Reduction of Refractive Index Contrast Threshold for Photonic Band-Gap in Square Lattices

The threshold of refractive index contrast （RIG） to open a photonic band gap can be reduced by symmetry breaking. For the case of square lattice composed by dielectric cylinders, the absolute band gap is demonstrated by inserting small rods in the centre of the lattices, and the threshold RIC is reduced to 3.8. As for the square lattices composed by air holes in dielectric, the minimal RIC required for an absolute band gap decreases to 2.20.     

The Scattering on the Bound State and the Two-Particle Resonances

In the given paper the scattering of a spinless particle by another spinless particle bound in the external field is considered in the three-dimensional case. The external field is represented by the rectangular well and the two-particle interaction is parametric. The influence of the single-particle basis and of the strength of the two-particle interaction on the resonance structure of the cross-section is investigated in the limit of weak coupling between channels. It is shown that the dependence of the number of resonances N_r on the number of single-particle levels N is given by the following formula: \documentclass{article}\pagestyle{empty}\begin{document}$ N_r = \frac{{N^2 + (N - 4)^2 }}{2}. $\end{document}. The scattering of a particle by another particle bound in the field of a core is considered.     

The Two-Particle Correlation Function for Systems with Long-Range Interactions

Pemantle and Steif provided a sharp threshold for the existence of a robust phase transition (RPT) for the continuous rotator model and the Potts model in terms of the branching number and the second eigenvalue of the transfer matrix whose kernel describes the nearest neighbor interaction along the edges of the tree. Here a RPT is said to occur if an arbitrarily weak coupling with symmetry-breaking boundary conditions suffices to induce symmetry breaking in the bulk. They further showed that for the Potts model RPT occurs at a different threshold than PT (phase transition in the sense of multiple Gibbs measures), and conjectured that RPT and PT should occur at the same threshold in the continuous rotator model. We consider the class of four- and five-state rotation-invariant spin models with reflection symmetry on general trees which contains the Potts model and the clock model with scalarproduct-interaction as limiting cases. The clock model can be viewed as a particular discretization which is obtained from the classical rotator model with state space \(S^1\). We analyze the transition between \(\hbox {PT}=\hbox {RPT}\) and \(\hbox {PT}\ne \hbox {RPT}\), in terms of the eigenvalues of the transfer matrix of the model at the critical threshold value for the existence of RPT. The transition between the two regimes depends sensitively on the third largest eigenvalue.     

Flow-equations for Hamiltonians

Flow-equations are introduced in order to bring Hamiltonians closer to diagonalization. It is characteristic for these equations that matrix-elements between degenerate or almost degenerate states do not decay or decay very slowly. In order to understand different types of physical systems in this framework it is probably necessary to classify various types of these degeneracies and to investigate the corresponding physical behavior. In general these equations generate many-particle interactions. However, for an n-orbital model the equations for the two-particle interaction are closed in the limit of large n. Solutions of these equations for a one-dimensional model are considered. There appear convergency problems, which are removed, if instead of diagonalization only a block-diagonalization into blocks with the same number of quasiparticles is performed.     

The twisting trick for double well Hamiltonians

We show that the use of a twisting trick allows a transparent geometrical analysis of the spectral properties of double well Hamiltonians. In particular one can provenorm resolvent convergence of the relevant Hamiltonians whenever one has two centres of force whose separationR diverges to infinity.     

Nonlinear Effects for Bose Einstein Condensates in Optical Lattices

Cold atoms and, more recently, Bose-Einstein condensates （BEC＇s） in optical lattices have attracted increasing interest since their first realization. In particular, the formal similarity between the wavefunction of a BEC inside the periodic potential of an optical lattice and of the electrons in a crystal lattice has triggered theoretical and experimental efforts alike. Many phenomena from condensed matter physics, such as Bloch oscillations and Landau-Zener tunneling have been shown to be observable also in optical lattices. An important difference between electrons in a crystal lattice and a BEC inside the periodic potential of an optical lattice is the strength of the self interaction and hence the magnitude of the nonlinearity of the system. Electrons in a metal are almost noninteracting, whereas atoms inside a BEC interact strongly. A＇ perturbation approach is appropriate in the former case while in the latter the full nonlinearity must be taken into account. From this feature new physics is expected. Most experiments to date have been carried out in the regime of shallow lattice depth, for which the system is well described by the mean field Gross-Pitaevskii equation with a periodic potential. Moreover, the nonlinearity induced by the mean-field of the condensate has been shown, both theoretically and experimentally, to give rise to instabilities in certain regions of the Brillouin zone. These instabilities are not present in the corresponding linear system, i.e. the electron system. Experimental and theoretical results on the subject of nonlinear Landau-Zener tunneling and nonlinearity-induced instabilities in a Bose-Einstein condensate interacting with an external periodic potential will be presented.     

