Abstract.We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Front. Phys. Chem. Chem. Phys. 2, 1 (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -40 in total to be exact -analytic solutions. Furthermore, an analogous analysis of the hyperbolic counterpart of the planar pendulum, the Razavy problem, which was shown to be also C-QES [Am. J. Phys. 48, 285 (1980)], confirmed that it is anti-isospectral with the pendular eigenproblem. Of key importance for both eigenproblems proved to be the topological index κ, as it determines the loci of the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless interaction parameters η and ζ. It also encapsulates the conditions under which analytic solutions to the two eigenproblems obtain and provides the number of analytic solutions. At a given κ, the anti-isospectrality occurs for single states only (i.e., not for doublets), like C-QES holds solely for integer values of κ, and only occurs for the lowest eigenvalues of the pendular and Razavy Hamiltonians, with the order of the eigenvalues reversed for the latter. For all other states, the pendular and Razavy spectra become in fact qualitatively different, as higher pendular states appear as doublets whereas all higher Razavy states are singlets.

We make use of the Quantum Hamilton-Jacobi (QHJ) theory to investigate conditional quasisolvability of the quantum symmetric top subject to combined electric fields (symmetric top pendulum). We derive the conditions of quasi-solvability of the time-independent Schrödinger equation as well as the corresponding finite sets of exact analytic solutions. We do so for this prototypical trigonometric system as well as for its anti-isospectral hyperbolic counterpart. An examination of the algebraic and numerical spectra of these two systems reveals mutually closely related patterns.The QHJ approach allows to retrieve the closed-form solutions for the spherical and planar pendula and the Razavy system that had been obtained in our earlier work via Supersymmetric Quantum Mechanics as well as to find a cornucopia of additional exact analytic solutions. * bretislav.friedrich@fhi-berlin.mpg.de † burkhard.schmidt@fu-berlin.de arXiv:1802.07939v1 [math-ph]

We determine the energy-momentum tensor of non-perfect fluids in thermodynamic equilibrium. To this end, we derive the constitutive equations for energy density, isotropic and anisotropic pressure as well as for heat-flux from the corresponding propagation equations and by drawing on Einstein's equations. Following Obukhov at this, we assume the corresponding space-times to be conformstationary and homogeneous. This procedure provides these quantities in closed form, i.e., in terms of the structure constants of the three-dimensional isometry group of homogeneity and, respectively, in terms of the kinematical quantities expansion, rotation and acceleration. In particular, we find a generalized form of the Friedmann equations. As special cases we recover Friedmann and Gödel models as well as non-tilted Bianchi solutions with anisotropic pressure. All of our results are derived without assuming any equations of state or other specific thermodynamic conditions a priori. For the considered models, results in literature are generalized to rotating fluids with dissipative fluxes.PACS numbers: 04.20.-q, 04.40.-b, 47.75.+f, 98.80.Jk * konscha@physik.tu-berlin.de † borzeszk@mailbox.tu-berlin.de ‡ tchrobok@mailbox.tu-berlin.de 1 We emphasize that we approach this without any specific thermodynamic conditions as done for instance in [16], i. e., we refrain from applying linear or extended thermodynamics.

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