Spinor with Schrödinger symmetry and non-relativistic supersymmetry

We construct the d-dimensional “half” Schrödinger equation, which is a kind of the root of the Schrödinger equation, from the (d+1)-dimensional free Dirac equation. The solution of the “half” Schrödinger equation also satisfies the usual free Schrödinger equation. We also find that the explicit transformation laws of the Schrödinger and the half Schrödinger fields under the Schrödinger symmetry transformation are derived by starting from the Klein-Gordon equation and the Dirac equation in d+1 dimensions. We derive the 3- and 4-dimensional super-Schrödinger algebra from the superconformal algebra in 4 and 5 dimensions. The algebra is realized by introducing two complex scalar and one (complex) spinor fields and the explicit transformation properties have been found.     

Holography with Schrödinger potentials

We examine the analogue one-dimensional quantum mechanics problem associated with bulk scalars and fermions in a slice of AdS₅. The “Schrödinger” potential can take on different qualitative shapes depending on the values of the mass parameters in the bulk theory. Several interesting correlations between the shape of the Schrödinger potential and the holographic theory exist. We show that the quantum mechanical picture is a useful guide to the holographic theory by examining applications from phenomenology.     

Dynamical barrier for the formation of solitary waves in discrete lattices

We consider the problem of the existence of a dynamical barrier of “mass” that needs to be excited on a lattice site to lead to the formation and subsequent persistence of localized modes for a nonlinear Schrödinger lattice. We contrast the existence of a dynamical barrier with its absence in the static theory of localized modes in one spatial dimension. We suggest an energetic criterion that provides a sufficient, but not necessary, condition on the amplitude of a single-site initial condition required to form a solitary wave. We show that this effect is not one-dimensional by considering its two-dimensional analog. The existence of a sufficient condition for the excitation of localized modes in the non-integrable, discrete, nonlinear Schrödinger equation is compared to the dynamics of excitations in the integrable, both discrete and continuum, version of the nonlinear Schrödinger equation.     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

On the role of symmetries in the theory of photonic crystals

We discuss the role of the symmetries in photonic crystals and classify them according to the Cartan–Altland–Zirnbauer scheme. Of particular importance are complex conjugation C$C$ and time-reversal T$T$, but we identify also other significant symmetries. Borrowing the jargon of the classification theory of topological insulators, we show that C$C$ is a “particle–hole-type symmetry” rather than a “time-reversal symmetry” if one considers the Maxwell operator in the first-order formalism where the dynamical Maxwell equations can be rewritten as a Schrödinger equation; The symmetry which implements physical time-reversal is a “chiral-type symmetry”. We justify by an analysis of the band structure why the first-order formalism seems to be more advantageous than the second-order formalism. Moreover, based on the Schrödinger formalism, we introduce a class of effective (tight-binding) models called Maxwell–Harper operators. Some considerations about the breaking of the “particle–hole-type symmetry” in the case of gyrotropic crystals are added at the end of this paper.     

Resonant control of solitons

It is shown that the effect of “scattering on resonance” can be used to control envelope solitons in the driven nonlinear Schrödinger equation. The control occurs by the frequency modulated driving with multiple crossing of the resonant frequency of the soliton.     

Numerical modeling of spatio-temporal solitons using an adaptive spherical Fourier Bessel split-step method

We present a novel technique to numerically solve pulsed optical beam or “light bullet” propagation in bulk nonlinear media based on the scalar nonlinear Schrödinger (NLS) equation in three dimensions with spherical symmetry. Using fast algorithms for spherical Fourier Bessel transforms along with adaptive longitudinal stepping and transverse grid management in a symmetrized split-step technique, it is possible to accurately study many nonlinear effects, including the possibility of spatio-temporal collapse, or the collapse-arresting mechanism due to saturable nonlinearity.     

Simple analytical expression for work function in the “nearest neighbour” approximation

Nonlocal operator of potential is suggested, based on the “nearest neighbour” approximation (NNA) for single electron wave function in metals. It is shown that Schrödinger equation with nonlocal potential leads to quite simple analytical expression for work function, which surprisingly well fits to experimental data.     

