Mathematical Methods of Physics

A theoretical study of a planar electronic waveguide with a uniformly curved section in the perpendicular homogeneous magnetic field B is presented within the envelope function approximation. Utilizing analytical solutions in each part of the waveguide, exact expressions are derived for the scattering and reflection matrices and for the transcendental equation defining bound-state energies. It is shown that in the magnetic field a propagation threshold in the continuously curved channel is always smaller than its counterpart for the straight arm which means that bound states in the uniform magnetic field always exist. Their energies do not depend on the direction of the field, and at high magnetic intensities they approach the lowest Landau level. For the transport in the fundamental mode an interaction of a quasibound level split off from the higher-lying threshold as a result of the bend, with its degenerate continuum counterpart, causes a dip in the transmission. In the magnetic field, contrary to the field-free case, conductance in the minimum G min , generally, ceases to be zero. It is shown that growing magnetic fields cause G min to saturate to 2e 2 / h which means that a quasibound level formed as a result of the bend is completely dissolved by the increasing B; however, this transformation is very different for the different bend angles and radii. In particular, quasibound states of the fundamental propagation mode survive stronger fields for the smaller bend angles which is explained by the larger total magnetic flux through the curved section where these levels are formed. Since a magnetic length l B = ͑ប / eB͒ 1/2 is inversely proportional to the square root of B, states for the waveguide with a smaller radius also survive stronger fields, and their asymptotic approach to the dissolution possesses nonmonotonic G min dependence on the magnetic field with minimum conductance again reaching zero for some special values of B. Vortex structure of the currents flowing in the waveguide near the resonance is strongly affected by the field. In particular, small magnetic intensities change zero-field vortices in the straight arms into the magnetic antivortices which correspond to the interacting with each other surface currents flowing along opposite walls of the channel. Increasing the magnetic field suppresses the formation of the vortices pushing currents to the outer ͑inner͒ walls in the straight ͑bent͒ section. For fields larger than the saturation magnetic intensity, the only consequence of the bend is a strong surface current near the convex wall of the bend, and the electronic flow along the junctions between straight and curved parts. FIG. 5. ͑Color online͒ Wave function ⌿͑x , y͒ ͑normalized to its maximum͒ of the bound state for 0 = 0.001 and the right angle for ͑a͒ B =0, ͑b͒ B = 10, ͑c͒ B = 20, and ͑d͒ B = 35. O. OLENDSKI AND L. MIKHAILOVSKA PHYSICAL REVIEW B 72, 235314 ͑2005͒ 235314-10

Properties of the two-dimensional ring and three-dimensional infinitely long straight hollow waveguide with unit width and inner radius ρ 0 in the superposition of the longitudinal uniform magnetic field B and the Aharonov-Bohm flux are analyzed within the framework of the scalar Helmholtz equation under the assumption that the Robin boundary conditions at the inner and outer confining walls contain extrapolation lengths Λ in and Λ out , respectively, with nonzero imaginary parts. As a result of this complexity, the selfadjointness of the Hamiltonian is lost, its eigenvalues E, in general, become complex too and the discrete bound states of the annulus, which are characteristic for the real Λ, turn into the corresponding quasibound states with their lifetime defined by the imaginary parts E i of the eigenenergies while the current along the wire exponentially amplifies/attenuates with the distance depending on the sign of E i . It is shown that, compared to the disk geometry, the annulus opens up additional possibilities of varying magnetization and currents by tuning imaginary components of the Robin parameters on each confining circumference; in particular, the possibility of restoring a lossless longitudinal flux by zeroing E i is discussed from mathematical and physical points of view. For each ρ 0 , the energy E turns real under a special correlation between the imaginary parts of Λ in and Λ out with the opposite signs being what physically corresponds to the equal transverse fluxes through the inner and outer interfaces of the annulus. In the asymptotic case of the very large radius, ρ 0 → ∞, simple expressions are derived and applied to the analysis of the dependence of the real energy E on Λ in and Λ out . New features also emerge in the magnetic field influence; for example, if, for the quantum disk, the imaginary energy E i is quenched by the strong intensities B, then for the annulus this takes place only when the inner Robin distance Λ in is real; otherwise, it almost quadratically depends on B with the corresponding enhancement of the reactive scattering. The closely related problem of the hole in the otherwise uniform medium is also addressed for the real and complex extrapolation lengths with the emphasis on the comparative analysis with its dot counterpart.

