The nonextensive entropic measure proposed by Tsallis [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] introduces a parameter, $q$, which is not defined but rather must be determined. The value of $q$ is typically determined from a piece of data and then fixed over the range of interest. On the other hand, from a phenomenological viewpoint, there are instances in which $q$ cannot be treated as a constant. We present two distinct approaches for determining $q$ depending on the form of the equations of constraint for the particular system. In the first case the equations of constraint for the operator $\widehat{O}$ can be written as $\mathrm{Tr}\left(F^q\widehat{O}\right)=C$, where $C$ may be an explicit function of the distribution function $F$. We show that in this case one can solve an equivalent maxent problem which yields $q$ as a function of the corresponding Lagrange multiplier. As an illustration the exact solution of the static generalized Fokker-Planck equation (GFPE) is obtained from maxent with the Tsallis enropy. As in the case where $C$ is a constant, if $q$ is treated as a variable within the maxent framework the entropic measure is maximized trivially for all values of $q$. Therefore $q$ must be determined from existing data. In the second case an additional equation of constraint exists which cannot be brought into the above form. In this case the additional equation of constraint may be used to determine the fixed value of $q$.

We examine phenomenological consequences of a scale-dependent gravity model in which the gravitational coupling runs with the cosmological scale factor as G(a)=G0an. In the companion paper, this running was shown to modify the Friedmann equation through effective starred couplings and to produce corresponding changes in the Wheeler–DeWitt minisuperspace potential. For values of n, we ask whether the same running can survive into observationally tested epochs. Applying three consistency tests—the expansion rate during Big-Bang nucleosynthesis, the growth of linear matter perturbations, and the CMB acoustic scale—we find that BBN and CMB consistency require n to be very small if the power-law form persists to nucleosynthesis or recombination. These tests constrain the epoch over which the pure power-law form can remain valid. The power-law running is best considered as an early-universe scaling regime that must shut off before nucleosynthesis — the question is not whether, but when and how. Any realistic extension of the strong-running regime must cut off, screen, or smoothly transition the gravitational coupling back to standard gravity before the nucleosynthesis era.

An algorithm for finding the bound-state eigenvalues and eigenfunctions of a Hermitian Hamiltonian operator using Green's method, developed by Waxman\cite{W98}, has been extended to include non-Hermitian Hamiltonian operators.

An algorithm for finding the bound-state eigenvalues and eigenfunctions of a Hermitian Hamiltonian operator using Green's method, developed by Waxman\cite{W98},has been extended to include non-Hermitian Hamiltonian operators.

We study an effective quantum cosmological model with a scale-dependent gravitational coupling $G\left(a\right)$. Starting from a Newtonian cosmological construction inspired by a corrected two-body gravitational potential, we derive an effective Friedmann-type equation and quantize the corresponding minisuperspace model to obtain a Wheeler–DeWitt equation. For the power-law running $G\left(a\right)=G_0{a}^n$, the small-scale behavior is controlled by the exponent n. In the flat dust model under the repulsive effective-potential prescription adopted here, with a quantum correction term $\mathrm{\Xi }>0$, the values $n=3/2$ and $n=2$ mark distinct analytic boundaries: $n=3/2$ marks the sign change of the quantum-correction coefficient, while $n=2$ marks the degeneracy of the leading powers of a. A genuine bounce occurs only when a root of $H^2\left(a\right)$ separates a forbidden small-a region from an allowed expanding branch, so that nonsingular sectors appear without spatial curvature, exotic matter, or nonstandard quantization prescriptions. We obtain Bessel, Airy, numerical, and WKB solutions, and show that the model separates genuine DeWitt suppression from regularity at the origin. The WKB analysis exhibits a forbidden region, a turning point, and an oscillatory semiclassical branch, while the conserved probability current distinguishes expanding and contracting sectors. A scalar field treated as a weakly backreacting internal clock yields wave packets that track the leading-order classical relational trajectory at a large scale factor; the WKB Hamilton–Jacobi limit recovers the modified Friedmann dynamics as the leading-order phase of the Wheeler–DeWitt wave function.

We study an effective quantum cosmological model with a scale-dependent gravitational coupling G(a). Starting from a quantum-corrected Newtonian cosmology, we derive a modified Friedmann equation and quantize the corresponding minisuperspace model to obtain a Wheeler–DeWitt equation. For the power-law running G(a)=G0an, the small-scale behavior is controlled by the exponent n. In the flat dust model with a positive quantum correction term Ξ, the values n=3/2 and n=2 mark distinct analytic boundaries: n=3/2 marks the sign change of the quantum-correction coefficient, while n=2 marks the degeneracy of the leading powers of a. A genuine bounce occurs only when a root of H2(a) separates a forbidden small-a region from an allowed expanding branch, so that nonsingular sectors appear without spatial curvature, exotic matter, or nonstandard quantization prescriptions. We obtain Bessel, Airy, numerical, and WKB solutions, and show that the model separates genuine DeWitt suppression from regularity at the origin. The WKB analysis exhibits a forbidden region, a turning point, and an oscillatory semiclassical branch, while the conserved probability current distinguishes expanding and contracting sectors. A scalar-field clock yields wave packets that track the classical trajectory at large scale factor; the WKB Hamilton–Jacobi limit recovers the modified Friedmann dynamics as the leading-order phase of the Wheeler–DeWitt wave function.