Operator approach to boundary Liouville theory

We propose new methods for calculation of the discrete spectrum, the reflection amplitude and the correlation functions of boundary Liouville theory on a strip with Lorentzian signature. They are based on the structure of the vertex operator V=e^-φ in terms of the asymptotic operators. The methods first are tested for the particle dynamics in the Morse potential, where similar structures appear. Application of our methods to boundary Liouville theory reproduces the known results obtained earlier in the bootstrap approach, but there can arise a certain extension when the boundary parameters are near to critical values. Namely, in this case we have found up to four different equidistant series of discrete spectra, and the reflection amplitude is modified, respectively.     

Horizon state, Hawking radiation, and boundary Liouville model

We demonstrate that the near-horizon physics, the Hawking radiation, and the reflection off the radial potential barrier can be understood entirely within a conformal field theory picture in terms of one- and two-point functions in the boundary Liouville theory. An important element in this demonstration is the notion of horizon state, the Hawking radiation being interpreted as a result of the transition of horizon state to the ordinary states propagating outside the black hole horizon.     

Wetting effects at a grain boundary

We consider a tier of weakened bonds along the center line of a two-dimensional Ising ferromagnet strip as a model of a grain boundary. When an interface traverses such a strip at an angle, whether or not there is a continuous pinning-depinning transition at subcritical temperature depends on this angle and the degree of bond weakening. We also study the relaxation of such a system to its equilibrium state using continuous time Monte Carlo simulation with Kawasaki dynamics; this reveals a matter transport mechanism confined to the grain boundary.     

One-point functions in perturbed boundary conformal field theories

We consider the one-point functions of bulk and boundary fields in the scaling Lee–Yang model for various combinations of bulk and boundary perturbations. The one-point functions of the bulk fields are analysed using the truncated conformal space approach and the form-factor expansion. Good agreement is found between the results of the two methods, though we find that the expression for the general boundary state given by Ghoshal and Zamolodchikov has to be corrected slightly. For the boundary fields we use thermodynamic Bethe ansatz equations to find exact expressions for the strip and semi-infinite cylinder geometries. We also find a novel off-critical identity between the cylinder partition functions of models with differing boundary conditions, and use this to investigate the regions of boundary-induced instability exhibited by the model on a finite strip.     

NLIE of Dirichlet sine-Gordon model for boundary bound states

We investigate boundary bound states of sine-Gordon model on the finite-size strip with Dirichlet boundary conditions. For the purpose we derive the nonlinear integral equation (NLIE) for the boundary excited states from the Bethe ansatz equation of the inhomogeneous XXZ spin 1/2 chain with boundary imaginary roots discovered by Saleur and Skorik. Taking a large volume (IR) limit we calculate boundary energies, boundary reflection factors and boundary Lüscher corrections and compare with the excited boundary states of the Dirichlet sine-Gordon model first considered by Dorey and Mattsson. We also consider the short distance limit and relate the IR scattering data with that of the UV conformal field theory.     

Renormalization group analysis of boundary conditions in potential scattering

We analyze how a short distance boundary condition for the Schrödinger equation must change as a function of the boundary radius by imposing the physical requirement of phase shift independence on the boundary condition. The resulting equation can be interpreted as a variable phase equation of a complementary boundary value problem. We discuss the corresponding infrared fixed points and the perturbative expansion around them generating a short distance modified effective range theory. We also discuss ultraviolet fixed points, limit cycles, and attractors with a given fractality which take place for singular attractive potentials at the origin. The scaling behavior of scattering observables can analytically be determined and is studied with some emphasis on the low energy nucleon-nucleon interaction via singular pion exchange potentials. The generalization to coupled channels is also studied.     

Classical dynamics of the supersymmetric Liouville theory

We solve the supersymmetric Liouville theory in the strip $\text{[0, π] ×}\text{R}$ with boundary conditions appropriate to the supersymmetric string dynamics recently discussed by Polyakov.     

