2,976 research outputs found
Schnol's theorem and spectral properties of massless Dirac operators with scalar potentials
The spectra of massless Dirac operators are of essential interest e.g. for
the electronic properties of graphene, but fundamental questions such as the
existence of spectral gaps remain open. We show that the eigenvalues of
massless Dirac operators with suitable real-valued potentials lie inside small
sets easily characterised in terms of properties of the potentials, and we
prove a Schnol'-type theorem relating spectral points to polynomial boundedness
of solutions of the Dirac equation. Moreover, we show that, under minimal
hypotheses which leave the potential essentially unrestrained in large parts of
space, the spectrum of the massless Dirac operator covers the whole real line;
in particular, this will be the case if the potential is nearly constant in a
sequence of regions.Comment: 18 page
Spherically symmetric Dirac operators with variable mass and potentials infinite at infinity
We study the spectrum of spherically symmetric Dirac operators in
three-dimensional space with potentials tending to infinity at infinity under
weak regularity assumptions. We prove that purely absolutely continuous
spectrum covers the whole real line if the potential dominates the mass, or
scalar potential, term. In the situation where the potential and the scalar
potential are identical, the positive part of the spectrum is purely discrete;
we show that the negative half-line is filled with purely absolutely continuous
spectrum in this case.Comment: 16 pages; submitted to Publ. RIM
Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment
We show that the point spectrum of the standard Coulomb-Dirac operator H_0 is
the limit of the point spectrum of the Dirac operator with anomalous magnetic
moment H_a as the anomaly parameter tends to 0. For negative angular momentum
quantum number kappa, this holds for all Coulomb coupling constants c for which
H_0 has a distinguished self-adjoint realisation. For positive kappa, however,
there are some exceptional values for c, and in general an index shift between
the eigenvalues of H_0 and the limits of eigenvalues of H_a appears,
accompanied with additional oscillations of the eigenfunctions of H_a very
close to the origin
On the resonances and eigenvalues for a 1D half-crystal with localised impurity
We consider the Schr\"odinger operator on the half-line with a periodic
potential plus a compactly supported potential . For generic , its
essential spectrum has an infinite sequence of open gaps. We determine the
asymptotics of the resonance counting function and show that, for sufficiently
high energy, each non-degenerate gap contains exactly one eigenvalue or
antibound state, giving asymptotics for their positions. Conversely, for any
potential and for any sequences (\s_n)_{1}^\iy, \s_n\in \{0,1\}, and
(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0, there exists a potential such that
\vk_n is the length of the -th gap, , and has exactly \s_n
eigenvalues and 1-\s_n antibound state in each high-energy gap. Moreover, we
show that between any two eigenvalues in a gap, there is an odd number of
antibound states, and hence deduce an asymptotic lower bound on the number of
antibound states in an adiabatic limit.Comment: 25 page
The HELP inequality on trees
We establish analogues of Hardy and Littlewood's integro-differential equation for Schrödinger-type operators on metric and discrete trees, based on a generalised strong limit-point property of the graph Laplacian
On unbounded positive definite functions
It is well known that positive definite functions are bounded, taking their maximum absolute value at 0. Nevertheless, there are unbounded functions, arising e.g. in potential theory or the study of (continuous) extremal measures, which still exhibit the general characteristics of positive definiteness. Taking a framework set up by Lionel Cooper as a motivation, we study the general properties of such functions which are positive definite in an extended sense. We prove a Bochner-type theorem and, as a consequence, show how unbounded positive definite functions arise as limits of classical positive definite functions, as well as that their space is closed under convolution. Moreover, we provide criteria for a function to be positive definite in the extended sense, showing in particular that complete monotonicity in conjunction with absolute integrability is sufficient
Statistical extraction of process zones and representative subspaces in fracture of random composite
We propose to identify process zones in heterogeneous materials by tailored
statistical tools. The process zone is redefined as the part of the structure
where the random process cannot be correctly approximated in a low-dimensional
deterministic space. Such a low-dimensional space is obtained by a spectral
analysis performed on pre-computed solution samples. A greedy algorithm is
proposed to identify both process zone and low-dimensional representative
subspace for the solution in the complementary region. In addition to the
novelty of the tools proposed in this paper for the analysis of localised
phenomena, we show that the reduced space generated by the method is a valid
basis for the construction of a reduced order model.Comment: Submitted for publication in International Journal for Multiscale
Computational Engineerin
A Discrete Hardy-Laptev-Weidl-Type Inequality and Associated Schrödinger-Type Operators
Although the classical Hardy inequality is valid only in the three- and higher dimensional case, Laptev and Weidl established a two-dimensional Hardy-type inequality for the magnetic gradient with an Aharonov-Bohm magnetic potential. Here we consider a discrete analogue, replacing the punctured plane with a radially exponential lattice. In addition to discrete Hardy and Sobolev inequalities, we study the spectral properties of two associated self-adjoint operators. In particular, it is shown that, for suitable potentials, the discrete Schrödingertype operator in the Aharonov-Bohm field has essential spectrum concentrated at 0, and the multiplicity of its lower spectrum satisfies a CLR-type inequality.Although the classical Hardy inequality is valid only in the three- and higher dimensional case, Laptev and Weidl established a two-dimensional Hardy-type inequality for the magnetic gradient with an Aharonov-Bohm magnetic potential. Here we consider a discrete analogue, replacing the punctured plane with a radially exponential lattice. In addition to discrete Hardy and Sobolev inequalities, we study the spectral properties of two associated self-adjoint operators. In particular, it is shown that, for suitable potentials, the discrete Schrödingertype operator in the Aharonov-Bohm field has essential spectrum concentrated at 0, and the multiplicity of its lower spectrum satisfies a CLR-type inequality
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