Related papers: Localization and landscape functions on quantum gr…
We consider the localization in the eigenfunctions of regular Sturm-Liouville operators. After deriving non-asymptotic and asymptotic lower and upper bounds on the localization coefficient of the eigenfunctions, we characterize the…
For a Hamiltonian ${\hat H}$ containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions $u_n(\vec{r})$ obeying ${\hat H} u_n(\vec{r}) = u_{n-1}(\vec{r})$ with $u_0(\vec{r}) = 1$. For $n \to…
We consider the localization landscape function $u$ and ground state eigenvalue $\lambda$ for operators on graphs. We first show that the maximum of the landscape function is comparable to the reciprocal of the ground state eigenvalue if…
In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for…
Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche \& Mayboroda showed that the function $u$ solving $(-\Delta + V)u = 1$ controls the behavior of eigenfunctions $(-\Delta + V)\phi = \lambda\phi$ via the…
We consider the localization of eigenfunctions for the operator $L=-\mbox{div} A \nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper…
It is speculated that the correct theory of fundamental physics includes a large landscape of states, which can be described as a potential which is a function of N scalar fields and some number of discrete variables. The properties of such…
The localization landscape gives direct access to the localization of bottom-of-band eigenstates in non-interacting disordered systems. We generalize this approach to eigenstates at arbitrary energies in systems with or without internal…
The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been…
We investigate the spatial statistics of the energy eigenfunctions on large quantum graphs. It has previously been conjectured that these should be described by a Gaussian Random Wave Model, by analogy with quantum chaotic systems, for…
Spectral function is a key tool for understanding the behavior of Bose-Einstein condensates of cold atoms in random potentials generated by a laser speckle. In this paper we introduce a new method for computing the spectral functions in…
A common goal of quantum control is to maximize a physical observable through the application of a tailored field. The observable value as a function of the field constitutes a quantum control landscape. Previous works have shown, under…
We study the effects of entanglement and control parameters on the energy landscape and optimization performance of the variational quantum circuit. Through a systematic analysis of the Hessian spectrum, we characterize the local geometry…
We generate random functions locally via a novel generalization of Dyson Brownian motion, such that the functions are in a desired differentiability class, while ensuring that the Hessian is a member of the Gaussian orthogonal ensemble…
Random, multifield functions can set generic expectations for landscape-style cosmologies. We consider the inflationary implications of a landscape defined by a Gaussian random function, which is perhaps the simplest such scenario. Many key…
Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schr\"odinger operators on domains. We review some known results obtained in the last ten years, unify several approaches used to achieve such bounds,…
We prove spectral localization for infinite metric graphs with a self-adjoint Laplace operator and a random potential. To do so we adapt the multiscale analysis (MSA) from the R^d-case to metric graphs. In the MSA a covering of the graph is…
A quantum control landscape is defined as the physical objective as a function of the control variables. In this paper the control landscapes for two-level open quantum systems, whose evolution is described by general completely positive…
We consider families of finite quantum graphs of increasing size and we are interested in how eigenfunctions are distributed over the graph. As a measure for the distribution of an eigenfunction on a graph we introduce the entropy, it has…
A quantum control landscape is defined as the observable as a function(al) of the system control variables. Such landscapes were introduced to provide a basis to understand the increasing number of successful experiments controlling quantum…