Semiclassical energy levels of sine-Gordon model on a strip with Dirichlet boundary conditions

We derive analytic expressions of the semiclassical energy levels of sine-Gordon model in a strip geometry with Dirichlet boundary condition at both edges. They are obtained by initially selecting the classical backgrounds relative to the vacuum or to the kink sectors, and then solving the Schrödinger equations (of Lamè type) associated to the stability condition. Explicit formulas are presented for the classical solutions of both the vacuum and kink states and for the energy levels at arbitrary values of the size of the system. Their ultraviolet and infrared limits are also discussed.     

Vibrations of an elastic strip with a rough boundary

A plane problem of steady-state forced vibrations of an elastic strip whose lower boundary contains a rough segment is considered. Using Green’s functions for a strip, the problem is reduced to a system of integral equations with integrals over the rough boundary, which is solved by the boundary-element method. The inverse problem of determining the shape of the rough boundary segment from the data on the displacement field of a certain part of the upper boundary is formulated. By the linearization procedure, the inverse problem is reduced to a Fredholm integral equation of the first kind with a smooth kernel, which is solved by Tikhonov’s regularization method.     

On some boundary value problems on a strip in the complex plane

The Sommerfeld integral inversion method for the Helmholtz equation in an angular region with different boundary values leads to boundary value problems in an infinite strip of the complex plane. We investigate a generic system for such boundary value problems and give the existence and uniqueness results with optimal growth estimates on the solution. We also give the solutions of the Dirichlet problem in a strip when the boundary functions grow exponentially.     

First-principle proof of the modified collision boundary conditions for the hard-sphere system

A fundamental issue lying at the foundation of classical statistical mechanics is the determination of the collision boundary conditions that characterize the dynamical evolution of multi-particle probability density functions (PDF) and are applicable to systems of hard-spheres undergoing multiple elastic collisions. In this paper it is proved that, when the deterministic N-body PDF is included in the class of admissible solutions of the Liouville equation, the customary form of collision boundary conditions adopted in previous literature becomes physically inconsistent and must actually be replaced by suitably modified collision boundary conditions.     

Influence of short roughness strip on the turbulent boundary layer structure

Hot-wire measurements have been undertaken in a turbulent boundary layer which is subjected to an impulse in form of a short roughness strip with the aim of examining its influence on the structure of the turbulent boundary layer. The results indicate that, while the energy containing motion is shifted from low wave number to high wave number near the wall due to the interfering of the roughness strip with the near-wall structure, the reverse is the case in the outer region. While the anisotropy at small scale changes appreciably, there is no discernable change at the large scale when distance from the wall is increased as reflected in the collapses of spectra shear correlation coefficient at the low wave number. It further shows that the roughness strip alters the flow dynamics of the boundary layer as shown in the changes in the mixing length distribution.     

Spectrum of local boundary operators from boundary form factor bootstrap

Using the recently introduced boundary form factor bootstrap equations, we map the complete space of their solutions for the boundary version of the scaling Lee–Yang model and sinh-Gordon theory. We show that the complete space of solutions, graded by the ultraviolet behaviour of the form factors can be brought into correspondence with the spectrum of local boundary operators expected from boundary conformal field theory, which is a major evidence for the correctness of the boundary form factor bootstrap framework.     

Form factors of boundary exponential operators in the sinh-Gordon model

Using the recently introduced boundary form factor bootstrap equations, the form factors of boundary exponential operators in the sinh-Gordon model are constructed. We also give a general method to evaluate the ultraviolet properties of boundary correlators by extending the bulk cumulant expansion to the boundary case. As an application, the ultraviolet scaling dimension and the normalization of the operators corresponding to the form factor solutions are checked against previously known results for boundary exponential operators. The construction presented in this paper can be applied to determine form factors of relevant primary boundary operators in general integrable boundary quantum field theories.     

