A Hamiltonian method for finding broadband modal eigenvalues

For shallow water waveguides over a layered elastic bottom, modal eigenvalues can be determined by searching the locations in the complex plane of the horizontal wave number at which the complex phase function is a multiple of π [C. T. Tindle and N. R. Chapman, J. Acoust. Soc. Am. 96, 1777-1782 (1994)]. In this paper, a Hamiltonian method is introduced for tracing the path in the complex plane along which the phase function keeps real. The Hamiltonian method can also be extended to compute the broadband modal eigenvalues or the modal dispersion curves in the Pekeris waveguide with fluid/elastic bottoms. For each proper or leaky normal mode, a different Hamiltonian is constructed in the complex plane and used to trace automatically the complex dispersion curve with the eigenvalue in a reference frequency as the initial value. In contrast to the usual methods, the dispersion curve for each mode is determined individually. The Hamiltonian method shows good performance by comparing with KRAKEN.     

A conformally invariant limit of the critical lattice Ising model

A detailed analysis of an expansion in powers of 1/N (N1) for the Hamiltonian and the transfer-matrix of the Ising chain consisting ofN sites is presented. A special attention is paid to a term proportional to 1/N closely related to the theory of free massless Majorana fermions. An effective Hamiltonian isomorphic to that in conformally invariant theory is derived. The eigenvectors of the Ising Hamiltonian are classified in the framework of conformal algebra representation theory. The momentum and the energy for these states are expressed through the central charge and conformal dimensions. A similar relation for the logarithms of the transfer-matrix eigenvalues is ascertained. These are complex-valued functions of a spectral parameter . A real part of such a function is shown to be proportional to sin , while an imaginary one - to cos. A direct geometrical interpretation of the lattice spectral parameter in the context of the conformally invariant theory for fermions inhabiting a torus is indicated. In other words, these fermions are represented as analytic anticommuting variables double (anti) periodic in a complex plane. The value of the spectral parameter coincides with an angle between these (anti) periods. A general scheme for the above expansion presumably suitable for a wide class of exactly solvable models is conjectured.     

Phases of wave functions and level repulsion

Avoided level crossings are associated with exceptional points which are the singularities of the spectrum and eigenfunctions, when considered as functions of a complex coupling parameter. It is shown that the wave function of one state changes sign but not the other, if the exceptional point is encircled in the complex plane. An experimental setup is suggested where this peculiar phase change could be observed. Received: 7 January 1999 / Received in final form: 15 March 1999     

The Resonance Phenomena Associated with the Time Asymmetry in Non-Hermitian Quantum Mechanics

Resonances are defined as the poles of the scattering matrix. The poles are associated with the complex eigenvalues of the Hamiltonian which are embedded in the lower half of the complex plane. The asymptotes of the corresponding eigenfunctions are exponentially diverged. Therefore, the resonance eigenfunctions are not embedded in the Hermitian domain of the Hamiltonian. The time asymmetric problem is discussed for these types of non-Hermitian Hamiltonians and several solutions of this problem are proposed.     

On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow-fast Hamiltonian systems

We consider a two degree of freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. We assume that at frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under rather general conditions, we prove that there are no stable periodic trajectories of any prescribed period inside the domain of separatrix crossings, except maybe for periodic trajectories passing anomalously close to the saddle point.     

PT symmetry and spontaneous symmetry breaking in a microwave billiard

We demonstrate the presence of parity-time (PT) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its PT symmetry. Spontaneous symmetry breaking occurs at the EP.     

Stability problem in nonlinear wave propagation

An explicit expression for the excitation spectrum of the stationary solutions of a nonlinear wave equation is obtained. It is found that all branches of many-valued solutions of a nonlinear wave equation between the (2K+1,2K+2) turning points (branch points in the complex plane of the nonlinearity parameter) are unstable. Some parts of branches between the (2K,2K+1) turning points are also unstable. The instability of the latter is related to the possibility that pairs of complex conjugate eigenvalues cross the real axis in the κ plane. Zh. éksp. Teor. Fiz. 114, 1487–1499 (October 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Gaussian-Random Ensembles of Pseudo-Hermitian Matrices

Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or well-known number sequences, may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enigmatic Riemann-zeros (1/2±it _n would rather correspond to some PT-symmetric (pseudo-Hermitian) Hamiltonian.     

Detecting level crossings without looking at the spectrum

In many physical systems it is important to be aware of the crossings and avoided crossings which occur when eigenvalues of a physical observable are varied using an external parameter. We have discovered a powerful algebraic method of finding such crossings via a mapping to the problem of locating the roots of a polynomial in that parameter. We demonstrate our method on atoms and molecules in a magnetic field, where it has implications in the search for Feshbach resonances. In the atomic case our method allows us to point out a new class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. In the case of molecules, it enables us to find curve crossings with practically no knowledge of the corresponding Born-Oppenheimer potentials.     

