Family Of Boundary Conditions Research Articles

Fusion hierarchies, T-systems and Y-systems for the dilute loop models on a strip

We study the dilute loop models on the geometry of a strip of width N. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its T- and Y-systems, for these four boundary conditions and with a generic crossing parameter λ. For rational and thus a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-N expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small N. The present work complements a previous study (Morin-Duchesne and Pearce 2019 J. Stat. Mech. 1909) that investigated the dilute loop models with periodic boundary conditions.

Topology in Shallow-Water Waves: A Spectral Flow Perspective

In the context of topological insulators, the shallow-water model was recently shown to exhibit an anomalous bulk-edge correspondence. For the model with a boundary, the parameter space involves both longitudinal momentum and boundary conditions, and exhibits a peculiar singularity. We resolve the anomaly in question by defining a new kind of edge index as the spectral flow around this singularity. Crucially, this edge index samples a whole family of boundary conditions, and we interpret it as a boundary-driven quantized pumping. Our edge index is stable due to the topological nature of spectral flow, and we prove its correspondence with the bulk Chern number index using scattering theory and a relative version of Levinson's theorem. The full spectral flow structure of the model is also investigated.

General Boundary Conditions for a Majorana Single-Particle in a Box in (1 + 1) Dimensions

We consider the problem of a Majorana single-particle in a box in (1 + 1) dimensions. We show that the most general set of boundary conditions for the equation that models this particle is composed of two families of boundary conditions, each one with a real parameter. Within this set, we only have four confining boundary conditions—but infinite not confining boundary conditions. Our results are also valid when we include a Lorentz scalar potential in this equation. No other Lorentz potential can be added. We also show that the four confining boundary conditions for the Majorana particle are precisely the four boundary conditions that mathematically can arise from the general linear boundary condition used in the MIT bag model. Certainly, the four boundary conditions for the Majorana particle are also subject to the Majorana condition.

Boundary conditions for General Relativity on AdS3 and the KdV hierarchy

It is shown that General Relativity with negative cosmological constant in three spacetime dimensions admits a new family of boundary conditions being labeled by a nonnegative integer $k$. Gravitational excitations are then described by "boundary gravitons" that fulfill the equations of the $k$-th element of the KdV hierarchy. In particular, $k=0$ corresponds to the Brown-Henneaux boundary conditions so that excitations are described by chiral movers. In the case of $k=1$, the boundary gravitons fulfill the KdV equation and the asymptotic symmetry algebra turns out to be infinite-dimensional, abelian and devoid of central extensions. The latter feature also holds for the remaining cases that describe the hierarchy ($k>1$). Our boundary conditions then provide a gravitational dual of two noninteracting left and right KdV movers, and hence, boundary gravitons possess anisotropic Lifshitz scaling with dynamical exponent $z=2k+1$. Remarkably, despite spacetimes solving the field equations are locally AdS, they possess anisotropic scaling being induced by the choice of boundary conditions. As an application, the entropy of a rotating BTZ black hole is precisely recovered from a suitable generalization of the Cardy formula that is compatible with the anisotropic scaling of the chiral KdV movers at the boundary, in which the energy of AdS spacetime with our boundary conditions depends on $z$ and plays the role of the central charge. The extension of our boundary conditions to the case of higher spin gravity and its link with different classes of integrable systems is also briefly addressed.

BPS monopole in the space of boundary conditions

The space of all possible boundary conditions that respect the self-adjointness of the Hamiltonian operator is known to be given by the group manifold U(2) in one-dimensional quantum mechanics. In this paper we study non-Abelian Berry’s connections in the space of boundary conditions in a simple quantum mechanical system. We consider a system for a free spinless particle on a circle with two point-like interactions described by the U(2) × U(2) family of boundary conditions. We show that, for a certain SU(2) ⊂ U(2) × U(2) subfamily of boundary conditions, all the energy levels become doubly-degenerate thanks to the so-called higher-derivative supersymmetry, and the non-Abelian Berry’s connection in the ground-state sector is given by the Bogomolny–Prasad–Sommerfield (BPS) monopole of SU(2) Yang–Mills–Higgs theory. We also show that, in the ground-state sector of this quantum mechanical model, the matrix elements of the position operator give the adjoint Higgs field that satisfies the BPS equation. It is also discussed that Berry’s connections in the excited-state sectors are given by non-BPS ’t Hooft–Polyakov monopoles.

The large level limit of Kazama-Suzuki models

Limits of families of conformal field theories are of interest in the context of AdS/CFT dualities. We explore here the large level limit of the two-dimensional N = (2,2) superconformal Wn+1 minimal models that appear in the context of the supersymmetric higher-spin AdS3/CFT2 duality. These models are constructed as Kazama-Suzuki coset models of the form SU(n+1)/U(n). We determine a family of boundary conditions in the limit theories, and use the modular bootstrap to obtain the full bulk spectrum of N = 2 super-Wn+1 primaries in the theory. We also confirm the identification of this limit theory as the continuous orbifold C n /U(n) that was discussed recently.

