The one-dimensional Schrödinger equation for the potential x 6 + αx 2 + l(l + 1)/x 2 has many interesting properties. For certain values of the parameters l and α the equation is in turn supersymmetric (Witten), quasi-exactly solvable (Turbiner), and it also appears in Lipatov's 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. Generalised 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 generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT -symmetric quantum-mechanical systems.

The appearances of complex eigenvalues in the spectra of PT -symmetric quantummechanical systems are usually associated with a spontaneous breaking of PT . In this letter we discuss a family of models for which this phenomenon is also linked with an explicit breaking of supersymmetry. Exact level-crossings are located, and connections with N -fold supersymmetry and quasi-exact solvability in certain special cases are pointed out.

This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT -symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

Using the well-known trigonometric six-vertex solution of the Yang-Baxter equation we derive an integrable pairing Hamiltonian with anyonic degrees of freedom. The exact algebraic Bethe ansatz solution is obtained using standard techniques. From this model we obtain several limiting models, including the pairing Hamiltonian with p + ip-wave symmetry. An in-depth study of the p + ip model is then undertaken, including a mean-field analysis, analytic and numerical solution of the Bethe ansatz equations, and an investigation of the topological properties of the ground-state wavefunction. Our main result is that the ground-state phase diagram of the p + ip model consists of three phases. There is the known boundary line with gapless excitations that occurs for vanishing chemical potential, separating the topologically trivial strong pairing phase and the topologically non-trivial weak pairing phase. We argue that a second boundary line exists separating the weak pairing phase from a topologically trivial weak coupling BCS phase, which includes the Fermi sea in the limit of zero coupling. The ground state on this second boundary line is the Moore-Read state.Combining these we have a single determinant expression

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miuratransformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed. For finite lattice models, the explicit diagonalisation of the An−1 cases has been performed through the algebraic Bethe ansatz by Schulz [21] and also by Babelon, de Vega and Viallet [22]. For Cn and Dn models, it has been done by Reshetikhin [23,24]. There is a shortcut to reach the same conclusions via the so-called analytic Bethe ansatz of Reshetikhin [25], and Wiegmann and Reshetikhin [26].† The constants {ma} are related to a particular matrix K ab emerging from the analysis of the Bethe ansatz. For simply-laced algebras, K ab is proportional to the Cartan matrix and v=(m1, m2, . . . , mr) is its Perron-Frobenius eigenvector.