Supersymmetric Biorthogonal Quantum Systems

We discuss supersymmetric biorthogonal systems, with emphasis given to the periodic solutions that occur at spectral singularities of PT symmetric models. For these periodic solutions, the dual functions are associated polynomials that obey inhomogeneous equations. We construct in detail some explicit examples for the supersymmetric pairs of potentials V_{+/-}(z) = -U(z)^2 +/- z(d/(dz))U(z) where U(z) = \sum_{k>0}u_{k}z^{k}. In particular, we consider the cases generated by U(z) = z and z/(1-z). We also briefly consider the effects of magnetic vector potentials on the partition functions of these systems.Comment: Changes are made to conform to the published version. In particular, some errors are corrected on pp 12-1

Absence of bound states for waveguides in 2D periodic structures

We study a Helmholtz-type spectral problem in a two-dimensional medium consisting of a fully periodic background structure and a perturbation in form of a line defect. The defect is aligned along one of the coordinate axes, periodic in that direction (with the same periodicity as the background), and bounded in the other direction. This setting models a so-called "soft-wall" waveguide problem. We show that there are no bound states, i.e., the spectrum of the operator under study contains no point spectrum.Comment: This is an updated version of our paper (in slightly different form in Journal of Mathematical Physics). An anonymous reviewer kindly made us aware that ref. 10 is not applicable in our situation. An application of the theorem in ref. 10 would have proved the absence of singular continuous spectrum also. Our result on the absence of point spectrum is not affected by thi

Kustaanheimo-Stiefel Regularization and the Quadrupolar Conjugacy

In this note, we present the Kustaanheimo-Stiefel regularization in a symplectic and quaternionic fashion. The bilinear relation is associated with the moment map of the $S^{1}$- action of the Kustaanheimo-Stiefel transformation, which yields a concise proof of the symplecticity of the Kustaanheimo-Stiefel transformation symplectically reduced by this circle action. The relation between the Kustaanheimo-Stiefel regularization and the Levi-Civita regularization is established via the investigation of the Levi-Civita planes. A set of Darboux coordinates (which we call Chenciner-F\'ejoz coordinates) is generalized from the planar case to the spatial case. Finally, we obtain a conjugacy relation between the integrable approximating dynamics of the lunar spatial three-body problem and its regularized counterpart, similar to the conjugacy relation between the extended averaged system and the averaged regularized system in the planar case.Comment: 19 pages, corrected versio

Fluctuations in Polymer Translocation

We investigate a model of chaperone-assisted polymer translocation through a nanopore in a membrane. Translocation is driven by irreversible random sequential absorption of chaperone proteins that bind to the polymer on one side of the membrane. The proteins are larger than the pore and hence the backward motion of the polymer is inhibited. This mechanism rectifies Brownian fluctuations and results in an effective force that drags the polymer in a preferred direction. The translocated polymer undergoes an effective biased random walk and we compute the corresponding diffusion constant. Our methods allow us to determine the large deviation function which, in addition to velocity and diffusion constant, contains the entire statistics of the translocated length.Comment: 20 pages, 6 figure

Biorthogonal Quantum Systems

Models of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporporate all the structure of PT symmetric models, and allow for generalizations, especially in situations where the PT construction of the dual space fails. The formalism is illustrated by a few exact results for models of the form H=(p+\nu)^2+\sum_{k>0}\mu_{k}exp(ikx). In some non-trivial cases, equivalent hermitian theories are obtained and shown to be very simple: They are just free (chiral) particles. Field theory extensions are briefly considered.Comment: 34 pages, 5 eps figures; references added and other changes made to conform to journal versio

Differential constraints and exact solutions of nonlinear diffusion equations

The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries

Classification of integrable discrete Klein-Gordon models

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein-Gordon type equations on quad-graphs. The list of equations passing the test is presented containing several well-known integrable models. A new integrable example is found, its higher symmetry is presented.Comment: 12 pages, submitted to Physica Script

Synthesis and characterization of core-shell structure silica-coated Fe29.5Ni70.5 nanoparticles

In view of potential applications of magnetic particles in biomedicine and electromagnetic devices, we made use of the classical Stober method base-catalysed hydrolysis and condensation of tetraethoxysilane (TEOS) to encapsulate FeNi nanoparticles within a silica shell. An original stirring system under high power ultrasounds made possible to disperse the otherwise agglomerated particles. Sonication guaranteed particles to remain dispersed during the Stober synthesis and also improved the efficiency of the method. The coated particles are characterized by electron microscopy (TEM) and spectroscopy (EDX) showing a core-shell structure with a uniform layer of silica. Silica-coating does not affect the core magnetic properties. Indeed, all samples are ferromagnetic at 77 K and room temperature and the Curie point remains unchanged. Only the coercive force shows an unexpected non-monotonous dependence on silica layer thickness.Comment: Regular paper submited to international peer-reveiwed journa