nonlinear stationary states of a wheeler--dewitt-type equation: a minisuperspace study with a self-interacting scalar clock

Quantum gravity phenomenology in black hole physics and cosmology demands frameworks that can bridge quantum effects with classical singularities while remaining amenable to observational constraints. motivated by effective quantum corrections in black hole thermodynamics and the quest for singularity resolution mechanisms analogous to those explored in loop quantum cosmology, we investigate a minimal nonlinear variant of the wheeler--dewitt framework in a flat frw minisuperspace where a homogeneous scalar field serves as an internal clock. after deparametrization, the physical wavefunction for the scale factor obeys a stationary nonlinear eigenvalue problem on the half-line, with a confining effective potential and a local cubic self-interaction that models quantum-gravitational backreaction or mean-field matter effects. we prove existence of a nodeless ground state under mild assumptions on the potential, develop a controlled perturbation theory at weak coupling, and compute solution families using high-accuracy chebyshev pseudo-spectral discretizations and continuation methods. across representative potentials we find a robust, sharply peaked probability density at a nonzero scale factor---a nonsingular quantum bounce---whose location shifts monotonically with the sign and strength of the nonlinearity. this behavior parallels bounce mechanisms in loop-inspired dynamics but emerges from a variational, local nonlinearity in the deparametrized equation. we discuss the interpretation within relational time, connections to effective quantum corrections in gravitational systems, and implications for excited states and time-dependent evolution, providing a transparent analytical and numerical baseline for exploring singularity avoidance in quantum cosmology.