The IR-truncated PT-symmetric V = ix3 model and its asymptotic spectral scaling graph


The IR-truncated PT-symmetric V = ix3 model and its asymptotic spectral scaling graph

Günther, U.; Stefani, F.

The PT-symmetric V = ix3 model over the real line is infra-red (IR) truncated and considered as Sturm-Liouville problem over a finite interval. Structures hidden in the Airy function setup of the V = ix3 model are combined with WKB techniques developed by Bender and Jones in 2012 for the derivation of the real part of the spectrum of theV = ix3 model. Via WKB and Stokes graph analysis, the location of the complex spectral branches of the ix3 model as well as those of more general V = -(ix)2n+1 models over finite intervals are obtained. Splitting the related action functions into purely real scale factors and scale invariant integrals allows to extract underlying asymptotic spectral scaling graphs. These (structurally very simple) scaling graphs are geometrically invariant and cutoff-independent so that the IR limit can be formally taken. Moreover an increasing length scale can be associated with a spectral UV-IR renormalization group flow on this scaling graph. It is shown that the eigenvalues of the IR-complete V = ix3 model can be bijectively mapped onto a finite segment of the scaling graph asymptotically approaching a (scale invariant) PT phase transition region. In this way, a simple heuristic picture and complementary explanation for the unboundedness of projector norms and C-operator for the ix3 model are provided and the lack of quasi-Hermiticity of the ix3 Hamiltonian over the real line appears physically plausible. Possible directions of further research are briefly sketched.

Keywords: PT Quantum Mechanics; PT phase transition; spectral branch points; exceptional points; ix3 model; WKB techniques; IR truncation; IR completion; asymptotic spectral scaling graphs; spectral UV-IR renormalization group flow

  • Invited lecture (Conferences)
    Discrete-18, organized by CERN and the Austrian Academy of Sciences, 26.-30.11.2018, Wien, Österreich

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Publ.-Id: 28787