Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

2009 ◽  
Vol 16 (2) ◽  
pp. 300-308 ◽  
Author(s):  
V. S. Rabinovich
Keyword(s):  
Pseudodifferential Operators ◽  
Square Root ◽  
Exponential Estimates ◽  
Klein Gordon

HADAMARD STATES, ADIABATIC VACUA AND THE CONSTRUCTION OF PHYSICAL STATES FOR SCALAR QUANTUM FIELDS ON CURVED SPACETIME

1996 ◽  
Vol 08 (08) ◽  
pp. 1091-1159 ◽  
Author(s):  
WOLFGANG JUNKER
Keyword(s):  
Pseudodifferential Operators ◽  
Cauchy Surface ◽  
Positive Energy ◽  
Hyperbolic Spacetime ◽  
Vacuum States ◽  
Mathematical Review ◽  
Scalar Quantum ◽  
Klein Gordon ◽  
Hadamard States ◽  
Adiabatic Vacuum

Quasifree states of a linear Klein-Gordon quantum field on globally hyperbolic spacetime manifolds are considered. After a short mathematical review techniques from the theory of pseudodifferential operators and wavefront sets on manifolds are used to develop a criterion for a state to be an Hadamard state. It is proven that ground- and KMS-states on certain static spacetimes and adiabatic vacuum states on Robertson-Walker spaces are Hadamard states. A counterexample is given which shows that the idea of instantaneous positive energy states w.r.t. a Cauchy surface does in general not yield physical states. Finally, the problem of constructing Hadamard states on arbitrary curved spacetimes is solved in principle.

scholarly journals Relativistic wave equations with fractional derivatives and pseudodifferential operators

2002 ◽  
Vol 2 (4) ◽  
pp. 163-197 ◽  
Author(s):  
Petr Závada
Keyword(s):  
Fractional Derivatives ◽  
Wave Equations ◽  
Pseudodifferential Operators ◽  
Relativistic Wave Equations ◽  
Green Functions ◽  
Relativistic Wave ◽  
Fractional Powers ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Symmetry Transformations

We study the class of the free relativistic covariant equations generated by the fractional powers of the d′Alembertian operator(□1/n). The equations corresponding ton=1and2(Klein-Gordon and Dirac equations) are local in their nature, but the multicomponent equations for arbitraryn>2are nonlocal. We show the representation of the generalized algebra of Pauli and Dirac matrices and how these matrices are related to the algebra ofSU (n)group. The corresponding representations of the Poincaré group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested.

scholarly journals Wave dispersion derived from the square-root Klein–Gordon–Poisson system

2013 ◽  
Vol 79 (4) ◽  
pp. 371-376 ◽  
Author(s):  
F. HAAS
Keyword(s):  
Relativistic Quantum ◽  
Relativistic Effects ◽  
Fundamental Aspect ◽  
Physical Parameters ◽  
Linear Wave ◽  
Relativistic Energy ◽  
Square Root ◽  
Maxwell System ◽  
Poisson System ◽  
Klein Gordon

AbstractRecently, there has been great interest around quantum relativistic models for plasmas. In particular, striking advances have been obtained by means of the Klein–Gordon–Maxwell system, which provides a first-order approach to the relativistic regimes of quantum plasmas. The Klein–Gordon–Maxwell system provides a reliable model as long as the plasma spin dynamics is not a fundamental aspect, to be addressed using more refined (and heavier) models involving the Pauli–Schrödinger or Dirac equations. In this work, a further simplification is considered, tracing back to the early days of relativistic quantum theory. Namely, we revisit the square-root Klein–Gordon–Poisson system, where the positive branch of the relativistic energy–momentum relation is mapped to a quantum wave equation. The associated linear wave propagation is analyzed and compared with the results in the literature. We determine physical parameters where the simultaneous quantum and relativistic effects can be noticeable in weakly coupled electrostatic plasmas.

scholarly journals "SQUARE ROOT" OF THE PROCA EQUATION: SPIN-3/2 FIELD EQUATION

2006 ◽  
Vol 21 (05) ◽  
pp. 1143-1155 ◽  
Author(s):  
S. I. KRUGLOV
Keyword(s):  
Projection Operators ◽  
Relativistic Wave Equation ◽  
Square Root ◽  
Local Equation ◽  
Proca Equation ◽  
Electromagnetic Interactions ◽  
Causal Propagation ◽  
Klein Gordon ◽  
Propagation Of Waves ◽  
Two Parameters

New equations describing particles with spin-3/2 are derived. The nonlocal equation with the unique mass can be considered as "square root" of the Proca equation in the same sense as the Dirac equation is related to the Klein–Gordon–Fock equation. The local equation describes spin-3/2 particles with three mass states. The equations considered involve fields with spin-3/2 and spin-1/2, i.e. multispin 1/2, 3/2. The projection operators extracting states with definite energy, spin, and spin projections are obtained. All independent solutions of the local equation are expressed through projection matrices. The first order relativistic wave equation in the 20-dimensional matrix form, the relativistically invariant bilinear form and the corresponding Lagrangian are given. Two parameters characterizing nonminimal electromagnetic interactions of fermions are introduced, and the quantum-mechanical Hamiltonian is found. It is proved that there is only causal propagation of waves in the approach considered.

scholarly journals Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 473
Author(s):  
Joshua Baines ◽  
Thomas Berry ◽  
Alex Simpson ◽  
Matt Visser
Keyword(s):  
Jacobi Equation ◽  
Gordon Equation ◽  
Rotation Parameter ◽  
Slow Rotation ◽  
Square Root ◽  
Killing Tensor ◽  
Constant Of Motion ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation ◽  
Klein Gordon Equation

Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Lense–Thirring spacetime, which has some particularly elegant features, including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics—the “rain” geodesics. At the linear level in the rotation parameter, this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton–Jacobi equation. Furthermore, we shall show that the Klein–Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing–Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.

scholarly journals Modified scattering for the nonlinear nonlocal Schrödinger equation in one-dimensional case

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Time ◽  
Pseudodifferential Operators ◽  
Third Order ◽  
Asymptotics Of Solutions ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Klein Gordon ◽  
Math Phys

AbstractWe study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity $$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}\left( u-\partial _{x}^{2}u\right) +\partial _{x}^{2}u-a\partial _{x}^{4}u=\lambda \left| u\right| ^{2}u,\text { } t>0,{\ }x\in {\mathbb {R}}\mathbf {,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,{\ }x\in {\mathbb {R}}\mathbf {,} \end{array} \right. \end{aligned}$$ i ∂ t u - ∂ x 2 u + ∂ x 2 u - a ∂ x 4 u = λ u 2 u , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where $$a>\frac{1}{5},$$ a > 1 5 , $$\lambda \in {\mathbb {R}}$$ λ ∈ R . We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the $${\mathbf {L}}^{2}$$ L 2 -boundedness of the pseudodifferential operators.

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