Related papers: Ballistic Transport for Discrete Multi-Dimensional…

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schr\"odinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative…

We consider the conjecture proposed in Matsumoto, Zhang and Schiebinger (2022) suggesting that optimal transport with quadratic regularisation can be used to construct a graph whose discrete Laplace operator converges to the…

The existence of absolutely continuous (a.c.) spectrum for the discrete Molchanov-Vainberg Schr\"odinger operator $D+V$ on $\ell^2(\mathbb{Z}^d)$, in dimensions $d\geq 2$, is further investigated for potentials $V$ satisfying the long range…

We consider one dimensional Schr\"{o}dinger operators $H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda$ with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. The potentials $U$ and…

We establish strong ballistic transport for a family of discrete quasiperiodic Schr\"odinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been, essentially, shown by Jitomirskaya and…

For electron transport in parallel-plane semiconducting structures, a model is developed that unifies ballistic and diffusive transport and thus generalizes the Drude model. The unified model is valid for arbitrary magnitude of the mean…

This is our first paper on the extension of our recent work on the Lax-Oleinik commutators and its applications to the intrinsic approach of propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations. We…

Let $-\im\Lie_\T$ (essentially Lie derivative with respect to $\T$, a smooth nowhere zero real vector field) and $P$ be commuting differential operators, respectively of orders 1 and $m\geq 1$, the latter formally normal, both acting on…

We prove a sharp H\"ormander multiplier theorem for Schr\"odinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$,…

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $H\varphi=-(c \varphi')'$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on…

I explore a theory of transport and optical properties of strange metallic carriers in strongly correlated systems that follows from assuming that the diffusion constant has reached its quantum limit $D=\hbar/m$, and that such quantum…

SrTiO$_3$-based heterointerfaces support quasi-two-dimensional (2D) electron systems that are analogous to III-V semiconductor heterostructures, but also possess superconducting, magnetic, spintronic, ferroelectric, and ferroelastic degrees…

We build on the formulation developed in Sridhar & Singh (JFM, 664, 265, 2010), and present a theory of the \emph{shear dynamo problem} for small magnetic and fluid Reynolds numbers, but for arbitrary values of the shear parameter.…

In this article, we consider the semiclassical Schr\"odinger operator $P = - h^{2} \Delta + V$ in $\mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $\lambda_{k} ( P )$ as $h \to…

For general second order evolution equations, we prove an optimal condition on the degree of unboundedness of the damping, that rules out finite-time extinction. We show that control estimates give energy decay rates that explicitly depend…

Consider the one-dimensional discrete Schr\"odinger operator $H_{\theta}$: $$(H_{\theta} q)_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n \ , \quad n\in Z \ ,$$ with $\omega\in R^d$ Diophantine, and $V$ a real-analytic function on $ T^d=(…

Let $\mathcal{L}=-\Delta+\mathit{V}(x)$ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse H\"{o}lder class $B_{q},…

We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional $s$-gradient constraint, $0<s<1$, associated to a general, possibly degenerate, linear fractional operator of the type, \begin{equation*}…

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular.…

We consider the family $\hat h_\mu:=\hat\varDelta\hat \varDelta - \mu \hat v,$ $\mu\in\mathbb{R}, $ of discrete Schr\"odinger-type operators in $d$-dimensional lattice $\mathbb{Z}^d$, where $\hat \varDelta$ is the discrete Laplacian and…