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Register a new userComplex plane representations and stationary states in cubic and quintic resonant systems
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Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives related to the physics of Bose-Einstein condensates and Anti-de Sitter spacetime) that admit special analytic solutions and an extra conserved quantity. Here, we develop and explore a complex plane representation for these systems modelled on the related cubic Szego and LLL equations. To demonstrate the power of this representation, we use it to give simple closed form expressions for families of stationary states bifurcating from all individual modes. The conservation laws, the complex plane representation and the stationary states admit furthermore a natural generalization from cubic to quintic nonlinearity. We demonstrate how two concrete quintic PDEs of mathematical physics fit into this framework, and thus directly benefit from the analytic structures we present: the quintic nonlinear Schroedinger equation in a one-dimensional harmonic trap, studied in relation to Bose-Einstein condensates, and the quintic conformally invariant wave equation on a two-sphere, which is of interest for AdS/CFT-correspondence.
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Resonant systems emerge as weakly nonlinear approximations to problems with highly resonant linearized perturbations. Examples include nonlinear Schroedinger equations in harmonic potentials and nonlinear dynamics in Anti-de Sitter spacetime. The classical dynamics within this class of systems can be very rich, ranging from fully integrable to chaotic as one changes the values of the mode coupling coefficients. Here, we initiate a study of quantum infinite-dimensional resonant systems, which are mathematically a highly special case of two-body interaction Hamiltonians (extensively researched in condensed matter, nuclear and high-energy physics). Despite the complexity of the corresponding classical dynamics, the quantum version turns out to be remarkably simple: the Hamiltonian is block-diagonal in the Fock basis, with all blocks of varying finite sizes. Being solvable in terms of diagonalizing finite numerical matrices, these systems are thus arguably the simplest interacting quantum field theories known to man. We demonstrate how to perform the diagonalization in practice, and study both numerical patterns emerging for the integrable cases, and the spectral statistics, which efficiently distinguishes the special integrable cases from generic (chaotic) points in the parameter space. We discuss a range of potential applications in view of the computational simplicity and dynamical richness of quantum resonant systems.
The Floydian trajectory method of quantum mechanics and the appearance of microstates of the Schr{o}dinger equation are reviewed and contrasted with the Bohm interpretation of quantum mechanics. The kinematic equation of Floydian trajectories is analysed in detail and a new definition of the variational derivative of kinetic energy with respect to total energy is proposed for which Floydian trajectories have an explicit time dependence with a frequency equal to the beat frequency between adjacent pairs of energy eigenstates in the case of bound systems. In the case of unbound systems, Floydian and Bohmian trajectories are found to be related by a local transformation of time which is determined by the quantum potential.
Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross-Pitaevskii equation, but not for any other cases.
In 1984 Michael Berry discovered that an isolated eigenstate of an adiabatically changing periodic Hamiltonian $H(t)$ acquires a phase, called the Berry phase. We show that under very general assumptions the adiabatic approximation of the phase of the zeta-regularized determinant of the imaginary-time Schrodinger operator with periodic Hamiltonian is equal to the Berry phase.
The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(lambda,k) associated with physically relevant classes N_{w,alpha}^gamma of nonlocal potentials in corresponding domains D_{gamma,alpha}^delta of the space C^2 of the complex angular momentum lambda and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Theta(lambda,k)/sigma(lambda,k) of two holomorphic functions in D_{gamma,alpha}^delta is obtained by using the Fredholm-Smithies theory for complex k, at first for lambda=l integer, and in a second step for lambda complex (Real(lambda)>-1/2). Finally, we justify the Watson resummation of the partial wave amplitudes in an angular sector of the lambda-plane in terms of the various components of the polar manifold of T with equation sigma(lambda,k)=0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of lambda, this unified representation of the singularities of T also provides an attractive possible description of antiresonances in the lower half-plane of lambda. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit-Wigner hard-sphere picture of antiresonances.
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