相关论文: The Ten Martini Problem

We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were…

For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states…

The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice $L$ with more than one element contains a…

Quasi-one-dimensional stochastic Dirac operators with an odd number of channels, time reversal symmetry but otherwise efficiently coupled randomness are shown to have one conducting channel and absolutely continuous spectrum of multiplicity…

It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely…

Let $\mu_{q, b}$ be the Cantor measure associated with the iterated function system $f_i(x)=x/b+i/q, 0\le i\le q-1$, where $2\le q, b/q\in \Z$. In this paper, we consider spectra and maximal orthogonal sets of the Cantor measure $\mu_{q,…

We study a version of the fractal uncertainty principle in the joint time-frequency representation. Namely, we consider Daubechies' localization operator projecting onto spherically symmetric $n$-iterate Cantor sets with an arbitrary base…

For $\lambda\in(0,1/2]$ let $K_\lambda \subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{\lambda x, \lambda x+1-\lambda\}$. Given $x\in(0,1/2)$, let $\Lambda(x)$ be the set of $\lambda\in(0,1/2]$ such…

An heuristic proof of the Riemman conjecture is proposed. It is based on the old idea of Polya-Hilbert. A discrete/fractal derivative self adjoint operator whose spectrum may contain the nontrivial zeroes of the zeta function is presented.…

We extend the exactly solvable Hamiltonian describing $f$ quantum oscillators considered recently by J. Dorignac et al. by means of a new interaction which we choose as quasi exactly solvable. The properties of the spectrum of this new…

Martin's Conjecture states that every definable function on the Turing degrees is either constant or increasing, and that every increasing function is an iterate of the Turing jump. This classification has already been corroborated for the…

We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K,…

We explicitly compute the spectral metric, torsion and Einstein tensors for a nontrivial spectral triple on a noncommutative torus, with the Dirac operator related to the fully equivariant Dirac by a partial conformal rescaling (as…

Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] claims a strong submultiplicative upper bound on the rank of a three-tensor obtained as an iterated Kronecker product of a constant-size base tensor. The conjecture, if true,…

We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit,…

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number $m$ generates a complete or incomplete Fourier…

From 1873 to 1897, Georg Cantor worked on developing set theory, and despite a strong initial resistance, it rapidly became accepted as the foundation of mathematics. In this work, however, we'll demonstrate that Cantor's use of infinity is…

We settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of such a system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the…

The Maurey-Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some L_p(\nu) which satisfies a certain vector-valued inequality even allows a weighted norm inequality. Continuing the work of…

We investigate some Weihrauch problems between $\mathsf{ATR}_2$ and $\mathsf{C}_{\omega^\omega}$ . We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not…