Systems of Transversal Sections near Critical Energy Levels of Hamiltonian Systems in ℝ⁴

2018 ◽  
Vol 252 (1202) ◽  
pp. 0-0
Author(s):  
Naiara de Paulo ◽  
Pedro Salomão
Keyword(s):  
Hamiltonian Systems ◽  
Energy Levels ◽  
Critical Energy

scholarly journals RESONANCES AND SPECTRAL SHIFT FUNCTION FOR THE SEMI-CLASSICAL DIRAC OPERATOR

2007 ◽  
Vol 19 (10) ◽  
pp. 1071-1115 ◽  
Author(s):  
ABDALLAH KHOCHMAN
Keyword(s):  
Dirac Operator ◽  
Upper Bound ◽  
Selfadjoint Operator ◽  
Energy Levels ◽  
Critical Energy ◽  
Spectral Shift ◽  
Approximation Formula ◽  
Shift Function ◽  
Spectral Shift Function ◽  
Electro Magnetic

We consider the selfadjoint operator H = H0+ V, where H0is the free semi-classical Dirac operator on ℝ3. We suppose that the smooth matrix-valued potential V = O(〈x〉-δ), δ > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of ℝ3. We establish an upper bound O(h-3) for the number of resonances in any compact domain. For δ > 3, a representation of the derivative of the spectral shift function ξ(λ,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.

scholarly journals The theory of electronic semi-conductors

1931 ◽  
Vol 133 (822) ◽  
pp. 458-491 ◽  
Keyword(s):  
Classical Theory ◽  
Energy Levels ◽  
Electron Gas ◽  
Mean Free Path ◽  
Pauli Principle ◽  
Critical Energy ◽  
Free Electrons ◽  
Conduction Electrons ◽  
Metallic Conduction ◽  
Direct Proportionality

The application of quantum mechanics to the problem of metallic conduction has cleared up many of the difficulties which were so apparent in the free electron theories of Drude and Lorentz. Sommerfeld* assumed that the valency electrons of the metallic atoms formed an electron gas which obeyed the FermiDirac statistics, instead of Maxwellian statistics, and, using in the main classical ideas, showed how the difficulty of the specific heat would be removed. He was, however, unable to determine the temperature dependence of the resistance, as his formulae contained a mean free path about which little could be said. F. Bloch took up the question of the mechanics of electrons in a metallic lattice, and showed that if the lattice is perfect an electron can travel quite freely through it. Therefore so long as the lattice is perfect the conductivity is infinite, and it is only when we take into account the thermal motion and the impurities that we obtain a finite value for the conductivity. On this view all the electrons in a metal are free, and we cannot assume, as we do in the classical theory, that only the valency electrons are free. This does not give rise to any difficulty in the theory of metallic conduction, as the direct proportionality between the conductivity and the number of free electrons no longer holds when the Pauli principle is taken into account. If there is no external electric field, the number of electrons moving in any direction is equal to the number moving in the opposite direction. The action of a field is to accelerate or retard the electrons, causing them to make transitions from one set of energy levels to another. This can only happen if the final energy levels are already unoccupied, and therefore only those electrons whose energies are near the critical energy of the Fermi distribution can make transitions and take part in conduction, as it is only in the neighbourhood of the critical energy that the energy levels are partly filled and partly empty. These electrons are few in number compared with the valency electrons, and are what should be called the conduction electrons. On the classical theory alone are the valency electrons, the free electrons and the conduction electrons the same.

On Anosov energy levels of convex Hamiltonian systems

1994 ◽  
Vol 217 (1) ◽  
pp. 367-376 ◽  
Author(s):  
Gabriel P. Paternain ◽  
Miguel Paternain
Keyword(s):  
Hamiltonian Systems ◽  
Energy Levels

Global surfaces of section in non-regular convex energy levels of Hamiltonian systems

