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Ivan Veselić

Publications and source records attributed to Ivan Veselić.

At least 19 recordsLinked to original sources
arXiv2016

Sampling inequality for $L^2$-norms of eigenfunctions, spectral projectors, and Weyl sequences of Schrödinger operators

We consider a Schrödinger operator with bounded, measurable potential in multidimensional Euclidean space. We prove for every $L^2$-eigenfunction a quantitative equidistribution estimate. It compares the total $L^2$-norm with the $L^2$-norm over an equidistributed collection of balls. Our estimate is explicit with respect to the radius of the balls, norm of the potential and the energy of the eigenfunction. Similar estimates also hold for Weyl sequences and for linear combinations of eigenfunctions, as long as the associated eigenvalues are sufficiently close.

math.AP↗
arXiv2016

Expansion of the almost sure spectrum in the weak disorder regime

The spectrum of random ergodic Schrödinger-type operators is almost surely a deterministic subset of the real line. The random operator can be considered as a perturbation of a periodic one. As soon as the disorder is switched on via a global coupling constant, the spectrum expands. We estimate how much the spectrum expands at its bottom for operators on $\ell^2(\mathbb Z^d)$.

math-ph↗
arXiv2015

Scale-free uncertainty principles and Wegner estimates for random breather potentials

We present new scale-free quantitative unique continuation principles for Schrödinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a prescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas-Molina & Veselić, and Klein. We apply the scale-free unique continuation principle to obtain a Wegner estimate for a random Schrödinger operator of breather type. It holds for arbitrarily high energies. Schrödinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challenges arising from this non-linear dependence.

math.AP↗
arXiv2015

Discrete alloy-type models: Regularity of distributions and recent results

We consider discrete random Schrödinger operators on $\ell^2 (\mathbb{Z}^d)$ with a potential of discrete alloy-type structure. That is, the potential at lattice site $x \in \mathbb{Z}^d$ is given by a linear combination of independent identically distributed random variables, possibly with sign-changing coefficients. In a first part we show that the discrete alloy-type model is not uniformly $τ$-Hölder continuous, a frequently used condition in the literature of Anderson-type models with general random potentials. In a second part we review recent results on regularity properties of spectral data and localization properties for the discrete alloy-type model.

math-ph↗
arXiv2015

Random walks across the sea: the origin of rogue waves?

Ocean rogue waves are large and suddenly appearing surface gravity waves, which may cause severe damage to ships and other maritime structures. Despite years of research, the exact origin of rogue waves is still disputed. Linear interference of waves with random phase has often been cited as one possible explanation, but apparently does not satisfactorily explain the probability of extreme events in the ocean. Other explanations therefore suggested a decisive role of a nonlinearity in the system. Here we show that linear interference of a finite and variable number of waves may very well explain the heavy tail in the wave height distribution. Our model can explain all prototypical ocean rogue waves reported so far, including the "three sisters" as well as rogue holes. We further suggest nonlinear time series analysis for estimation of the characteristic number of interfering waves for a given sea state. If ocean dynamics is ruled by interference of less than ten waves, rogue waves cannot appear as a matter of principle. In contrast, for larger numbers, their appearance is much more likely than predicted by parameterless models or longterm observation. The pronounced threshold behavior of our model enables effective forecasting of extreme ocean waves.

physics.ao-ph↗
arXiv2015

Multiscale unique continuation properties of eigenfunctions

Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g. random Schrödinger operators and control theory. We review recent results and announce new ones regarding quantitative unique continuation principles for partial differential equations with an underlying multiscale structure. They concern Schrödinger and second order elliptic operators. An important feature is that the estimates are scale free and with quantitative dependence on parameters. These unique continuation principles apply to functions satisfying certain `rigidity' conditions, namely that they are solutions of the corresponding elliptic equations, or projections on spectral subspaces. Carleman estimates play an important role in the proofs of these results. We also present an explicit Carleman estimate for second order elliptic operators.

math.AP↗
arXiv2014

Lifshitz asymptotics for percolation Hamiltonians

We study a discrete Laplace operator $Δ$ on percolation subgraphs of an infinite graph. The ball volume is assumed to grow at most polynomially. We are interested in the behavior of the integrated density of states near the lower spectral edge. If the graph is a Cayley graph we prove that it exhibits Lifshitz tails. If we merely assume that the graph has an exhausting sequence with positive $δ$-dimensional density, we obtain an upper bound on the integrated density of states of Lifshitz type.

math-ph↗
arXiv2013

Minami's estimate: beyond rank one perturbation and monotonicity

In this note we prove Minami's estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami's estimate to prove Poisson statistics for the energy level spacing. Our result is valid for random potentials which are in a certain sense sufficiently close to the standard Anderson potential (rank one perturbations coupled with i.i.d. random variables).

math.SP↗
arXiv2012

Unbounded quantum graphs with unbounded boundary conditions

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self-adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self-adjoint operators. We then characterize the lower bounded self-adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions.

math-ph↗
arXiv2011

A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying graph. In this setting it is natural to expect that the integrated density of states (IDS), or spectral distribution function, exists. We show that it can be defined as the uniform limit of approximants associated to finite matrices. The proof is based on a Banach space valued ergodic theorem which even allows explicit convergence estimates. Our result applies to a variety of group structures and colouring types, in particular to periodic operators and percolation-type Hamiltonians.

math-ph↗
arXiv2011

Discrete Schrödinger operators with random alloy-type potential

We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schrödinger operators $H_ω= - Δ+ V_ω$ on $\ell^2 (\ZZ^d)$ where $Δ$ is the discrete Laplacian and $V_ω$ the multiplication by the function $V_ω(x) = \sum_{k \in \ZZ^d} ω_k u(x-k)$. Here $ω_k$, $k \in \ZZ^d$, are i.i.d. random variables and $u \in \ell^1 (\ZZ^d ; \RR)$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $H_ω$ depend in a non-monotone way on the random parameters $ω_k$. This requires new methods at certain stages of the localization proof.

math-ph↗
arXiv2011

$L^p$-approximation of the integrated density of states for Schrödinger operators with finite local complexity

We study spectral properties of Schrödinger operators on $\RR^d$. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in $\ZZ^d$, with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e.the normalised eigenvalue counting functions. The convergence holds in the space $L^p(I)$ where $I$ is any finite energy interval and $1\leq p< \infty$ is arbitrary.

math.SP↗
arXiv2010

Localization via fractional moments for models on $\mathbb{Z}$ with single-site potentials of finite support

One of the fundamental results in the theory of localization for discrete Schrödinger operators with random potentials is the exponential decay of Green's function and the absence of continuous spectrum. In this paper we provide a new variant of these results for one-dimensional alloy-type potentials with finitely supported sign-changing single-site potentials using the fractional moment method.

math-ph↗
arXiv2010

Wegner estimate for discrete alloy-type models

We study discrete alloy-type random Schrödinger operators on $\ell^2(\mathbb{Z}^d)$. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds. The bound is polynomial in the volume of the box and thus applicable as an ingredient for a localisation proof via multiscale analysis.

math.SP↗
arXiv2010

Spectral properties of discrete alloy-type models

We discuss recent results on spectral properties of discrete alloy-type random Schrödinger operators. They concern Wegner estimates and bounds on the fractional moments of the Green's function.

math.SP↗
arXiv2009

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov-Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.

math.SP↗
arXiv2008

Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs

In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum. The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem.

math.SP↗