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Register a new userA Nonlinear Adiabatic Theorem for Coherent States
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Authors
Remi Carles
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We consider the propagation of wave packets for a one-dimensional nonlinear Schrodinger equation with a matrix-valued potential, in the semi-classical limit. For an initial coherent state polarized along some eigenvector, we prove that the nonlinear evolution preserves the separation of modes, in a scaling such that nonlinear effects are critical (the envelope equation is nonlinear). The proof relies on a fine geometric analysis of the role of spectral projectors, which is compatible with the treatment of nonlinearities. We also prove a nonlinear superposition principle for these adiabatic wave packets.
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We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.
Nonlinear fermions of degree $n$ ($n$-fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation $AA^dagger + {A^dagger}^n A^n = 1$. The ($n+1$)-order nilpotency of these operators follows from the existence of unique $A$-vacuum. Supposing appropreate ($n+1$)-order nilpotent para-Grassmann variables and integration rules the sets of $n$-fermion number states, right and left ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The $(n+1)times(n+1)$ matrix realization of the related para-Grassmann algebra is provided. General $(n+1)$-order nilpotent ladder operators of finite dimensional systems are expressed as polynomials in terms of $n$-fermion operators. Overcomplete sets of (normalized) right and left eigenstates of such general ladder operators are constructed and their properties briefly discussed.
We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization. We also indicate how the content of the present paper allows to extend our previous results for the massive case [5] to more general nonlinearities.
We study ground state solutions for linear and nonlinear elliptic PDEs in $mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.,g. higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szego inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy-Littlewood majorant problem in $mathbb{R}^n$ recently obtained by the last two authors of the present paper.
We explore squeezed coherent states of a 3-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered $X_1$-Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three mode squeezed coherent states resolve the identity operator. We evaluate Mandels $Q$-parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Further, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate Wigner function of these three mode squeezed coherent states and demonstrate squeezing property explicitly.
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