiently large dimension and replace the translation and rotation group by a more general variable coefficient family of transformations. Interesting arithmetic issues arise in the consideration of sharpness examples. These results are partly motivated by those in \cite{EIT11} and \cite{IJL10} where in the former the classical regular value theorem from differential geometry was investigated in a fractal setting and in the latter discrete incidence theory is explored from an analytic standpoint. Fourier Integral Operator bounds and other techniques of harmonic analysis play a crucial role in our investigation. We also consider, in the spirit of the Furstenberg conjecture, inverse problems for intersections by asking how small a dimension of a set can be given that the dimension of its intersections with a suitably well-curved family of manifolds is bounded from below by a given threshold. Finally, we shall discuss applications of our estimates to the problem of estimating the number of solutions of systems of diophantine equations over integers.

math.CA↗

arXiv2015

Strong Scarring of Logarithmic Quasimodes

We consider a semiclassical (pseudo)differential operator on a compact surface $(M,g)$, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit $γ$ at some energy $E_0$. For any $ε>0$, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order $ε\frac{h}{|\log h|}$ in the semiclassical limit, but which still exhibit a "strong scar" on the orbit $γ$, i.e. that these states have a positive weight in any microlocal neighbourhood of $γ$. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.

math.AP↗

arXiv2015

Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces

We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

math.AP↗

arXiv2014

Smooth billiards with a large Weyl remainder

The celebrated Hardy-Landau lower bound for the error term in the Gauss's circle problem can be viewed as an estimate from below for the remainder in Weyl's law on a square, with either Dirichlet or Neumann boundary conditions. We prove an analogous estimate for smooth star-shaped planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. Our results confirm a prediction of P. Sarnak who proved a similar statement for surfaces without boundary. We also obtain lower bounds on the error term in higher dimensions. In this case, the main contribution to the Weyl remainder typically comes from the "big" singularity at zero of the wave trace. However, for certain domains, such as the Euclidean ball, the dimension of the family of periodic trajectories is large enough to dominate the contribution of the singularity at zero.

math.AP↗

arXiv2014

Expected values of eigenfunction periods

Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2Δ_g ϕ_{h_j} = ϕ_{h_j}$, whose eigenvalues satisfy $h h_j^{-1} \in (1, 1 + hD]$ for $D>0$ a large enough constant. Let $\mathbf{P}_h$ be a uniform probability measure on the $L^2$ unit-sphere $S_h$ of this cluster of eigenfunctions and take $u \in S_h$. Given a closed curve $γ\subset M$, there exists $C_{1}(γ, M), C_{2}(γ, M) > 0$ and $h_0>0$ such that for all $h \in (0, h_0],$ \begin{equation*} C_1 h^{1/2} \leq \mathbf{E}_{h} \bigg[ \big| \int_γ u \, d σ\big| \bigg] \leq C_2 h^{1/2} . \end{equation*} This result contrasts the deterministic $\mathcal{O}(1)$ upperbounds obtained by Chen-Sogge \cite{CS}, Reznikov \cite{Rez}, and Zelditch \cite{Zel}. Furthermore, we treat the higher dimensional cases and compute large deviation estimates. Under a measure zero assumption on the periodic geodesics in $S^*M$, we can consider windows of small width $D=1$ and establish a $\mathcal{O}(h^{1/2})$ estimate. Lastly, we treat probabilistic $L^q$ restriction bounds along curves.

math.AP↗

arXiv2012

Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions

Let (M,g) be a n-dimensional compact Riemannian manifold. We consider the magnetic deformations of semiclassical Schrodinger operators on M for a family of magnetic potentials that depends smoothly on $k$ parameters $u$, for $k \geq n$, and satisfies a generic admissibility condition. Define the deformed Schrodinger eigenfunctions to be the $u$-parametrized semiclassical family of functions on M that is equal to the unitary magnetic Schrodinger propagator applied to the Schrodinger eigenfunctions. The main result of this article states that the $L^2$ norms in $u$ of the deformed Schrodinger eigenfunctions are bounded above and below by constants, uniformly on $M$ and in $\hbar$. In particular, the result shows that this non-random perturbation "kills" the blow-up of eigenfunctions. We give, as applications, an eigenfunction restriction bound and a quantum ergodicity result.

math.SP↗

arXiv2011

Microlocal analysis of scattering data for nested conormal potentials

Working in the time domain, we show that the location of the singularities and the principal symbol of a potential that is conormal to nested submanifolds $S_2 \subset S_1 \subset \mathbb{R}^n$, for $n \geq 3$, can be recovered from the backscattering as well as from the restriction of the far-field pattern to more general determined sets of scattering data. This extends the work of Greenleaf and Uhlmann where the potentials considered are conormal to a single submanifold $S \subset \mathbb{R}^n$. We utilize the microlocal analysis of the wave operator $\square=\partial_t^2 - \triangle_x$ and multiplication by a nested conormal distribution in order to study their action on spaces of conormal-like distributions.

math.AP↗

arXiv2011

Fourier integral operators, fractal sets and the regular value theorem

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{\mathcal H}({\mathcal E})$, and $ϕ$ is a sufficiently regular function, then the upper Minkowski dimension of the set $$ \{w \in {\mathcal E}: ϕ_l(w)=t_l; 1 \leq l \leq m \}$$ does not exceed $dim_{\mathcal H}({\mathcal E})-m$, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier Integral Operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are in general sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.

math.CA↗