Asymptotic behaviour of solutions to the stationary Navier–Stokes equations in two-dimensional exterior domains with zero velocity at infinity

We investigate analytically and numerically the existence of stationary solutions converging to zero at infinity for the incompressible Navier–Stokes equations in a two-dimensional exterior domain. Physically, this corresponds for example to fixing a propeller by an external force at some point in a two-dimensional fluid filling the plane and to ask if the solution becomes steady with the velocity at infinity equal to zero. To answer this question, we find the asymptotic behaviour for such steady solutions in the case where the net force on the propeller is nonzero. In contrast to the three-dimensional case, where the asymptotic behaviour of the solution to this problem is given by a scale invariant solution, the asymptote in the two-dimensional case is not scale invariant and has a wake. We provide an asymptotic expansion for the velocity field at infinity, which shows that, within a wake of width |x|2/3, the velocity decays like |x|-1/3, whereas outside the wake, it decays like |x|-2/3. We check numerically that this behaviour is accurate at least up to second order and demonstrate how to use this information to significantly improve the numerical simulations. Finally, in order to check the compatibility of the present results with rigorous results for the case of zero net force, we consider a family of boundary conditions on the body which interpolate between the nonzero and the zero net force case.

Path integral on star graph

In this paper we study path integral for a single spinless particle on a star graph with N edges, whose vertex is known to be described by U(N) family of boundary conditions. After carefully studying the free particle case, both at the critical and off-critical levels, we propose a new path integral formulation that correctly captures all the scale-invariant subfamily of boundary conditions realized at fixed points of boundary renormalization group flow. Our proposal is based on the folding trick, which maps a scalar-valued wave function on star graph to an N-component vector-valued wave function on half-line. All the parameters of scale-invariant subfamily of boundary conditions are encoded into the momentum independent weight factors, which appear to be associated with the two distinct path classes on half-line that form the cyclic group Z2. We show that, when bulk interactions are edge-independent, these weight factors are generally given by an N-dimensional unitary representation of Z2. Generalization to momentum dependent weight factors and applications to worldline formalism are briefly discussed.

Vacuum Boundary Effects

The Casimir effect is highly dependent on the shape and structure of space boundaries. This dependence is encoded in the variation of vacuum energy with the different types of boundary conditions. We analyze from a global perspective the properties of the Casimir energy as a function on the largest space of the consistent boundary conditions \(\mathcal{M}_{F}\) for a massless scalar field confined between to homogeneous parallel plates. In particular, we analyze the analytic properties of this function and point out the existence of a third order phase transition at periodic boundary conditions. We also characterize the boundary conditions which give rise to attractive or repulsive Casimir forces. In the interface between both regimes we find a very interesting family of boundary conditions without Casimir effect, and fully characterize the boundary conditions which do not induce any type of Casimir force.

Running Boundary Condition

In this paper we argue that boundary condition may run with energy scale. As an illustrative example, we consider one-dimensional quantum mechanics for a spinless particle that freely propagates in the bulk yet interacts only at the origin. In this setting we find the renormalization group flow of U(2) family of boundary conditions exactly. We show that the well-known scale-independent subfamily of boundary conditions are realized as fixed points. We also discuss the duality between two distinct boundary conditions from the renormalization group point of view. Generalizations to conformal mechanics and quantum graph are also discussed.

Dimension reduction of homoclinic orbits of buckled beams via the non-linear normal modes technique

The problem of detecting the homoclinic orbits of an initially straight buckled beam is addressed. Two families of boundary conditions are identified and investigated in detail. For the first family, the homoclinic orbits belong to a planar invariant manifold, and are easily computed in closed form. For the second family, the manifold is no longer planar, and is detected via the non-linear normal modes technique by obtaining approximate expressions which are sufficient to highlight the effects of the non-flatness. A hierarchy of reduced order, single degree of freedom, models is determined. These are obtained by taking into account increasing degrees of non-linearity in the potential energy, which allow for a more and more refined computation of the homoclinic solution. The various models are compared with each other and discussed in detail, and the non-planarity of the manifold is illustrated through examples.

Extracting information behind the veil of the horizon

In AdS/CFT duality, it is often argued that information behind the event horizon is encoded even in boundary correlators. However, its implication is not fully understood. We study a simple model which can be analyzed explicitly. The model is a two-dimensional scalar field propagating on the s-wave sector of the BTZ black hole formed by the gravitational collapse of a null dust. Inside the event horizon, we placed an artificial timelike singularity where one-parameter family of boundary conditions is permitted. We compute two-point correlators with two operators inserted on the boundary to see if the parameter can be extracted from the correlators. In a typical case, we give an explicit form of the boundary correlators of an initial vacuum state and show that the parameter can be read off from them. This does not immediately imply that the asymptotic observer can extract the information of the singularity since one cannot control the initial state in general. Thus, we also study whether the parameter can be read off from the correlators for a class of initial states.

