Patrick Dorey

Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics

The one-dimensional Schrodinger equation for the potential x6 + αx2 + l(l + 1)/x2 has many interesting properties. For certain values of the parameters l and α the equation is in turn supersymmetric (Witten) and quasi-exactly solvable (Turbiner), and it also appears in Lipatovs approach to high-energy QCD. In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. These relationships are obtained via a recently observed connection between the theories of ordinary differential equations and integrable models. Generalized supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. Finally we generalize slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain -symmetric quantum mechanical systems.

Affine Toda field theory and exact S-matrices

Abstract The masses and three-point couplings for all affine Toda theories are calculated. The exact factorisable S-matrices are conjectured on the basis of the classical masses and couplings and found, in the case of theories based on simply-laced algebras, to give consistent solutions to the bootstrap. An investigation of the properties of the exact S-matrices in perturbation theory is begun but non-perturbative methods will be required to understand the conjectured duality between weak and strong coupling which appears to be a striking feature of these theories.

Excited states by analytic continuation of TBA equations

Abstract We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the thermodynamic Bethe ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two-particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically.

Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations

The spectral determinant D(E) of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the A3-related Y-system emerging in the treatment of a certain perturbed conformal field theory, allowing us to give an alternative integral expression for D(E). Generalizing this result, we conjecture a relationship between the x2M anharmonic oscillators and the A2M-1 thermodynamic Bethe ansatz systems. Finally, spectral determinants for general |x| potentials are mapped onto the solutions of nonlinear integral equations associated with the (twisted) XXZ and sine-Gordon models.

Root systems and purely elastic S-matrices

Abstract Starting from a recently-proposed general formula, various properties of the ADE series of purely elastic S-matrices are rederived in a universal way. In particular, the relationship between the pole structure and the bootstrap equations is shown to follow from properties of root systems. The discussion leads to a formula for the signs of the three-point couplings in the simply-laced affine Toda theories, and a simple proof of a result due to Klassen and Melzer of relevance to Thermodynamic Bethe Ansatz calculations.

Classically integrable boundary conditions for affine Toda field theories

Abstract Boundary conditions compatible with classical integrability are studied both directly, using an approach based on the explicit construction of conserved quantities, and indirectly by first developing a generalisation of the Lax pair idea. The latter approach is closer to the spirit of earlier work by Sklyanin and yields a complete set of conjectures for permissible boundary conditions for any affine Toda field theory.

Affine Toda field theory on a half-line

Abstract The question of the integrability of real-coupling affine toda field theory on a half-line is addressed. It is found, by examining low-spin conserved charges, that the boundary conditions preserving integrability are strongly constrained. In particular, for the a n ( n >1) series of models there can be no free parameters introduced by the boundary condition; indeed the only remaining freedom (apart from choosing the simple condition ∂ 1 φ =0, resides in a choice of signs. For a special case of the boundary condition, it is argued that the classical boundary bound state spectrum is closely related to a consistent set of reflection factors in the quantum field theory.

On the relation between Stokes multipliers and the T-Q systems of conformal field theory

Abstract The vacuum expectation values of the so-called Q -operators of certain integrable quantum field theories have recently been identified with spectral determinants of particular Schrodinger operators. In this paper we extend the correspondence to the T -operators, finding that their vacuum expectation values also have an interpretation as spectral determinants. As byproducts we give a simple proof of an earlier conjecture of ours, proved by another route by Suzuki, and generalise a problem in PT symmetric quantum mechanics studied by Bender and Boettcher. We also stress that the mapping between Q -operators and Schrodinger equations means that certain problems in integrable quantum field theory are related to the study of Regge poles in non-relativistic potential scattering.

Excited states in some simple perturbed conformal field theories

The method of analytic continuation is used to find exact integral equations for a selection of finite-volume energy levels for the non-unitary minimal models M2,2N+3 perturbed by their ϕ13 operators. The N = 2 case is studied in particular detail. Along the way, we find a number of general results which should be relevant to the study of excited states in other models.

Extended Toda Field Theory and Exact S Matrices

Abstract The existence of exact unitary crossing symmetric S -matrices associated with the a, d, e series of Toda field theories is conjectured and the main features illustrated within a discussion of the d 4 case.