The Gaussian Model on the Inhomogeneous Fractal Lattices

The decimation real-space renormalization group and spin-rescaling methods are applied to the study of phase transition of the Gaussian model on fractal lattices. It is found that the critical point K^* equals b/2 ( b is the distribution constant of Gaussian model) on nonbranching Koch curves. For inhomogeneous fractal lattices, it is proposed that the b is replaced with b_qi (qi is the coordination number of the site i) and satisfies a certain relation b_qi/b_qj = q_i/q_j. Under this supposition we find that the critical point of the Gaussian model on a branching Koch curve can be expressed uniquely as K^* = b_qi/q_i.     

CPT-Frames for Non-Hermitian Hamiltonians

Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C^d, it is proved that H has unbroken PT- symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.     

CPT-Frames for Non-Hermitian Hamiltonians

Based on the P T-symmetric quantum theory,the concepts of P T-frame,P T-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed.It is proved that the spectrum and point spectrum of a P T-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken P T-symmetric operator are real.For a linear operator H on Cd,it is proved that H has unbroken P Tsymmetry if and only if it has d diferent eigenvalues and the corresponding eigenstates are eigenstates of P T.Given a C P T-frame on K,a new positive inner product on K is induced and called C P T-inner product.Te relationship between the CP T-adjoint and the Dirac adjoint of a densely defined linear operator is derived,and it is proved that an operator which has a bounded CP T-frame is CP T-Hermitian if and only if it is T-symmetric,in that case,it is similar to a Hermitian operator.The existence of an operator C consisting of a CP T-frame is discussed.These concepts and results will serve a mathematical discussion about P T-symmetric quantum mechanics.     

Construction of Covariant Bases for Two-Particle Reactions

A method for the construction of the Lorentz-covariant bases occurring in the decomposition of spinor amplitudes (M-functions) of four massive particle processes is given. Each particle of spin- is described by a spinor state of the type-(s, 0) and/or (0, s), but the essential parts of our method apply whatever the type of spinor state used, and the necessary relations between the hitherto familiar types of spinor states are given, by means of which the relationship between the resulting sets of invariant amplitudes can be made explicit. By way of illustration four worked examples are presented.     

Symmetry of a Two-Particle Equation for Parastates

Abstract

We study hidden symmetry of a two-particle system of equations for parastates. Invariance operators are described for various potentials.     

The self-adjointness of Hermitian Hamiltonians

For several examples of Hermitian operators, the issues involved in their possible self-adjoint extension are shown to conform with recognizable properties in the solutions to the associated classical equations of motion. This result confirms the assertion made in an earlier paper (Ref. 1) that there are sufficient classical symptoms to diagnose any quantum illness.     

Two-Particle Resonances in the Complex Energy Plane

An evaluation of two-particle resonances in the complex energy plane is presented. The representation used in the method consists of bound and antibound single-particle states, Gamow resonances and scattering waves on the complex energy plane. Within this representation the structure of halo nuclei is studied.     

Higher-order Hamiltonians for the trigonometric Gaudin model

Letters in Mathematical Physics - We consider the trigonometric classical r-matrix for $$\mathfrak {gl}_N$$ and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit...     

Cryptohermitian Hamiltonians on Graphs

A family of nonhermitian quantum graphs is proposed and studied via their discretization.     

On the Mourre estimates for Floquet Hamiltonians

Letters in Mathematical Physics - In the spectral and scattering theory for a Schrödinger operator with a time-periodic potential $$H(t)=p^2/2+V(t,x)$$ , the Floquet Hamiltonian $$K=-i\partial...     

Path Integrals for Pseudo-Hermitian Hamiltonians

Inner products in pseudo-Hermitian quantum theories depend on the details of the Hamiltonians themselves, which makes them difficult to calculate. We shall see that, for some questions, the functional integrals for such theories can be calculated without needing to determine the inner product metric. The reason is that their derivation is based on the Heisenberg equations of motion and the canonical commutation relations, which are unchanged. In particular, this can greatly simplify the derivation of Hermitian theories that are equivalent to these pseudo-Hermitian systems.     

Positivity Representations for Non-Hermitian Hamiltonians

We show how 1-dimensional, non-Hermitian, Sturm-Liouville systems, with rational fraction potentials, define a Moment Problem positivity representation, enabling the generation of (numerical and algebraic) converging lower and upper bounds to the (real and imaginary parts of) complex eigenenergies.