Radial confinement bounds

Correct radial operator

Consider the scalar operator $H=-\Delta+V(r)$ on ordinary $\mathbb R^d$, $d\ge2$, in units $\hbar^2/(2m)=1$. For a normalized spherical harmonic with angular eigenvalue $\ell(\ell+d-2)$, write $\psi(x)=R(r)Y_\ell(\hat x)$ and $u(r)=r^{(d-1)/2}R(r)$. This transformation sends the radial norm to $\int_0^\infty|u|^2dr$. Differentiating $R=r^{-(d-1)/2}u$ yields

$$H_\ell u=-u''+ \left[V(r)+\frac{(\ell+(d-2)/2)^2-1/4}{r^2}\right]u.$$

The term $(d-1)(d-3)/(4r^2)$ is contributed by the radial measure. Use the Friedrichs realization inherited from the full spatial quadratic form. In particular the regular branch has $u\sim r^{\ell+(d-1)/2}$; the critical $d=2,\ell=0$ endpoint selects the regular, nonlogarithmic branch. An arbitrary half-line boundary condition describes a different operator.

A form bound on attraction

For $d\ge3$, integration by parts for smooth compactly supported $\psi$ gives

$$0\le\int\left|\nabla\psi+a\frac{x}{r^2}\psi\right|^2dx =\int|\nabla\psi|^2dx+ [a^2-a(d-2)]\int\frac{|\psi|^2}{r^2}dx.$$

Choose $a=(d-2)/2$ and extend by form closure:

$$\boxed{\int|\nabla\psi|^2dx\ge \frac{(d-2)^2}{4}\int\frac{|\psi|^2}{r^2}dx.}$$

This is Hardy's inequality; see Nonnenmacher, Proposition 3.2.7.

As an application, measure lengths in a stated range $L$ and energies in $\hbar^2/(2mL^2)$. For the dimensionless operator

$$H_{\rm sat}=-\Delta+\lambda\frac{r^2}{1+r^2},\qquad \lambda>0,$$

subtract its limiting potential $\lambda$. Since $(1+r^2)^{-1}\le r^{-2}$,

$$\langle H_{\rm sat}-\lambda\rangle \ge\left[\frac{(d-2)^2}{4}-\lambda\right] \int\frac{|\psi|^2}{r^2}dx.$$

Thus $\lambda\le(d-2)^2/4$ excludes every spectral value below $\lambda$. The bound is sufficient; exceeding it alone does not establish a bound state. A first-order Dirac operator needs its own analysis.

• Confining oscillator domain — treats an unbounded harmonic potential with compact resolvent.

• Coulomb exponential profiles — illustrates how normalized spatial profiles enter energy bounds.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.