Nonlinear von neumann equations for quantum dissipative systems

For pure states nonlinear Schrödinger equations, the so-called Schrödinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed.Supported by Deutsche ForschungsgemeinschaftSupported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project no. 3569     

Relative to the interpretation of terms of the Schrödinger perturbation theory series

An approach different from those ordinarily expounded is proposed for the construction of the Schrödinger perturbation theory. This approach sets up a functional dependence between the energy corrections and the coefficients characterizing the corrections to the wave functions. It is shown that within the framework of the Schrödinger theory these coefficients are determined by extremum conditions of fourth-order corrections to the energy. In the Schrödinger approximation the fourth-order energy corrections reach their extremal values. The approach developed includes a proof of the Dalgarno and Stewart theorem that says that the energy can be estimated to the order (2n + 1) if the wave function is known to n-th-order accuracy.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 18–20, February, 1982.     

Quantization in classical mechanics and its relation to the Bohmian Ψ-field

Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of Hamilton systems in the form of the Schrödinger equation.It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with the Bohm “quantum” potential. Within the frame-work of Bohmian quantum mechanics supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Schrödinger equation is equivalent semantically and syntactically to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters. The conditions for the correctness of trajectory interpretation of quantum mechanics are discussed.     

Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.     

Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations. II

This is our second paper devoted to the study of some non-linear Schrödinger equations with random potential. We study the non-linear eigenvalue problems corresponding to these equations. We exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails. Contrary to our results in the first paper, the results established in the present paper hold for an arbitrary strength of the non-linear (cubic) term in the non-linear Schrödinger equation.     

Analytic bounds on transmission probabilities

We develop some new analytic bounds on transmission probabilities (and the related reflection probabilities and Bogoliubov coefficients) for generic one-dimensional scattering problems. To do so we rewrite the Schrödinger equation for some complicated potential whose properties we are trying to investigate in terms of some simpler potential whose properties are assumed known, plus a (possibly large) “shift” in the potential. Doing so permits us to extract considerable useful information without having to exactly solve the full scattering problem.     

Optimally stabilized beam pairs

In this paper, evolution of two co-axially co-propagating beams with the propagation distance in a Kerr type nonlinear medium has been analyzed for all possible physical parameters and situations. For the first time, the paper analyzes useful “optimally stabilized” (minimum width oscillations of beams with distance of propagation) pairing of bright-bright, bright-dark, dark-dark beams of same/different wavelengths and of same/different widths using coupled nonlinear Schrödinger equation (NLSE) that has been solved using split-step Fourier method. Soliton pairs of two unequal beam widths are known to be impossible, however, it is shown that “optimally stabilized” pairing is possible in such cases. Existence surfaces for optimally stabilized pairing have also been revealed for different interaction coefficient of the coupled beams.     

Geometric dequantization

Dequantization is a set of rules which turn quantum mechanics (QM) into classical mechanics (CM). It is not the WKB limit of QM. In this paper we show that, by extending time to a 3-dimensional “supertime,” we can dequantize the system in the sense of turning the Feynman path integral version of QM into the functional counterpart of the Koopman-von Neumann operatorial approach to CM. Somehow this procedure is the inverse of geometric quantization and we present it in three different polarizations: the Schrödinger, the momentum and the coherent states ones.     

Klein-Gordon-Wheeler-DeWitt-Schrödinger equation

We start from the Einstein-Hilbert action for the gravitational field in the presence of a “point particle” source, and cast the action into the corresponding phase space form. The dynamical variables of such a system satisfy the point particle mass shell constraint, the Hamilton and the momentum constraints of the canonical gravity. In the quantized theory, those constraints become operators that annihilate a state. A state can be represented by a wave functional Ψ that simultaneously satisfies the Klein-Gordon and the Wheeler-DeWitt-Schrödinger equation. The latter equation, besides the term due to gravity, also contains the Schrödinger like term, namely the derivative of Ψ with respect to time, that occurs because of the presence of the point particle. The particle?s time coordinate, X⁰, serves the role of time. Next, we generalize the system to p-branes, and find out that for a quantized spacetime filling brane there occurs an effective cosmological constant, proportional to the expectation value of the brane?s momentum, a degree of freedom that has two discrete values only, a positive and a negative one. This mechanism could be an explanation for the small cosmological constant that drives the accelerated expansion of the universe.