Spectral Asymptotics Research Articles

The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.

Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian

We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (-Delta )^s in bounded open Lipschitz sets in the small order limit s rightarrow 0^+. While it is easy to see that all eigenvalues converge to 1 as s rightarrow 0^+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log |xi |. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L^2-normalized Dirichlet eigenfunctions of (-Delta )^s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L^2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.

Spectral dimensions of Kreĭn–Feller operators and Lq-spectra

We study the spectral dimensions and spectral asymptotics of Kreĭn–Feller operators for arbitrary finite Borel measures on (0,1). Connections between the spectral dimension, the Lq-spectrum, the partition entropy and the optimized coarse multifractal dimension are established. In particular, we show that the upper spectral dimension always corresponds to the fixed point of the Lq-spectrum of the corresponding measure. Natural bounds reveal intrinsic connections to the Minkowski dimension of the support of the associated Borel measure. Further, we give a sufficient condition on the Lq-spectrum to guarantee the existence of the spectral dimension. As an application, we confirm the existence of the spectral dimension of self-conformal measures with or without overlap as well as of certain measures of pure point type. We construct a simple example for which the spectral dimension does not exist and determine explicitly its upper and lower spectral dimension.

Spectral Asymptotics of Two-Term Even Order Operators with Involution

Spectral Asymptotics of Two-Term Even Order Operators with Involution

Diagonalization of elliptic systems via pseudodifferential projections

Consider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with A, we construct an almost-unitary pseudodifferential operator that diagonalizes A modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between the spectrum of A and the spectrum of its approximate diagonalization, and discuss the implications at the level of spectral asymptotics.

Pointwise Spectral Asymptotics out of the Diagonal near Degeneration Points

Мы установим равномерные (по $x$, $y$) квазиклассические асимптотики и оценки для ядра Шварца $e_h(x,y;\tau)$ спектрального проектора для эллиптического оператора второго порядка внутри области, удовлетворяющего условию микрогиперболичности (но не $\xi$-микрогиперболичности). Хотя такие асимптотики для его сужения на диагональ $e_h(x,x,\tau)$ и особенно для его следа $\mathsf N_h(\tau)=\int e_h(x,x,\tau) dx$ хорошо известны, внедиагональные асимптотики исследованы гораздо хуже, особенно равномерные. Библиография: 3 названия.

Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov—Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.

On the spectral asymptotics for the buckling problem

We provide a direct proof of Weyl’s law for the buckling eigenvalues of the biharmonic operator on domains of Rd of finite measure. The proof relies on asymptotically sharp lower and upper bounds that we develop for the Riesz mean R2(z). Lower bounds are obtained by making use of the so-called “averaged variational principle.” Upper bounds are obtained in the spirit of Berezin–Li–Yau. Moreover, we state a conjecture for the second term in Weyl’s law and prove its correctness in two special cases: balls in Rd and bounded intervals in R.

Spectral Asymptotics and Lamé Spectrum for Coupled Particles in Periodic Potentials

We make two observations on the motion of coupled particles in a periodic potential. Coupled pendula, or the space-discretized sine-Gordon equation is an example of this problem. Linearized spectrum of the synchronous motion turns out to have a hidden asymptotic periodicity in its dependence on the energy; this is the gist of the first observation. Our second observation is the discovery of a special property of the purely sinusoidal potentials: the linearization around the synchronous solution is equivalent to the classical Lamè equation. As a consequence, all but one instability zones of the linearized equation collapse to a point for the one-harmonic potentials. This provides a new example where Lamé’s finite zone potential arises in the simplest possible setting.

Spectral invariants of Dirichlet-to-Neumann operators on surfaces

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For non-zero potentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which give rise to the parameter-dependent Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators.

Spectral Asymptotics of Solutions of a $$2\times 2$$ System of First-Order Ordinary Differential Equations

Spectral Asymptotics of Solutions of a $$2\times 2$$ System of First-Order Ordinary Differential Equations

Moment Estimates of the Cloud of a Planar Measure

With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class.

Schatten-von Neumann classes of integral operators

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in Rn of finite measure, and conditions for operators on Rn given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

On a class of anharmonic oscillators

In this work we study a class of anharmonic oscillators within the framework of the Weyl-Hörmander calculus. A prototype is an operator on Rn of the form (−Δ)ℓ+|x|2k for k,ℓ integers ≥1. We obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers and derive from them estimates on the rate of growth for the eigenvalues of the anharmonic oscillator (−Δ)ℓ+|x|2k. In particular we give a simple proof for the main term of the spectral asymptotics of these operators. We also study some examples of anharmonic oscillators arising from the analysis on Lie groups.

Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures * *The authors are supported in part by the National Natural Science Foundation of China, Grant 11771136, and Construct Program of the Key Discipline in Hunan Province. The first author is also supported in part by the Hunan Province Hundred Talents Program and a Faculty Research Scholarly

The spectral dimension of a fractal Laplacian encodes important geometric, analytic, and measure-theoretic information. Unlike standard Laplacians on Euclidean spaces or Riemannian manifolds, the spectral dimension of fractal Laplacians are often non-integral and difficult to compute. The computation is much harder in higher-dimensions. In this paper, we set up a framework for computing the spectral dimension of the Laplacians defined by a class of graph-directed self-similar measures on (d ⩾ 2) satisfying the graph open set condition. The main ingredients of this framework include a technique of Naimark and Solomyak and a vector-valued renewal theorem of Lau et al.

A statistical mechanism for operator growth

It was recently conjectured that in generic quantum many-body systems, the spectral density of local operators has the slowest high-frequency decay as permitted by locality. We show that the infinite-temperature version of this ‘universal operator growth hypothesis’ holds for the quantum Ising spin model in d ⩾ 2 dimensions, and for the chaotic Ising chain (with longitudinal and transverse fields) in one dimension. Moreover, the disordered chaotic Ising chain that exhibits many-body localization can have the same high-frequency spectral density asymptotics as thermalizing models. Our argument is statistical in nature, and is based on the observation that the moments of the spectral density can be written as a sign-problem-free sum over paths of Pauli string operators.

On the spectral asymptotics of waves in periodic media with Dirichlet or Neumann exclusions

Summary We consider homogenization of the scalar wave equation in periodic media at finite wavenumbers and frequencies, with the focus on continua characterized by: (a) arbitrary Bravais lattice in $\mathbb{R}^d$, $d \geqslant 2$, and (b) exclusions, that is, ‘voids’ that are subject to homogeneous (Neumann or Dirichlet) boundary conditions. Making use of the Bloch-wave expansion, we pursue this goal via asymptotic ansatz featuring the ‘spectral distance’ from a given wavenumber-eigenfrequency pair (situated anywhere within the first Brillouin zone) as the perturbation parameter. We then introduce the effective wave motion via projection(s) of the scalar wavefield onto the Bloch eigenfunction(s) for the unit cell of periodicity, evaluated at the origin of a spectral neighborhood. For generality, we account for the presence of the source term in the wave equation and we consider—at a given wavenumber—generic cases of isolated, repeated, and nearby eigenvalues. In this way, we obtain a palette of effective models, featuring both wave- and Dirac-type behaviors, whose applicability is controlled by the local band structure and eigenfunction basis. In all spectral regimes, we pursue the homogenized description up to at least first order of expansion, featuring asymptotic corrections of the homogenized Bloch-wave operator and the homogenized source term. Inherently, such framework provides a convenient platform for the synthesis of a wide range of intriguing wave phenomena, including negative refraction and topologically protected states in metamaterials and phononic crystals. The proposed homogenization framework is illustrated by approximating asymptotically the dispersion relationships for (i) Kagome lattice featuring hexagonal Neumann exclusions and (ii) square lattice of circular Dirichlet exclusions. We complete the numerical portrayal of analytical developments by studying the response of a Kagome lattice due to a dipole-like source term acting near the edge of a band gap.

On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

N-Laplacians on Metric Graphs and Almost Periodic Functions: I

The spectra of n-Laplacian operators (-Delta )^n on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics are analysed, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced, and its uniqueness is proved using the theory of almost periodic functions. To achieve this, new results concerning roots of functions close to almost periodic functions are proved. The results obtained on almost periodic functions are of general interest outside the theory of differential operators.

Regularized Trace of a Sturm–Liouville Operator on a Curve with a Regular Singularity on the Chord

For a Sturm–Liouville operator on a piecewise smooth curve, we study the effect that the spectrum of a nonintegrable singularity of the potential on the segment joining the endpoints of this curve has on the operator asymptotics. It is shown that in the case where the singular point does not give rise to the branching of solutions in its neighborhood (the case of trivial monodromy), the spectral asymptotics and the formula for the regularized trace look the same as for the classical Sturm–Liouville operator on a segment with smooth potential. Further, it is shown that in the case of nontrivial monodromy the spectral asymptotics substantially depends on the commensurability of the parts into which the segment is partitioned by the singular point $$\theta $$ : if $$\theta $$ is rational, then the spectrum is divided into finitely many series, each going to infinity along “its own” parabola. In this case, the regularized trace formula is significantly more complicated and does not show any similarity with the classical formula.