2006 ◽  
Vol 255 (2) ◽  
pp. 323-334
Author(s):  
C. Grotta-Ragazzo ◽  
Pedro A. S. Salomão
Keyword(s):  
Hamiltonian Systems ◽  
Energy Levels ◽  
Regular Convex

scholarly journals EQUILIBRIUM TIMES FOR THE MULTICANONICAL METHOD

2004 ◽  
Vol 15 (03) ◽  
pp. 471-478 ◽  
Author(s):  
M. L. GUERRA ◽  
J. D. MUÑOZ
Keyword(s):  
Ising Model ◽  
Markov Process ◽  
Sampling Method ◽  
Energy Levels ◽  
Critical Energy ◽  
System Size ◽  
Sampling Procedure ◽  
Perfect Agreement ◽  
Slowing Down ◽  
New Criterion

This work measures the time to equilibrium for the multicanonical method on the 2D-Ising system by using a new criterion, proposed here, to find the time to equilibrium, t eq , of any sampling procedure based on a Markov process. Our new procedure gives the same results that the usual one, based on the magnetization, for the canonical Metropolis sampling on a 2D-Ising model at several temperatures. For the multicanonical method we found a power-law relationship with the system size, L, of t eq =0.27(15) L2.80(13) and with the number of energy levels to explore, k E , of [Formula: see text], in perfect agreement with the above result. In addition, a kind of critical slowing down was observed around the critical energy. Our new procedure is completely general, and can be applied to any sampling method based on Markov processes.

Convex energy levels of Hamiltonian systems

2004 ◽  
Vol 4 (2) ◽  
pp. 439-454 ◽  
Author(s):  
Pedro A. S. Salomão
Keyword(s):  
Hamiltonian Systems ◽  
Energy Levels

scholarly journals The combination of nitrogen and hydrogen activated by electrons

1927 ◽  
Vol 115 (772) ◽  
pp. 684-700 ◽  
Keyword(s):  
Chemical Reactions ◽  
Reaction Rate ◽  
Energy Levels ◽  
Monochromatic Light ◽  
Critical Energy ◽  
Band Spectrum ◽  
Ionisation Potential ◽  
Subsequent Work ◽  
Physical Measurements ◽  
Production Of Ammonia

Although the chemical reactions of active molecules have been used to fix their critical potentials, little attention has been paid to the converse process of analysing the mechanism of a reaction by studying the rates of reaction of molecules activated to the energy-levels already known from physical measurements. The method is analogous to photochemistry, with the advantages that a stream of electrons with a nearly uniform velocity is easier to obtain than quasi-monochromatic light, and that the mechanism is a more general one. The following is an account of experiments on the rate of production of ammonia from nitrogen and hydrogen as a function of the energy of thermions used to activate molecules and atoms. No attempt was made to measure the critical energy increments with great exactness, since that can be accomplished more easily by physical means; the object in the present case was to recognise the critical potentials of significance in the reaction. Previous Work . Heidemann described the production of ammonia even at the lowest voltages, but subsequent work by Andersen and Storch and Olson did not confirm this. They detected no combination until the molecular ionisation potential of N 2 (circa 17 V) was reached, after which the reaction rate increased abruptly every 4-7 V. The mechanism proposed was that H 2 + and N 2 + appearing at 16 V and 17 V respectively gave H and N atoms on collision, and that increased combination was due to the activation of H by 4 V electrons. Later Kwei found that the NH 3 band spectrum was not excited in hydrogen and nitrogen mixtures until 23 V was reached. This voltage corresponds to the second jump in Storch and Olson’s curve. In a subsequent note Olson explained the failure of Kwei to detect ammonia at 17 V by postulating that NH 3 + must be present for the spectrum to appear. Thus at 17 V the reactions were considered to be N 2 + + e → N' 2 , N' 2 + N 2 → N 2 + 2N, the nitrogen atoms then combining with H 2 or H produced by the reaction N 2 + H 2 → 2N + 2H; while at 23 V the voltage at which N + begins to appear, NH 3 + is obtained in the same way.

scholarly journals The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

2017 ◽  
Vol 17 (1) ◽  
Author(s):  
Alberto Abbondandolo ◽  
Luca Asselle ◽  
Gabriele Benedetti ◽  
Marco Mazzucchelli ◽  
Iskander A. Taimanov
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Cotangent Bundle ◽  
Energy Levels ◽  
High Energy ◽  
Intermediate Range ◽  
Magnetic Flows ◽  
Almost All

AbstractWe consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range

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