Patrick Dorey, Clare Dunning, Roberto Tateo
A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators with order greater than two. In particular, links with Bender-Dunne polynomials and resonances between independent solutions are observed for certain second-order cases, and extended to the higher-order problems.
http://arxiv.org/abs/1209.4736
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
Joe Watkins
We study the spectral zeta functions associated to the radial Schrodinger problem with potential \(V(x)=x^{2M}+\alpha x^{M-1} +(\lambda^2-1/4)/x^2\). Using the quantum Wronskian equation, we provide results such as closed-form evaluations for some of the second zeta functions i.e. the sum over the inverse eigenvalues squared. Also we discuss how our results can be used to derive relationships and identities involving special functions, using a particular \({}_5F_4\) hypergeometric series as an example. Our work is then extended to a class of related \({\cal PT}\)-symmetric eigenvalue problems. Using the fused quantum Wronskian we give a simple method for calculating the related spectral zeta functions. This method has a number of applications including the use of the ODE/IM correspondence to compute the (vacuum) nonlocal integrals of motion \(G_n\) which appear in the associated integrable quantum field theory.
http://arxiv.org/abs/1110.2004
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th)
Paulo E. G. Assis
A class of non-Hermitian quadratic su(2) Hamiltonians that fulfil an anti-linear symmetry is constructed. If unbroken this anti-linear symmetry leads to a purely real spectrum and the Hamiltonian can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here Lie-algebraic methods which were used to investigate the generalised Swanson Hamiltonian is used to construct a class of quadratic Hamiltonians that allow for such a simple mapping to the Hermitian counterpart. While for the linear su(2) Hamiltonian every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su(2) generators, the situation is more complicated for quadratic Hamiltonians. The existence of finite dimensional representations for the su(2) Hamiltonian, as opposed to the su(1,1) studied before, allows for comparison with explicit diagonalisation results for finite matrices. The possibility of more elaborate similarity transformations, including quadratic exponents, is also discussed. Finally, the similarity transformations are compared with the analogue of Swanson’s method of diagonalising the problem.
http://arxiv.org/abs/1012.0194
Quantum Physics (quant-ph); Mathematical Physics (math-ph)