On the spectrum of the sinh-Gordon model in finite volume

We derive a characterization of the spectrum of the sinh-Gordon model in terms of certain nonlinear integral equations. There exists a large class of solutions to these equations which allows a continuation between the infrared and the ultraviolet limits, respectively. We present nontrivial evidence for the claim that the class of solutions in question describes the spectrum of the sinh-Gordon model completely in both of these limits. The evidence includes some nontrivial relations to Liouville theory.     

Symmetry boundary conditions

A simple approach to energy conserving boundary conditions using exact symmetries is described which is especially useful for numerical simulations using the finite difference method. Each field in the simulation is normally either symmetric (even) or antisymmetric (odd) with respect to the simulation boundary. Another possible boundary condition is an antisymmetric perturbation about a nonzero value. One of the most powerful aspects of this approach is that it can be easily implemented in curvilinear coordinates by making the scale factors of the coordinate transformation symmetric about the boundaries. The method is demonstrated for magnetohydrodynamics (MHD), reduced MHD, and a hybrid code with particle ions and fluid electrons. These boundary conditions yield exact energy conservation in the limit of infinite time and space resolution. Also discussed is the interpretation that the particle charge reverses sign at a conducting boundary with boundary normal perpendicular to the background magnetic field.     

Bulk and boundary effects in a one-dimensional water-like fluid

Sites of a one-dimensional lattice may be either vacant or occupied by molecules with two orientational states, a ‘monomer’ state taking up one site and a ‘dimer’ state taking up two sites. There is a nearest-neighbour energy — ε for all molecules and an additional bonding energy — ω for adjacent dimers. Hence low temperatures and pressures favour an open structure with molecules predominantly in the dimer state. Bulk behaviour is investigated by both matrix and combinatorial methods and maxima in density isobars and minima in isothermal compressibility curves are found at certain parameter values. These are anomalous properties similar to those occurring in fluid water.

Boundary effects in a chain with end sites are calculated by matrix methods for several types of boundary condition. The effect on the open structure is measured by the change in the number of dimer-dimer bonds due to the presence of the boundary. For a ‘structure-breaking’ boundary, which forces the adjacent molecule into the monomer state, occupation probabilities near the boundary are compared with bulk values. It is concluded that boundary effects in the fluid are confined to the first few layers of molecules.     

Effects of boundary conditions on spatially periodic states

We investigate analytically and numerically the influence of linear homogeneous boundary conditions on the stationary solutions of a simple model for cellular pattern formation in one dimension. For all boundary conditions there exists in a reduced wavenumber band at least one static solution where the amplitude falls below its bulk value near the boundary (“Type-I” solution). A linear stability analysis of the uniform state at threshold reveals that Type-I solutions are often unstable. Then there exists in the full Eckhaus-stable band, a static solution where the amplitude rises above its bulk value near the boundary (“Type-II” solution), or a limit-cycle solution where the amplitude near the boundary oscillates. These solutions bifurcate from the homogeneous state below the bulk threshold and therefore remain finite at threshold.     

Diffusion,effusion, and chaotic scattering: An exactly solvable Liouvillian dynamics

We study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area-preserving mapping of an infinitely extended strip onto itself. This exactly solvable mapping sustains chaotic behaviors and diffusion processes. The relationship between the diffusion coefficient, the Lyapunov exponent, and the entropy per unit time is derived. The long-lived classical resonances of the Liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation. In this sense, there is a quasi-isomorphism between the resonance spectrum of the Liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation. Furthermore, we show that a fractal repeller is associated to each non-equilibrium state in the isolated and finite multibaker chain. The nonequilibrium states are all unstable with respect to the equilibrium, validating a weak form of the second principle of thermodynamics for the present dynamical system. Consequences of nonequilibrium fractals on classical measurements are discussed. We then describe the open multibaker chain as a scattering system. Fractal properties of chaotic scattering are here shown to be related to diffusion in the chain.