A New Family of Self-Adjoint Quantum Operators with Logarithmic Potential: From Self-Adjointness to Sharp Spectral Asymptotics and Green’s Function Analysis

The Schrödinger operator with a combined singular, confining, and logarithmic potential arises in diverse physical contexts—from quantum dots and Bose–Einstein condensates to effective field theories. Despite its broad relevance, a rigorous spectral analysis of this operator family has remained elusive due to the simultaneous presence of three challenging features: a $1/x^{2}$ singularity at the origin, an unbounded logarithmic term, and harmonic confinement at infinity.
We introduce and rigorously analyze the one-parameter family of operators
\[H = -\frac{d^2}{dx^2} + \frac{\nu(\nu + 1)}{x^2} + \omega^2 x^2 + \beta \ln x,\qquad \nu > 0,\;\beta \in \mathbb{R},\;\omega > 0,\]
acting on $L^{2}(0,\infty)$. Self-adjointness is established through a combination of Weyl's limit-point/limit-circle classification and the KLMN theorem for quadratic forms, circumventing the failure of standard Kato–Rellich perturbation theory. The spectrum is shown to be purely discrete and bounded below, with simple eigenvalues that depend analytically on the logarithmic coupling $\beta$.
The central result of this work is the sharp asymptotic expansion of the eigenvalues for large quantum numbers:
\[E_{n} = 2\omega n + \beta \ln n + \gamma_{0} + \frac{\beta}{2n} + \mathcal{O}\left(\frac{1}{n^{2}}\right),\]
where the constant
\[\gamma_{0} = \omega \left(\nu + \tfrac{3}{2}\right) + \beta \left(\ln \sqrt{\tfrac{2}{\omega}} - \tfrac{1}{2}\right)\]
is determined explicitly from the Langer-corrected WKB quantization condition. This expansion reveals a distinctive logarithmic growth that modifies the standard linear Weyl law, yielding the spectral counting function
\[N(E) = \frac{E}{2\omega} - \frac{\beta}{2\omega}\ln E + \mathcal{O}(1).\]
To complement the asymptotic analysis, we construct the exact Green's function and prove the Hilbert–Schmidt property of the resolvent. Numerical diagonalization using a logarithmic grid confirms the asymptotic formula with relative errors below $0.02\%$ for all $n \geq 5$, validating both the analytic predictions and the underlying WKB methodology.
These results provide the first complete spectral characterization of Schrödinger operators with logarithmic potentials in the presence of singular and confining terms, with potential applications to few-body quantum systems, spectral geometry, and the analysis of trace formulae.