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18
Localisation for nonmonotone Schrödinger operators. arXiv:1201.2211v3 [mathph
, 2012
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Geometrical structure of Laplacian eigenfunctions
, 2013
"... We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and com ..."
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Cited by 11 (3 self)
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We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is placed onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.
Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models
 J. Spectr. Theory
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Quantum harmonic oscillator systems with disorder
, 2012
"... We study manybody properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zerovelocity LiebRobinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective oneparticle Hamiltonian. W ..."
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We study manybody properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zerovelocity LiebRobinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective oneparticle Hamiltonian. We show how stateoftheart techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of oneparticle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective oneparticle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.
Disorderassisted error correction in Majorana chains
 Comm. Math.Phys
"... It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions the simplest toy model of a quantum m ..."
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It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions the simplest toy model of a quantum memory. Disorder takes the form of a random sitedependent chemical potential. The corresponding oneparticle problem is a onedimensional Anderson model with disorder in the hopping amplitudes. We focus on the zerotemperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an errorcorrection step. Assuming dynamical localization of the oneparticle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the oneparticle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.
An Area Law for the Bipartite Entanglement of Disordered Oscillator Lattices
, 2013
"... We prove an upper bound proportional to the surface area for the bipartite entanglement of the ground state and thermal states of harmonic oscillator systems with disorder, as measured by the logarithmic negativity. Our assumptions are satisfied for some standard models that are almost surely gapl ..."
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We prove an upper bound proportional to the surface area for the bipartite entanglement of the ground state and thermal states of harmonic oscillator systems with disorder, as measured by the logarithmic negativity. Our assumptions are satisfied for some standard models that are almost surely gapless in the thermodynamic limit.
Solving Multilinear Systems via Tensor Inversion
 SIAM Journal on Matrix Analysis and Applications
, 2013
"... Part of the Engineering Commons This Article is brought to you for free and open access by the Mechanical Engineering at Wyoming Scholars Repository. It has been accepted for inclusion in Mechanical Engineering Faculty Publications by an authorized administrator of Wyoming Scholars Repository. For m ..."
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Part of the Engineering Commons This Article is brought to you for free and open access by the Mechanical Engineering at Wyoming Scholars Repository. It has been accepted for inclusion in Mechanical Engineering Faculty Publications by an authorized administrator of Wyoming Scholars Repository. For more information, please contact
Identities and exponential bounds for transfer matrices
 J. Phys. A
"... Abstract. This paper is about analytic properties of single transfer matrices originating from general blocktridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and ..."
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Abstract. This paper is about analytic properties of single transfer matrices originating from general blocktridiagonal or banded matrices. Such matrices occur in various applications in physics and numerical analysis. The eigenvalues of the transfer matrix describe localization of eigenstates and are linked to the spectrum of the block tridiagonal matrix by a determinantal identity. If the block tridiagonal matrix is invertible, it is shown that half of the singular values of the transfer matrix have a lower bound exponentially large in the length of the chain, and the other half have an upper bound that is exponentially small. This is a consequence of a theorem by Demko, Moss and Smith on the decay of matrix elements of inverse of banded matrices.