A straight quasi-one-dimensional Dirichlet wave guide with a Neumann window of length L on one or two confining surfaces is considered theoretically with and without perpendicular homogeneous magnetic field B. It is shown that for the field-free case, a bound state in the continuum (BIC) for one Neumann window exists for some critical lengths only, while for the two Neumann segments symmetrically located on the opposite walls, due to the restored transverse symmetry of the system, BICs exist for the arbitrary L. Bound states lying below the fundamental propagation threshold of the Dirichlet strip survive any strength of the uniform magnetic field and do not depend on its direction. Moreover, an increasing field induces new bound states regularly arranged with the levels present at B = 0. For two Neumann windows, strong magnetic fields lead to the degeneracy of the adjacent odd and even bound states with their energies almost equal to each other and to their corresponding counterpart for one Neumann segment, which is explained by mapping the problem onto the field-free one or two purely attractive one-dimensional quantum wells with field-dependent depth. Miscellaneous magnetotransport characteristics of the structures are also considered; in particular, it is demonstrated that small fields applied to the channel with two Neumann windows destroy BICs by coupling them to the continuum states. This is manifested in the conductance-Fermi energy dependence by Fano resonances. Currents flowing in the wave guide are investigated too, and it is shown that current density patterns near the resonances form vortices which change their chirality as energy sweeps through the resonant region. Generalizations to any other arbitrary combination of the boundary conditions are provided. Comparison with other structures such as window-coupled Dirichlet wave guides, a bent strip or straight Dirichlet channel with electrostatic impurity inside, is performed.

A theoretical study of a waveguide with a uniformly curved section and an embedded quantum dot is presented within the envelope function approximation. For the quantum dot being extremely localized in the direction of the electron propagation, exact form of the scattering matrix of the system is derived, and a conductance of the waveguide is calculated. It is shown that if, for the straight wire with dot, in the fundamental mode conductance is a monotonically increasing function of the Fermi energy, then, after bending the waveguide, one has Fano resonances on the conductance-Fermi energy dependence. The resonances appear as a result of mixing by the bend of the longitudinal and transverse electron motion in the straight parts of the waveguide and concomitant intersubband interaction. For the small bend angle ͑large bend radius͒ the width of the resonance grows narrower with decreasing the angle ͑increasing the radius͒, until in the limit of the zero angle ͑infinite radius͒ one recovers true bound state in the continuum and the corresponding monotonic conductance of the straight channel. A dependence of the resonance width on the parameters of the bend and the dot is investigated; in particular, it is shown that, as a result of coherent resonant phenomena in the superposition of the bend and the dot, true bound states in the continuum can be formed also for the nonzero bend angle and the finite bend radius. Miscellaneous cases of bending the waveguide which, if uncurved, already exhibits the Fano resonance, are also investigated; for example, it is shown that in this situation, as a result of the bend, resonances can collapse too, again producing true bound states in the continuum. The case of inversion of the Fano resonances, i.e., a change of their minimum and maximum mutual location, is also analyzed. Mathematical and physical interpretation of the obtained results is given, and characteristic features of the critical parameters at which the Fano resonances collapse, are discussed. It is demonstrated that currents flowing in the waveguide, near the Fano resonances drastically change their behavior from laminar to vortical structure, and an evolution of the vortices is described. Parallels are drawn to the other types of the guiding structures, such as electromagnetic waveguides and acoustic ducts.

Abstract: We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable $ z $(spectral parameter) and the other a recurrence relation in $ n $(the lattice variable). For the Jacobi weight

We present a theoretical study of a planar waveguide with a uniformly curved section. Opposite sides of the channel satisfy different boundary conditions. It is shown that if the Dirichlet condition is applied to the inner side of the strip and the Neumann one to the outer wall, then properties of such a system in many respects resemble those with the Dirichlet requirements on both surfaces. Namely, in both cases a propagation threshold for the curved section is smaller than its counterpart for the straight channel. As a consequence, a localized mode exists with its energy below the propagation threshold of the straight waveguide. Analysis of such states is presented as a function of the bend parameters. For the transport in the fundamental mode an interaction of a quasibound level split off from the higher-lying threshold, with its degenerate continuum counterpart, causes a dip in the transmission. Such a resonance is characterized by a location of its zero minimum E min and the half width ⌫. Changing the bend angle and radius, one varies E min and ⌫. In particular, for some critical parameters of the bend it is possible to turn the half width to zero, i.e., to eliminate the dip in the transmission. This corresponds to the absence of the interaction between the split-off level and the continuum, and, consequently, to the formation of the true bound state in the continuum. Vortex structure of the currents flowing in the waveguide near the resonance is also shown to strongly resemble the analogous results for the Dirichlet case. It is pointed out that the properties of the waveguide with the Neumann inner condition and the Dirichlet outer one mimic the duct with the Neumann requirements on the two sides, since for both these cases the propagation threshold in the curved section is greater than in the straight channel.

