Quantum black hole as a harmonic oscillator from the perspective of the minimum uncertainty approach

Starting from the eigenvalue equation for the mass of a black hole derived by Mäkelä and Repo, we show that, by reparametrizing the radial coordinate and the wave function, it can be rewritten as the eigenvalue equation of a quantum harmonic oscillator. We then study the interior of a Schwarzschild black hole using two quantization approaches. In the standard quantization, the area and mass spectra are discrete, characterized by a quantum number \(n\), but the wave function is not square-integrable, limiting its physical interpretation. In contrast, a minimal-uncertainty quantization approach yields an area spectrum that grows as \(n^2\), and consequently the mass \(M\) also increases. In this framework, the wave function is finite and square-integrable, with convergence requiring that the deformation parameter \(\beta \) be regulated by a discrete quantum number \(m\). The wave function exhibits quantum tunneling connecting the black hole interior with both its exterior and a white hole region, effects that disappear in the limit \(\beta \rightarrow 0\). These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole.