Localization properties of eigenstates in waveguides and cavities with corrugated boundaries

In this paper, we investigate the localization properties of eigenstates for the infinite periodic and closed versions of the generalized ripple billiard (exemplifying corrugated waveguides and corrugated cavities, respectively) having Dirichlet and Neumann boundary conditions. In particular, we demonstrate the existence of strongly energy-localized eigenstates that suffer repulsion, in configuration space, from highly corrugated billiard boundaries. Thus, exhibiting the repulsion effect as a universal property present in billiards with corrugated boundaries. We also characterize this repulsion effect (both, in energy and in configuration space) and provide heuristic expressions for the repelled eigenstate profiles in configuration representation.     

Effects of spatial dispersion at intraband transitions in multiple quantum well structures

Effects of spatial dispersion in reflection of light from multiple quantum well structures of different symmetry have been investigated. It has been shown that with inclined incidence of linearly polarized light on the system of symmetry C _∞v , the reflected wave starts to manifest the circular polarization. Upon the incidence of s(p)-polarized light on the structure of the D _2d symmetry, the reflected wave starts to manifest p(s) component, while in the case of the point symmetry C _2v , this phenomenon also occurs for the normal incidence. The magneto-spatial dispersion in the magnetic field lying in the structure plane leads to the same conversion of polarization. Dependences of the polarization-sensitive reflection coefficients on the incidence angle are calculated. The microscopically gyrotropic contributions to the dielectric permittivity of the multiple quantum well structures are calculated for the intraband frequency range. Evaluations show that the effects of spatial dispersion in such systems can be observed experimentally.     

Circumferential and helical normal waves of a cylindrical waveguide: Helical waves in a free space

Properties of circumferential and helical normal waves of a cylindrical waveguide, which appear as aperiodic (in angle) solutions to the Helmholtz equation, are considered. The dispersion characteristics and eigenfunctions of these wave are determined for the Dirichlet and Neumann boundary conditions, and the spatial structure of these waves is described in detail. The properties of helical waves in a free space with cylindrical symmetry are considered.     

Instanton approximation to the graded nonlinear sigma model for the integer quantum hall effect

We construct the multi-instanton solutions for the graded nonlinear σ model with symmetry U(1,1/2)/U(1/1) ⊗ U(1/1), and we calculate the quantum fluctuations around these solutions. The determinant of the fluctuation operator for a fixed multi-instanton solution turns out to be UV finite. However, the integration over instanton parameters contains an integral, ∫d|a| |a|⁻³, over the size, |a|, of each instanton, which is quadratically singular at |a|=0. It is shown that these quadratic divergences cancel exactly in the calculation of all Green functions. The applicability of the present results to the integer quantum Hall effect is discussed.     

The Yang-Mills Heat Semigroup on Three-Manifolds with Boundary

Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space H ₁. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Marini boundary conditions. The last is a nonlinear boundary condition, specified by setting the normal component of the curvature to zero on the boundary. The Yang-Mills heat equation is a weakly parabolic nonlinear equation. We use gauge symmetry breaking to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation. Apriori estimates are developed by first establishing a gauge invariant version of the Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for solutions is also established. Uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity. This work is motivated by possible applications to quantum field theory.     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001     

Entanglement in the supermolecular dimer [Mn4]2

This paper investigates the entanglement in the supermolecular dimer [Mn4]2 consisting of a pair of single molecular magnets with antiferromagnetic exchange-coupllng J. The conventional yon Neumann entropy as a function of the exchange-coupling is calculated explicitly for all eigenstates with the quantum number range from M = M1 ＋ M2 = -9 to 0. It is shown that the yon Neumann entropy is not a monotonic function of the coupling strength. However, it is significant that the entropy of entanglement has the maximum values and the minimum values for most eigenstates, which is extremely useful in the quantum computing. It also presents the time-evolution of entanglement from various initial states. The results are useful in the design of devices based on the entanglement of two molecular magnets.     

Quantum Mechanical Analysis of the Equilateral Triangle Billiard: Periodic Orbit Theory and Wave Packet Revivals

Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by T_rev=9μa²/4?π where a and μ are the length of one side and the mass of the point particle, respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral (60°-60°-60°) triangle, such as the half equilateral (30°-60°-90°) triangle and other “foldings,” which have related energy spectra and revival structures.     

Electron states near surfaces of solids

The mathematical theory of electron eigenstates near the surfaces of solids is developed in stages. The first stage is the study of the eigenstates of solids which terminate at a surface but are otherwise unperturbed; it is proved that near the surface the three-dimensional band edges are “softened” and that van Hove's singularities in the density of states are eliminated. A set of “vacuum states” lying primarily outside the solid is constructed out of plane waves orthogonalized to the eigenstates of the solid. This set of vacuum states is not orthonormal, and it must be orthonormalized by a procedure different from that of Schmidt (which is unsuitable). Effects of surface perturbations are studied. An exact method is elaborated for obtaining eigenfunctions in closed form, for a variety of perturbing potentials that extend arbitrary distances from the surface and include interband matrix elements. It consists of calculating the effects of one surface layer at a time and cumulating the results. The intrinsic instability of certain surfaces against the formation of bands of surface states is shown to be the consequence of the vanishing of a bulk quantity α_zz, one of the components of the inverse-effective mass tensor, at certain points in the B.Z.     

The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

The spectral properties of operators formed from generators of the q-Onsager non-Abelian infinite-dimensional algebra are investigated. Using a suitable functional representation, all eigenfunctions are shown to obey a second-order q-difference equation (or its degenerate discrete version). In the algebraic sector associated with polynomial eigenfunctions (or their discrete analogues), Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz approach is not applicable in related massive quantum integrable models, the eigenfunctions are also described. The spin-half XXZ open spin chain with general integrable boundary conditions is reconsidered in light of this approach: all the eigenstates are constructed. In the algebraic sector which corresponds to special relations among the parameters, known results are recovered.     

Reaction matrix for Dirichlet billiards with attached waveguides

The reaction matrix of a cavity with attached waveguides connects scattering properties to properties of a corresponding closed billiard for which the waveguides are cut off by straight walls. On the one hand, this matrix is directly related to the S-matrix, on the other hand it can be expressed by a spectral sum over all eigenfunctions of the closed system. However, in the physically relevant situation where these eigenfunctions vanish on the impenetrable boundaries of the closed billiard, the spectral sum for the reaction matrix, as it was used before, fails to converge and does not reliably reproduce the scattering properties. We derive here a convergent representation of the reaction matrix in terms of eigenmodes satisfying Dirichlet boundary conditions and demonstrate its validity in the rectangular and the Sinai billiards.     

Relativistic New Yukawa-Like Potential and Tensor Coupling

We approximately solve the Dirac equation for a new suggested generalized inversely quadratic Yukawa potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number ${\kappa}$ . In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions, in closed form, by using the parametric Nikiforov–Uvarov method. The numerical results show that the Coulomb-like tensor interaction, ?T/r, removes degeneracies between spin and p-spin state doublets. The Dirac solutions in the presence of exact spin symmetry are reduced to Schr?dinger solutions for Yukawa and inversely quadratic Yukawa potentials.     

Spin eigenfunctions and operators for the Dn groups

An automated procedure for determining symmetry-adapted spin eigenfunctions is developed for the D_n symmetry groups with n equivalent spins for arbitrary size of I and for n=3,…,6. These eigenfunctions are also eigenfunctions of the vector sum of the n equivalent nuclear spins. Generalized hyperfine spin operators are developed that have real matrix elements and that exploit the full symmetry. These spin operators can be combined with operators for other spins and molecular rotation using only real arithmetic.     

Axially symmetric multi-baryon solutions and their quantization in the chiral quark soliton model

We study axially symmetric solutions with B=2-5 in the chiral quark soliton model. In the background of axially symmetric chiral fields, the quark eigenstates and profile functions of the chiral fields are computed self-consistently. The resultant quark bound spectrum are doubly degenerate due to the symmetry of the chiral field. Upon quantization, various observable spectra of the chiral solitons are obtained. Taking account of the Finkelstein-Rubinstein constraints, we show that the quantum numbers of our solitons coincide with the physical observations for B=2 and 4 while B=3 and 5 do not.     

Phase transition of the two-dimensional classicalXY-model

The two dimensional classicalXY-model, representative for two dimensional models with continuous symmetry, is shown to undergo a phase transition of continuous order in a low temperature region. In particular: 1. The correlation functions decay with a power law at large distance in a temperature range belowT′ with a characteristic singular exponentη which varies continuously between 0 and ∞; 2. For magnetic fieldB → 0 the free energy develops a singular part in the form of a power with a singular exponentκ which varies continuously between 1 and ∞ atT _∞ (<T′); 3. The singular exponentsκ, η, and a correlation length exponent are related to one another by scaling laws and are determined by a dispersion coefficient for low lying excitations.     

The Gaussian Free Field and SLE<Subscript>4</Subscript> on Doubly Connected Domains

The level lines of the Gaussian free field are known to be related to SLE₄. It is shown how this relation allows to define chordal SLE₄ processes on doubly connected domains, describing traces that are anchored on one of the two boundary components. The precise nature of the processes depends on the conformally invariant boundary conditions imposed on the second boundary component. Extensions of Schramm’s formula to doubly connected domains are given for the standard Dirichlet and Neumann conditions and a relation to first-exit problems for Brownian bridges is established. For the free field compactified at the self-dual radius, the extended symmetry leads to a class of conformally invariant boundary conditions parametrised by elements of SU(2). It is shown how to extend SLE₄ to this setting. This allows for a derivation of new passage probabilities à la Schramm that interpolate continuously from Dirichlet to Neumann conditions.     

The Generalized <Emphasis Type="Italic">PT</Emphasis>-Symmetric Sinh-Gordon Potential Solvable within Quantum Hamilton–Jacobi Formalism

The generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. The quasi exact solutions of energy eigenvalues and eigenfunctions of the generalized Sinh-Gordon potential are found for n=0,1 states.     

Quantisations of Piecewise Parabolic Maps on the Torus and their Quantum Limits

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so-called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking.