Probabilistic Weyl laws for quantized tori

A remainder estimate for Weyl's law on Liouville tori

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 revolution as special cases. Using separation of variables, we reduce the eigenvalue counting problem to the problem of counting lattice points in certain planar domains. This allows us to improve the remainder estimate in Weyl's law on a large class of Liouville tori. For flat metrics, such an estimate has been known for more than a century due to classical results…

Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two

We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. An asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant Lagrangian tori, is established. The Weyl law is given in terms of the long time averages of the leading non-selfadjoint perturbation along the classical flow of the unperturbed part.

Almost sure Weyl law for quantized tori

We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean $0$, variance $1$ and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves in particular a conjecture by T. Christiansen and M. Zworski.

平坦环面上带奇异势的薛定谔算子的精确逐点Weyl定律

Sharp Pointwise Weyl Laws for Schrödinger Operators with Singular Potentials on Flat Tori

针对低维平坦环面拉普拉斯算子Weyl定律精确误差项这一公开难题，本文证明一类奇异扰动下任意维数都存在带精确误差项的逐点Weyl定律，验证了作者此前相关定理的精确性并将三维结果推广至任意维。

Quantized mirror curves and resummed WKB

Based on previous insights, we present an ansatz to obtain quantization conditions and eigenfunctions for a family of difference equations which arise from quantized mirror curves in the context of local mirror symmetry of toric Calabi-Yau threefolds. It is a first principles construction, which yields closed expressions for the quantization conditions and the eigenfunctions when $\hbar/2π\in \mathbb Q$. The key ingredient is the modular duality structure of the underlying quantum integrable system. We use our ansatz to write down explicit results in some examples, which are successfully check…

Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials

Thanks to the Birman-Schwinger principle, Weyl's laws for Birman-Schwinger operators yields semiclassical Weyl's laws for the corresponding Schrödinger operators. In a recent preprint Rozenblum established quite general Weyl's laws for Birman-Schwinger operators associated with pseudodifferential operators of critical order and potentials that are product of $L\log L$-Orlicz functions and Alfhors-regular measures supported on a submanifold. In this paper, for matrix-valued $L\log L$-Orlicz potentials supported on the whole manifold, Rozenblum's results are direct consequences of the Cwikel-typ…