The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function ${\hspace{0pt}}_2F_1(a, b;c;z)$ in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of ${\hspace{0pt}}_2F_1(a, b;c;z)$ have been provided in the literature, they are all limited by a narrow domain of validity, either in the complex plane of the variable or in the parameter space. Overcoming this restriction, we provide new asymptotic expansions for the hypergeometric function with two large parameters, which are valid for the entire complex plane of z except for a few specific points. We show that these expansions work well even when we approach the possible singularity of ${\hspace{0pt}}_2F_1(a, b;c;z)$ , $\vert z\vert =1$ , where the current expansions typically fail. Using our results we determine asymptotically the partition function of a lattice gas in a field of traps in the different possible physical limits of few/many particles and few/many traps, illustrating the applicability of the derived asymptotic expansions of the HGF in physics.

A 3-bracket variant of the Virasoro–Witt algebra is constructed through the use of su(1, 1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.

The main intension of this paper is to extract new and further general analytical wave solutions to the (2 þ 1)dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation in the sense of conformable derivative by implementing the advanced exp ðÀϕ ðξÞÞ-expansion method. This method is a particular invention of the generalized exp ðÀϕ ðξÞÞ-expansion method. By the virtue of the advanced exp ðÀϕ ðξÞÞ-expansion method, a series of kink, singular kink, soliton, combined soliton, and periodic wave solutions are constructed to our preferred space time-fractional (2 þ 1)-dimensional AKNS equation. An extensive class of new exact traveling wave solutions are transpired in terms of, hyperbolic, trigonometric, and rational functions. To express the underlying propagated features, some attained solutions are exhibited by making their three-dimensional (3D), twodimensional (2D) combined, and 2D line plot with the help of computational packages MATLAB. All plots are given to show the proper wave features through the founded solutions to the studied equation with particular preferring of the selected parameters. Moreover, it may conclude that the attained solutions and their physical features might be helpful to comprehend the water wave propagation in water wave mechanics.

We study complex integrable systems on quiver varieties associated with the cyclic quiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero-Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type $A_{n−1}$ and $B_n$, as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.

A theoretical study of a waveguide with a uniformly curved section is presented within the envelope function approximation. Utilizing analytical solutions in each part of the waveguide, exact expression of the scattering matrix of the system is derived. Based on it, a conductance of the waveguide is calculated for the wide range of the bend angle and radius. It is shown that a quasibound state formed as a result of the bend, at some critical parameters of the curve becomes a true bound state with infinite lifetime. It has its degenerate continuum counterpart, but does not interact with it. As a result of a constructive resonant interference in the bend, dip in the conductance, which is an essential property for the noncritical values, vanishes with full transmission being observable instead. Mathematical and physical interpretation of these results is given, and characteristic features of the critical parameters are discussed. Comparison with quantum waveguides with other types of nonuniformity is performed.

In this paper we reproduce the continuum model of electro-osmotic oscillations at a non-charged porous membrane and study their onset with a focus on the singular nature of this transmission (singular Hopf bifurcation), resulting in a rapid transformation of the oscillations, harmonic at the onset, into the jerky ones of relaxation type.

The dynamics of even topological open membranes relies on Nambu brackets. Consequently, such 2p-branes can be quantized through the consistent quantization of the underlying Nambu dynamical structures. This is a summary construction relying on the methods detailed in Refs. [New J. Phys. 4 (2002) 83.1; Phys. Rev. D 68 (2003)].

The exponential solutions of the Darboux equations for conjugate nets is considered. It is shown that rank-one constraints over the right derivatives of invertible operators on an arbitrary linear space give solutions of the Darboux system, which can be understood as a vectorial Darboux transformation of the exponential background. The method is extended further to obtain vectorial Darboux transformations of the Darboux system.

