Abstract
New observations in Gel'fand triplets are studied. An interesting one is vortex phenomena that stem from zero energy solutions of two-dimensional Schrödinger equations with central potentials V(ρ) ∝ ρr(ρ2 = x 2 + y 2 and r ≠ −2), which are eigenstates of conjugate spaces of Gel'fand triplets. The zero energy solutions for all the potentials V(ρ) are shown to have the same structure with infinite degeneracy by making use of the conformal transformation ξα = z α with z = x + iy. The infinite degeneracy is observed as variety of vortex patterns in real physical phenomena. Some simple vortex patterns such as vortex lines and vortex lattices are presented. Such a new freedom on the Gel'fand triplets can be treated in a statistical mechanics. In the theory a new entropy being different from the so-called Boltzmann entropy appears. Transitions between the two entropies occur in thermal nonequilibrium phenomena, where energy emissions are observed.
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Kobayashi, T. New Aspects in Physics on Gel'fand Triplets. International Journal of Theoretical Physics 42, 2265–2283 (2003). https://doi.org/10.1023/B:IJTP.0000005958.57221.3f
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DOI: https://doi.org/10.1023/B:IJTP.0000005958.57221.3f