Construction of model hamiltonians in framework of Rayleigh-Schrödinger perturbation theory

A theory of the model Hamiltonians within the framework of Rayleigh-Schrödinger perturbation theory is elaborated. The approach of a model Hamiltonian is based on the assumption that if it is diagonalized in a chosen model space it will yield eigenvalues of the original Hamiltonian in the entire Hilbert space. The theory of the model Hamiltonians may be fruitful as a theoretical background for the study of effective Hamiltonians and as natural extension of the standard Rayleigh-Schrödinger perturbation theory.     

A critique of Zwaan-Stueckelberg phase integral techniques

The Zwaan-Stueckelberg technique, based on semi-classical J.W.K.B. phase integrals and their analytic continuation in the complex plane, is reviewed. Stueckelberg's derivation of Jeffreys' connection formula is discussed, as are his connection formulae for strongly coupled, non-adiabatic collisions involving adiabatic crossings or diabatic non-crossings. His choice of branch cut is clarified. Avoided adiabatic crossings are described using both physical and non-physical branch cuts. Other limitations and defects of the Stueckelberg treatment are examined in detail and most are eliminated. New formulae for the general non-crossing case are presented. They describe perturbed symmetric resonance, reducing in an exactly correct manner for the Rosen-Zener model and also for exact resonance. Within a dynamical adiabatic treatment they also describe perturbed united-atom degeneracy, giving rise to strong rotational coupling in σ-π transitions. The limitation of the Landau-Zener theory to σ-σ transitions is thus avoided.     

Quantum damped oscillator I: Dissipation and resonances

Quantization of a damped harmonic oscillator leads to so called Bateman’s dual system. The corresponding Bateman’s Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.     

New structure in the energy spectrum of reggeon quantum mechanics with quartic couplings

We find general that the energy spectrum of the reggeon hamiltonian with cubic plus quartic interactions at D=0 is more complicated than previously assumed. We observe the phenomenon of “ground state oscillation”, which suggests, for D ≠ 0, the existence of multiple phases. We find that for certain values of the parameters of the theory complex energy eigenvalues occur. We also find complicated level crossings among higher eigenvalues as the quartic coupling strength is increased.     

Topological Invariants in Braid Theory

Many invariants of knots and links have their counterparts in braid theory. Often, these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is the winding number – the net number of times two strings in a braid wrap about each other. But other, higher-order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher-order invariants using only elementary differential geometry.Some of the higher-order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.Some of the higher-order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.     

The Analytic Eigenvalue Structure of the 1+1 Dirac Oscillator

We study the analytic structure for the eigenvalues of the one-dimensional Dirac oscillator,by analytically continning its frequency on the complex plane.A twofold Riemann surface is found,connecting the two states of a pair of particle and antiparticle.One can,at least in principle,accomplish the transition from a positive energy state to its antiparticle state by moving the frequency continuously on the complex plane,without changing the Hamiltonian after transition.This result provides a visual explanation for the absence of a negative energy state with the quantum number n=0.     

Theory of many-body localization in periodically driven systems

We present a theory of periodically driven, many-body localized (MBL) systems. We argue that MBL persists under periodic driving at high enough driving frequency: The Floquet operator (evolution operator over one driving period) can be represented as an exponential of an effective time-independent Hamiltonian, which is a sum of quasi-local terms and is itself fully MBL. We derive this result by constructing a sequence of canonical transformations to remove the time-dependence from the original Hamiltonian. When the driving evolves smoothly in time, the theory can be sharpened by estimating the probability of adiabatic Landau–Zener transitions at many-body level crossings. In all cases, we argue that there is delocalization at sufficiently low frequency. We propose a phase diagram of driven MBL systems.     

Probability phenomena due to separatrix crossing

The effect of a small or slow perturbation on a Hamiltonian system with one degree of freedom is considered. It is assumed that the phase portrait ("phase plane") of the unperturbed system is divided by separatrices into several regions and that under the action of the perturbations phase points can cross these separatrices. The probabilistic phenomena are described that arise due to these separatrix crossings, including the scattering of trajectories, random jumps in the values of adiabatic invariants, and adiabatic chaos. These phenomena occur both in idealized problems in classical mechanics and in real physical systems in planetary science and plasma physics contexts.     

The complex scaling method for Dirac resonances

The method of complex scaling, usually used in atomic and molecular resonance calculations, is generalized to the Dirac equation. It is shown that Dirac resonances are associated with nonreal eigenvalues of the scaled Dirac Hamiltonian. The perturbation theory for the resonance parameters is also discussed.     

Interaction of eigenvalues in multi-parameter problems

The paper presents a new general theory of interaction of eigenvalues of matrix operators depending on parameters. Both complex and real eigenvalues are considered. Strong and weak interactions are distinguished, and their geometric interpretation on the complex plane is given. Mechanical examples are presented and discussed in detail.