Geometry of quantum Riemannian Hamiltonian evolution

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

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Detecting Topological Transitions in Two Dimensions by Hamiltonian Evolution

Physical Review Letters ◽

10.1103/physrevlett.119.197401 ◽

2017 ◽

Vol 119 (19) ◽

Cited By ~ 4

Author(s):

Wei-Wei Zhang ◽

Barry C. Sanders ◽

Simon Apers ◽

Sandeep K. Goyal ◽

David L. Feder

Keyword(s):

Two Dimensions ◽

Topological Transitions ◽

Hamiltonian Evolution

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SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891606000781 ◽

2006 ◽

Vol 03 (02) ◽

pp. 247-267 ◽

Cited By ~ 10

Author(s):

JOHN K. HUNTER

Keyword(s):

Rayleigh Waves ◽

Evolution Equations ◽

Hyperbolic Surface ◽

Smooth Solutions ◽

Scale Invariant ◽

Weakly Nonlinear ◽

Hamiltonian Evolution ◽

Short Time ◽

Nonlinear Scale ◽

Time Existence

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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Operations involving momentum variables in non-Hamiltonian evolution equations

Il Nuovo Cimento B ◽

10.1007/bf02828713 ◽

1988 ◽

Vol 101 (3) ◽

pp. 333-355 ◽

Cited By ~ 24

Author(s):

F. Benatti ◽

G. C. Ghirardi ◽

A. Rimini ◽

T. Weber

Keyword(s):

Evolution Equations ◽

Hamiltonian Evolution

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NONCOMMUTATIVE KdV HIERARCHY

Reviews in Mathematical Physics ◽

10.1142/s0129055x07003036 ◽

2007 ◽

Vol 19 (07) ◽

pp. 677-724 ◽

Cited By ~ 5

Author(s):

FRANÇOIS TREVES

Keyword(s):

Evolution Equations ◽

Vries Equation ◽

Kdv Equation ◽

Completely Integrable Systems ◽

Maximal Abelian Subalgebra ◽

Abelian Subalgebra ◽

Kdv Hierarchy ◽

Noncommutative Version ◽

Constants Of Motion ◽

Hamiltonian Evolution

The noncommutative version of the Korteweg–de Vries equation studied here is shown to admit infinitely many constants of motion and to give rise to a hierarchy of higher-order Hamiltonian evolution equations, each one the noncommutative version of the commutative KdV equation of the same order. The noncommutative KdV polynomials span, topologically, a maximal Abelian subalgebra of the Lie algebra of noncommutative Bäcklund transformations. Two classes of examples of "completely integrable" systems of evolution equations to which the theory applies are described in the last two sections.

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