We describe the relation between operators of invariant differentiation and invariant operators on orbits of Lie group actions. We propose a new effective method for finding differential invariants and operators of invariant differentiation and present examples.

We construct the general and N-soliton solutions of an integro-differential Schrödinger equation with a nonlocal nonlinearity. We consider integrable nonlinear integro-differential equations on the manifold of an arbitrary connected unimodular Lie group. To reduce the equations on the group to equations with a smaller number of independent variables, we use the method of orbits in the coadjoint representation and the generalized harmonic analysis based on it. We demonstrate the capacities of the algorithm with the example of the SO(3) group.

It is shown that if the frequency of oscillations in the wells between three barriers satisfy the condition hw~E"-E'(where E' and E" are sublevels formed as a result of the overlap ofwavefunctions in adjacent wells), the probability of passage through one ofthe inelastic channels has a prominent maximum. The resemblance and distinction of rectangular and trapezoidal barriers are considered. Ways to increase the frequency are discussed.

In this paper we have tried to deduce the possible origin of particle and evolution of their intrinsic properties through spiral dynamics.  We consider some of the observations which include exponential mass function of particles following a sequence when fitted on logarithmic potential spiral, inwardly rotating spiral dynamics in Reaction-Diffusion System, the separation of Electron’s Spin-Charge-Orbit into quasi-particles. The paper brings a picture of particles and their Anti Particles in spiral form and explains how the difference in structure varies their properties. It also explains the effects on particles in Accelerator deduced through spiral dynamics.

The Nambu-bracket quantization of the hydrogen atom is worked out as an illustration of the general method. The dynamics of topological open branes is controlled classically by Nambu brackets. Such branes then may be quantized through the consistent quantization of the underlying Nambu brackets: properly defined, the quantum Nambu-brackets com- prise an associative structure, although the naive derivation property is mooted through operator entwinement. For superintegrable systems, such as the hydrogen atom, the re- sults coincide with those furnished by Hamiltonian quantization — but the method is not limited to Hamiltonian systems.

The problem of coupling between spin and torsion is analysed from a Lyra's manifold background for scalar and vector massive fields using the Duffin-Kemmer-Petiau (DKP) theory. We found the propagation of the torsion is dynamical, and the minimal coupling of DKP field corresponds to a non-minimal coupling in the standard Klein-Gordon-Fock and Proca approaches. The origin of this difference in the couplings is discussed in terms of equivalence by surface terms.

A new 'wave-particle non-dualistic interpretation of quantum mechanics at a single-quantum level' is presented by interpreting the Schrödinger wave function as an 'instantaneous resonant spatial mode' (IRSM) to which a quantum is confined and moves akin to the case of a test particle in the curved space-time of the general theory of relativity. This union of the wave and particle natures into a single entity is termed as non-duality. Using quantum formalism, the IRSM is shown to induce dual-vectors at the boundaries and interacts according to the inner-product. The overall phase associated with the state vector, which never contributes to the inner-product, is related to a particular eigenstate of an observable where the particle resides. This eigenstate becomes the natural outcome during observation of a single-quantum's single event. Observation over a large number of identical quanta, differing only by overall phases, results in the relative frequency of detection which yields Born's rule as a limiting case proving the absence of measurement problem in quantum mechanics. The non-duality not only provides the actual mechanism for the 'wave function collapse' but also statistically becomes equivalent to the Copenhagen interpretation. An explicit derivation for the Born rule using individual quantum events is provided by considering an example of spin-1/2 particles in the Stern-Gerlach experiment. In this regard, a generalized representation for the SU (2) algebra, facilitating the description of single-quantum events, is constructed without any deviations from the quantum formalism.

Some interesting inter-connections between solitons, non-linear sigma-models, and gravity (in two and four dimensions) are discussed. Certain sigma-models and non-constant scalar curvature metrics are constructed from generalized solitons. Speculation is presented whether such metrics can be transformed (by a suitable change of coordinates) to black hole metrics.

