October 2011
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Month October 2011

Spectral zeta functions of a 1D Schrödinger problem

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.

Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th)

PT-symmetric quantum models living in an auxiliary Pontryagin space

Miloslav Znojil

An extension of the scope of quantum theory is proposed in a way inspired by the recent heuristic as well as phenomenological success of the use of non-Hermitian Hamiltonians which are merely required self-adjoint in a Krein space with an indefinite metric (chosen, usually, as the operator of parity). In nuce, the parity-like operators are admitted to represent the mere indefinite metric in a Pontryagin space. A constructive version of such a generalized quantization strategy is outlined and, via a non-numerical illustrative example, found feasible.

Mathematical Physics (math-ph)

PT-symmetry, indefinite damping and dissipation-induced instabilities

Oleg N. Kirillov

When gain and loss are in perfect balance, dynamical systems with indefinite damping can obey the exact PT-symmetry and therefore be marginally stable with a pure imaginary spectrum. At an exceptional point where the exact PT-symmetry is spontaneously broken, the stability is lost via a Krein collision of eigenvalues just as it happens at the Hamiltonian Hopf bifurcation. In the parameter space of a general dissipative system, marginally stable PT-symmetric ones occupy singularities on the boundary of the asymptotic stability domain. To observe how the singular surface governs dissipation-induced destabilization of the PT-symmetric system when gain and loss are not matched, an extension of recent experiments with PT-symmetric LRC circuits is proposed.

Mathematical Physics (math-ph); Other Condensed Matter (cond-mat.other); Spectral Theory (math.SP); Quantum Physics (quant-ph)