Related papers: Sharp Pointwise Weyl Laws for Schr\"odinger Operat…

We consider the Schr\"odinger operators $H_V=-\Delta_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that…

We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise…

We obtain generalizations of classical versions of the Weyl formula involving Schr\"odinger operators $H_V=-\Delta_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials $V$. In particular, we extend the…

The classical Weyl Law says that if $N_M(\lambda)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal to $\lambda$, then $$…

Weyls law is a fundamental result governing the asymptotic behaviour of the eigenvalues of teh Laplacian. It states that for a compact d dimensional manifold M (without boundary), the eigenvalue counting function has an asymptotic growth,…

We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $\lambda_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For…

We give a short proof of a strong form of Weyl's law for $\text{SO}(N)$ using well known facts of the theory of modular forms. The exponent of the error term is sharp when the rank is at least~$4$. We also discuss the cases with smaller…

We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schr\"odinger operators $ H=\Delta+V $ on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard…

We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on the two-dimensional torus. Accurate upper and lower bounds on the number of such…

The paper is concerned with the asymptotic distribution of Laplace eigenvalues on Liouville tori. Liouville metrics are the largest known class of integrable metrics on two-dimensional tori; they contain flat metrics and metrics of…

We discuss the validity of the Weyl asymptotics -- in the sense of two-sided bounds -- for the size of the discrete spectrum of (discrete) Schr\"odinger operators on the $d$--dimensional, $d\geq 1$, cubic lattice $\mathbb{Z}^{d}$ at large…

Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$, a notion analogous to full Lebesgue measure which, in particular, implies density. We show that…

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be…

Let X=Sl(3,Z)\Sl(3,R)/SO(3,R). Let N(lambda) denote the dimension of the space of cusp forms with Laplace eigenvalue less than lambda. We prove that N(lambda)=C lambda^(5/2)+O(lambda^2) where C is the appropriate constant establishing…

Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schr\"odinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp}…

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a…

We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds, or pseudodifferential operators of Shubin type on…

Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on $M$ defined by…

We study the spectral asymptotics of wave equations on certain compact spacetimes where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime $S^1 \times S^2$. For the Laplacian on $S^1 \times S^2$ the Weyl…

In this paper, we investigate the singular values of a natural family of transfer operators twisted by large random permutation matrices. In the large N limit, we obtain a Weyl law for its singular values, valid asymptotically almost surely…