1,720,973 research outputs found
Self-adjointness of Quantum Hamiltonians with Symmetries
Full text linkThis thesis discusses the general problem of the self-adjoint realisation of formal Hamiltonians with a focus on a number of quantum mechanical models of actual relevance in the current literature, which display certain symmetries.
In the first part we analyse the general extension theory of (possibly unbounded) linear operators on Hilbert space, and in particular we revisit the Kre\u{i}n-Vi{s}ik-Birman theory that we are going to use in the applications. We also discuss the interplay between extension theory and presence of discrete symmetries, which is the framework of the present work.
The second part of the thesis contains the study of three explicit quantum models, two that are well-known since long and a more modern one, each of which is receiving a considerable amount of attention in the recent literature as far as the identification and the classification of the extensions is concerned. First we characterise all self-adjoint extensions of the Hydrogen Hamiltonian with point-like interaction in the origin and of the Dirac-Coulomb operators. For these two operators we also provide an explicit formula for the eigenvalues of every self-adjoint extension and a characterisation of the domain of respective operators in term of standard functional spaces.
Then we investigate the problem of geometric quantum confinement for a particle constrained on a Grushin-type plane: this yields the analysis of the essential self-adjointness for the Laplace-Beltrami operator on a family of Riemannian manifolds
Two dimensional NLS ground states with attractive Coulomb potential and point interaction
Full text linkWe study the existence and the properties of ground states at fixed mass for a focusing nonlinear Schroedinger equation in dimension two with a point interaction, an attractive Coulomb potential and a nonlinearity of power type. We prove that for any negative value of the Coulomb charge, for any positive value of the mass and for any L2-subcritical power nonlinearity, such ground states exist and exhibit a logarithmic singularity where the interaction is placed. Moreover, up to multiplication by a phase factor, they are positive, radially symmetric and decreasing. An analogous result is obtained also for minimizers of the action restricted to the Nehari manifold, getting the existence also in the L2-critical and supercritical cases
Energy cascade and Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain
Full text linkThe dynamics of initial long-wavelength excitations of the Fermi-Pasta-Ulam-Tsingou chain has been the subject of intense investigations since the pioneering work of Fermi and collaborators. We have recently found a regime where the spectrum of the Fourier modes decays with a power law and we have interpreted this regime
as a transient turbulence associated with the Burgers equation. In this paper we present the full derivation of the latter equation from the lattice dynamics using an infinite-dimensional Hamiltonian perturbation theory. This
theory allows us to relate the time evolution of the Fourier spectrum E_k of the Burgers equation to that of the Fermi-Pasta-Ulam-Tsingou (FPUT) chain. As a consequence, we derive analytically both the shock time and
the power law −8/3 of the spectrum at this time. Using the shock time as a unit, we follow numerically the time evolution of the spectrum and observe the persistence of the power −2 over an extensive time window. The exponent −2 has been widely discussed in the literature on the Burgers equation. The analysis of the Burgers
equation in Fourier space also gives information on the time evolution of the energy of each single mode which, at short time, is also a power law depending on the kth wavenumber E_k ~ t^{2k−2}. This approach to the FPUT
dynamics opens the way to a wider study of scaling regimes arising from more general initial conditions
Hamiltonian field theory close to the wave equation: from Fermi-Pasta-Ulam to water waves
No full textIn the present work we analyse the structure of the Hamiltonian field theory
in the neighbourhood of the wave equation . We show that,
restricting to ``graded'' polynomial perturbations in , and their
space derivatives of higher order, the local field theory is equivalent, in the
sense of the Hamiltonian normal form, to that of the Korteweg-de Vries
hierarchy of second order. Within this framework, we explain the connection
between the theory of water waves and the Fermi-Pasta-Ulam system.Comment: 37 pages; published version with minor changes. Added some remarks
and a concluding sectio
Singularity: A Seventh Memo?
No full textIn this paper we explore the relationships between Calvino's memos and Mathematics. In the first part, we discuss how Lightness, Quickness, Exactitude, Visibility, Multiplicity are present in the mathematical language, reasoning and in the work of the mathematician. In addiction, we follow a similar path for the topics of Calvino's lecture of which we only know the title or some notes. In the final part, we explain why `Singularity' could be chosen as a possible topic for Calvino's seventh lectur
Burgers Turbulence in the Fermi-Pasta-Ulam-Tsingou Chain
Full text linkWe prove analytically and show numerically that the dynamics of the Fermi-Pasta-Ulam-Tsingou chain is characterized by a transient Burgers turbulence regime on a wide range of time and energy scales. This regime is present at long wavelengths and energy per particle small enough that equipartition is not reached on a fast timescale. In this range, we prove that the driving mechanism to thermalization is the formation of a shock that can be predicted using a pair of generalized Burgers equations. We perform a perturbative calculation at small energy per particle, proving that the energy spectrum of the chain Ek decays as a power law, Ek & SIM; k-zeta ot thorn , on an extensive range of wave numbers k. We predict that zeta ot thorn takes first the value 8=3 at the Burgers shock time, and then reaches a value close to 2 within two shock times. The value of the exponent zeta 1/4 2 persists for several shock times before the system eventually relaxes to equipartition. During this wide time window, an exponential cutoff in the spectrum is observed at large k, in agreement with previous results. Such a scenario turns out to be universal, i.e., independent of the parameters characterizing the system and of the initial condition, once time is measured in units of the shock time
Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors
Full text linkWe investigate the properties of self-adjointness of a two-dimensional Dirac
operator on an infinite sector with infinite mass boundary conditions and in
presence of a Coulomb-type potential with the singularity placed on the vertex.
In the general case, we prove the appropriate Dirac-Hardy inequality and
exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential,
we describe the self-adjoint extensions for all the intensities of the
potential relying on a radial decomposition in partial wave subspaces adapted
to the infinite-mass boundary conditions. Finally, we integrate our results
giving a description of the spectrum of these operators
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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