In this short note we demonstrate that the definition of the density of states of a Schr{o}dinger operator with bounded potential in general depends on the choice of the domain undergoing the thermodynamic limit.
We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asympt
otics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(mathcal{A},H,D)$ and self-adjoint $V in mathcal{A}$, there holds [lim_{hdownarrow 0} h^pmathrm{Tr}(chi_{(-infty,0)}(h^2D^2+V)) = int V_-^{frac{p}{2}}|ds|^p.] where $int$ is Connes noncommutative integral.
We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular with the use of wavelet techniques. We are able to provide a simple proof of norm estimates
for integral operators with kernel in $B^{frac{1}{p}-frac{1}{2}}_{p,p}(mathbb R,L_2(mathbb R))$. This recovers, extends and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals, with bounded symbol in $B^{frac{1}{p}-frac{1}{2}}_{frac{2p}{2-p},p}(mathbb R,L_infty(mathbb R))$, which extends Birman and Solomyaks result to symbols without compact domain.
We study the class of functions $f$ on $mathbb{R}$ satisfying a Lipschitz estimate in the Schatten ideal $mathcal{L}_p$ for $0 < p leq 1$. The corresponding problem with $pgeq 1$ has been extensively studied, but the quasi-Banach range $0 < p < 1$ is
by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class $dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})$ obey the estimate $$ |f(A)-f(B)|_{p} leq C_{p}(|f|_{L_{infty}(mathbb{R})}+|f|_{dot{B}^{frac{1}{p}}_{frac{p}{1-p},p}(mathbb{R})})|A-B|_{p} $$ for all bounded self-adjoint operators $A$ and $B$ with $A-Bin mathcal{L}_p$. In the case $p=1$, our methods recover and provide a new perspective on a result of Peller that $f in dot{B}^1_{infty,1}$ is sufficient for a function to be Lipschitz in $mathcal{L}_1$. We also provide related Holder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on $mathbb{R}$ are not Lipschitz in $mathcal{L}_p$ for any $0 < p < 1$. This gives counterexamples to a 1991 conjecture of Peller that $f in dot{B}^{1/p}_{infty,p}(mathbb{R})$ is sufficient for $f$ to be Lipschitz in $mathcal{L}_p$.
A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrodinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of s
tates. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.
We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case.
We study the topic of quantum differentiability on quantum Euclidean $d$-dimensional spaces (otherwise known as Moyal $d$-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differential to have
decay $O(n^{-alpha})$ for $0 < alpha leq frac{1}{d}$. This result is substantially more difficult than the analogous problems for Euclidean space and for quantum $d$-tori.