The Einstein theory of relativistic gravity encoded in the General Relativity Theory (GRT) is investigated from a holographic statistical geometrophysical viewpoint done so here for the first time. In so doing, the arguments are carried out systematically and the four laws of geometrodynamics are enunciated with a proper reasonable development. To do so, new objects characterizing the quantum geometry christened “geomets” are proposed to exist and it is also proposed that there exist geometrodynamic states that these geomets occupy. The geometrodynamic states are statistical states of different curvature and when occupied determine the geometry of the spacetime domain under scrutiny and thereby tell energy-momentum how to behave and distribute. This is a different take and the theory is developed further by developing the idea that the quantities appearing in the Einstein field equations are in fact physically realistic and measurable quantities called the geometrodynamic state functions. A complete covariant geometrodynamic potential theory is then developed thereafter. Finally a new quantity called the “collapse index” is defined and how the spacetime geometry curves is shown as a first order geometrodynamic phase transition using information bit saturation instead of the concept of temperature. Relationship between purely and only geometry and information is stressed throughout. . The statistical formula relating curvature and probability is inverted and interpretation is provided. This is followed by a key application in the form of a correspondence between the Euler-Poincaré formula and the proposed “extended Gibbs formula”. In an appendix the nub of the proof of the Maldacena conjecture is provided.

In solving for static electric and magnetic fields in the presence of an object one usually assumes for simplicity the object to be of infinite extent in the direction perpendicular or parallel to the field applied. This leads to the paradoxes of finite electric and magnetic fields inside the superconductor. Their origin is traced to the unphysical assumption of an infinite extent and serves as a reminder that one has to take care in forming limits.

The fully developed steady velocity field in pressure gradient driven laminar flow of non-linear viscoelas-tic fluids with instantaneous elasticity constitutively represented by a class of single mode, non-affine quasilinear constitutive equations is investigated in straight pipes of arbitrary contour. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour. The analytical method presented is capable of predicting the velocity field in tubes with arbitrary cross-section. The base flow is the Newtonian field and is obtained at O(1). Field variables are expanded in asymptotic series in terms of the Weissenberg number Wi. The analysis does not place any restrictions on the smallness of the driving pressure gradients which can be large and applies to dilute and weakly elastic non-linear vis-coelastic fluids. The velocity field is investigated up to and including the third order in Wi. The Newtonian field in general arbitrary contours is obtained and longitudinal velocity field components due to shear-thinning and to non-linear viscoelastic effects are identified. Third order analysis shows a further contribution to the longitudinal field driven by first normal stress differences. Secondary flows driven by unbalanced second normal stresses in the cross-section manifest themselves as well at this order. Longitudinal equal velocity contours, the secondary flow field structure, the first and the second normal stress differences as well as wall shear stress variations are discussed for several non-circular contours some for the first time.

Having adopted the fundamental distinction between Perceptible space and Geometrical space, we select the Projective space as Geometrical space of choice for the description of the correlations between the natural world's elements. This description is attempted by the Theory of the harmonicity of the field of light, which always introduces a localized observer or a scientific instrument. As the speed of light is finite, the results of our observations and measurements of a linearly moving element of matter always refer not to its present position, but rather to a previous one, which we call Conjugate position, (the " shadow " in the Plato's Allegory of the Cave). In the Projective Geometrical space the Conjugate positions, in a linear motion, are two which are harmonically connected through the observer. As a consequence of the axioms of the Projective geometry these Conjugate positions are both accepted. Moreover those axioms introduce automatically the Principle of Duality of the Geometrical space's fundamental elements, which leads to the well known Duality in Physics. A result of the Geometrical Principle of Duality is the Inverse square law in the Gravitational field, as well in the Electric field. Thus, this work produces the Inverse square law from First Principles. Theory of the harmonicity of the field of light, Duality, Inverse square law, Briefest Computation Principle (BCP).

Transcendental analytics as an absolute Kantian understanding of things can be extrapolated into the infinitesimal calculus in order to demonstrate the nature of movements as well as the nature of the space. The space per se, containing a domain mathematically explained as a constitutive of a function, gives a light into the considerations of the mathematical infinite as we know it.

The principles to use variables and mathematical methodologies in physics are addressed. A set of refined definitions with designated variables are used to derive the Velocity and Acceleration Theories in Distance Field and Vector Space. Mathematical methodologies such as Linear Algebra and Vector Calculus are used systematically in a step by step derivation process. The proof of the theories can be easily achieved by substitution of the designated variables with a set of parameters that matches the same assumptions and conditions in every step of the derivation process.