Related papers: Lower bounds on Anderson-localised eigenfunctions …
We prove rapid decay (even exponential decay under some stronger assumptions) of the eigenfunctions associated to discrete eigenvalues, for a class of self-adjoint operators in $L^2(\mathbb{R}^d)$ defined by ``magnetic'' pseudodifferential…
The Agmon estimate shows that eigenfunctions of Schr\"odinger operators, $ -\Delta \phi + V \phi = E \phi$, decay exponentially in the `classically forbidden' region where the potential exceeds the energy level $\left\{x: V(x) > E…
An important result by Agmon implies that an eigenfunction of a Schr\"{o}dinger operator in $\mathbb{R}^n$ with eigenvalue $E$ below the bottom of the essential spectrum decays exponentially if the associated classically allowed region $\{x…
Lyapunov equations with low-rank right-hand sides often have solutions whose singular values decay rapidly, enabling iterative methods that produce low-rank approximate solutions. All previously known bounds on this decay involve quantities…
It is proven that the inverse localization length of an Anderson model on a strip of width $L$ is bounded above by $L/\lambda^2$ for small values of the coupling constant $\lambda$ of the disordered potential. For this purpose, a formalism…
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by F\"urstenberg's theorem. That is, a…
We study the distribution of eigenvalues of the Schr\"odinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.
Following the proof given by Froese and Herbst in [FH82] with another conjugate operator, we show for a class of real potential that possible eigenfunction of the Schr\"odinger operator has to decay sub-exponentially. We also show that, for…
We consider Schr\"odinger operator with random decaying potential on $\ell^2 ({\bf Z}^d)$ and showed that, (i) IDS coincides with that of free Laplacian in general cases, and (ii) the set of extremal eigenvalues, after rescaling, converges…
A local time decay estimate of fractional Schr\"odinger operators with slowly decaying positive potentials are studied. It is shown that its resolvent is smooth near zero and the time propagator has fast local time decay which is very…
We consider random Schr\"{o}dinger operators on $\ell^2(\mathbb{Z}^d)$ when the distribution of single site potentials is $\alpha$-H\"{o}lder continuous ($0<\alpha\leq 1$). In localized regime we study the distribution of eigenfunctions…
This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and…
We investigate the dispersive properties of solutions to the Schr\"odinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schr\"odinger flow on…
We study decaying half-line Schr\"odinger operators and the local eigenvalue spacing of their Dirichlet restrictions. While absolutely continuous spectrum is strongly associated with bulk universality and clock behavior, singular spectral…
The Lyapunov exponents for Anderson localization are studied in a one dimensional disordered system. A random Gaussian potential with the power law decay $\sim 1/|x|^q$ of the correlation function is considered. The exponential growth of…
We report our results on the scaling limit of the eigenvalues and the corresponding eigenfunctions for the 1-d random Schr\"odinger operator with random decaying potential. The formulation of the problem is based on the paper by…
We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as…
We study the decay of eigenfunctions of the non self-adjoint matrix operator $\calH = (\begin{smallmatrix} -\Delta +\mu+U & W \W & \Delta -\mu -U \end{smallmatrix})$, for $\mu>0$, corresponding to eigenvalues in the strip $-\mu<\re E <\mu$.
We study fluctuations of polynomial linear statistics for discrete Schr\"odinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth…
We study how the spectral properties of ergodic Schr\"odinger operators are reflected in the asymptotic properties of its periodic approximation as the period tends to infinity. The first property we address is the asymptotics of the…