Analytical solutions in the problem of the optimal control of the rotation of an axisymmetric body

The problem of the optimal control of the rotation of an axisymmetric rigid body is investigated. An integral functional, characterizing the power consumption to carry out a manoeuvre is chosen as the criterion, and the boundary conditions for the angular velocity vector are arbitrary. The principal moment of the applied external forces serves as the control. The necessary conditions of the maximum principle are used to solve the problem in the case of a fixed completion time. New non-trivial first integrals are established for the canonical system of direct and conjugate differential equations obtained, which enable the set of all extremals to be parametrized. Hence, the optimal-control problem is reduced to a problem of non-linear mathematical programming. It is shown that there cannot be more than two different solutions in the latter, and a family of boundary conditions is established when the optimum rotation is determined in a uniquely explicit form.

Riesz basis property of mode shapes for aircraft wing model (subsonic case)

The present paper is devoted to the Riesz basis property of the mode shapes for an aircraft wing model in an inviscid subsonic airflow. The model has been developed in the Flight Systems Research Center of the University of California at Los Angeles in collaboration with NASA Dryden Flight Research Center. The model has been successfully tested in a series of flight experiments at Edwards Airforce Base, CA, and has been extensively studied numerically. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The system of equations of motion is equivalent to a single operator evolution–convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is a finite—meromorphic operator—valued function of the spectral parameter. Its poles are precisely the aeroelastic modes. In the author's previous works, it has been shown that the set of aeroelastic modes asymptotically splits into two disjoint subsets called the β -branch and the δ -branch, and precise spectral asymptotics with respect to the eigenvalue number have been derived for both branches. The asymptotical approximations for the mode shapes have also been obtained. In the present work, the author proves that the set of the mode shapes forms an unconditional basis (the Riesz basis) in the Hilbert state space of the system. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.

EXCLUSION PROCESSES AND BOUNDARY CONDITIONS

A family of boundary conditions corresponding to exclusion processes is introduced. This family is a generalization of the boundary conditions corresponding to the simple exclusion process, the drop-push model, and the one-parameter solvable family of pushing processes with certain rates on the continuum.1–3 The conditional probabilities are calculated using the Bethe ansatz, and it is shown that at large times they behave like the corresponding conditional probabilities of the family of diffusion-pushing processes introduced in Refs. 1–3.

A Second Order Three-Point Boundary Value Problem with Mixed Nonlinear Boundary Conditions

We apply the generalized quasilinearization method to a second order three-point boundary value problem involving mixed nonlinear boundary conditions and obtain a monotone sequence of approximate solutions converging to the unique solution of the problem possessing a convergence of order k(k� 2). 1. Introduction. The method of quasilinearization developed by Bellman and Kalaba (1) and generalized by Lakshmikantham (2-3) later on, has been studied and extended in several diverse disciplines. In fact, it is generating a rich history and an extensive bibliography can be found in (4-10). Multi-point nonlinear boundary value problems, which refer to a different family of boundary conditions in the study of disconjugacy theory (11), have been addressed by many authors, for example, see (12-14). In particular, Eloe and Gao (15) dis- cussed the quasilinearization method for a three-point boundary value problem. In this paper, we study the generalized quasilinearization method for a second order three-point boundary value problem with mixed nonlinear boundary conditions. In fact, a sequence of approximate solutions converging monotonically to a solution of the nonlinear three-point problem with the order of convergence k(k ≥ 2) has been presented.

Boundary conditions in path integrals from point interactions for the path integral of the one-dimensional Dirac particle

A proposal is given, of how to implement point interactions and boundary conditions in the path integral. The starting point is a path-integral formulation of a Dirac particle in one dimension. The implementation of a point interaction yields, by means of a perturbation expansion, the corresponding Green function for a relativistic particle. In the non-relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen. By a proper combination of the various possibilities of implementing the point interaction, the whole range of the four-parameter family of boundary conditions for point interactions can be exploited, in the relativistic case as well as in the non-relativistic limit. In addition, making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line. In particular, Dirichlet and Neumann boundary conditions emerge. The method is illustrated with some examples.

General boundary conditions for a Dirac particle in a box and their non-relativistic limits

The most general relativistic boundary conditions (BCs) for a `free' Dirac particle in a one-dimensional box are discussed. It is verified that in the Weyl representation there is only one family of BCs, labelled with four parameters. This family splits into three sub-families in the Dirac representation. The energy eigenvalues as well as the corresponding non-relativistic limits of all these results are obtained. The BCs which are symmetric under space inversion P and those which are CPT invariant for a particle confined in a box, are singled out.

Boundary value problems compatible with symmetries

Boundary value problems for nonlinear differential equations are considered from the point of view of symmetry. In addition to the known ones new families of boundary conditions are found for integrable equations like the Harry-Dym, KdV and mKdV.

Families of boundary conditions for nonlinear ordinary differential equations

Families of boundary conditions for nonlinear ordinary differential equations