Threshold Phenomena

A threshold is an energy where the structure of propagating channels changes. Even when scattering is complete at positive energy, the resolvent may have a different expansion at a threshold. In three dimensions, zero energy already distinguishes a regular point, a nonnormalizable resonance, and a genuine eigenstate. These distinctions govern low-energy limits and can change late-time decay.

Required background. Stationary Scattering Theory explains why its positive-energy estimates exclude zero. Resonances distinguishes momentum and energy continuation. Scattering-length and effective-range formulas remain with Low-Energy Scattering.

A regular momentum expansion can be singular in energy

For the free three-dimensional Hamiltonian $H_0=-\Delta$, the localized outgoing resolvent has kernel

$$\mathcal R_0(k;x,y) =\frac{e^{ik|x-y|}}{4\pi|x-y|} =\frac1{4\pi|x-y|}+\frac{ik}{4\pi}+O(k^2).$$

On fixed compact source and observation regions, this is an ordinary analytic expansion in $k$ near zero. In the energy variable $z=k^2$, however, the linear term is proportional to $\sqrt z$. Thus a branch point in energy does not require a pole in momentum.

This distinction persists in more complicated channel problems. Near a channel threshold $E_j$, its momentum behaves like $\sqrt{E-E_j}$, and the number or character of open channels changes. An expansion valid away from all thresholds need not be uniform as one is approached.

The three-dimensional formula is not universal. The one-dimensional free resolvent contains a $1/k$ term, and two-dimensional expansions contain logarithms. Dimension and boundary conditions belong in the statement of every threshold classification.

Zero-energy resonance versus zero-energy eigenstate

For real bounded compactly supported $V$ on $\mathbb R^3$, consider distributional solutions of

$$(-\Delta+V)\psi=0.$$

A zero-energy eigenstate is a nonzero $L^2$ solution in the operator domain. A zero-energy resonance, in the convention used here, is a nonzero solution in $H^2_{\rm loc}$ satisfying

$$\psi\in L^2_{-s}\quad\text{for every }s>1/2, \qquad \psi\notin L^2.$$

The weights are defined by $\|\psi\|_{L^2_{-s}}=\|\langle x\rangle^{-s}\psi\|_2$. In this compact-support setting a resonant solution has a nonzero $1/r$ leading tail. It is locally regular but its unweighted norm diverges.

If neither a zero eigenstate nor a zero resonance exists, the threshold is called regular. The free constant function is not a three-dimensional resonance in this definition: it fails the weight condition for $1/2<s\leq3/2$.

The classification is equivalent to analyzing the zero-energy obstruction in the local Lippmann–Schwinger operator. If that operator is invertible at zero, analytic inversion yields the regular localized expansion. A nontrivial kernel requires a separate finite-dimensional analysis and can produce negative powers of $k$. This is why simply substituting zero into a positive-energy inverse is unsafe.

A square well exactly at a resonance threshold

Let $V(r)=-q^2$ for $r<R$ and zero for $r>R$, with $q>0$ and $\hbar=2m=1$. For an s-wave write $\psi(r)=u(r)/r$. Regularity at the origin gives

$$u(r)=A\sin(qr),\qquad r<R.$$

A zero-energy resonant exterior has $u(r)=B$, so $\psi(r)=B/r$. Matching $u$ and $u'$ at $R$ requires

$$B=A\sin(qR),\qquad qA\cos(qR)=0.$$

A nonzero resonant solution therefore occurs at $qR=(n+\tfrac12)\pi$. In particular the first threshold occurs at $qR=\pi/2$.

Its tail is not square-integrable:

$$\int_R^\infty |\psi(r)|^2r^2\,dr =|B|^2\int_R^\infty dr=\infty,$$

but adding the weight $\langle r\rangle^{-2s}$ makes it integrable for $s>1/2$. The near-origin limit of $\sin(qr)/r$ is finite, so the divergence comes entirely from infinity.

This is the threshold test, not a second derivation of the full square-well scattering amplitude. The physical matching and cross sections are developed in Square-Well Scattering.

Threshold spectral weight produces power-law tails

A simple exact spectral model demonstrates the time consequence without asserting a universal decay rate. Let a normalized state have energy density on $E\geq0$

$$\rho(E)= \frac{E^\alpha e^{-E/\Lambda}} {\Gamma(\alpha+1)\Lambda^{\alpha+1}}, \qquad \alpha>-1,\quad\Lambda>0.$$

Its survival amplitude is the elementary Laplace transform

$$A(t)=\int_0^\infty e^{-itE/\hbar}\rho(E)\,dE =(1+i\Lambda t/\hbar)^{-\alpha-1}.$$

Consequently $|A(t)|^2=[1+(\Lambda t/\hbar)^2]^{-\alpha-1}$. The threshold exponent determines the late-time power, while the high-energy exponential cutoff makes the model well-defined.

For a regular free three-dimensional packet with nonzero smooth Fourier amplitude at zero, the local spectral density behaves like $E^{1/2}$. Threshold obstructions can change such powers, but proving a decay statement for an interacting Hamiltonian requires precise weighted resolvent expansions and remainder estimates. The gamma-density model explains the mechanism; it is not a theorem assigning the same exponent to every potential or preparation.

Exercises

1. Why can a higher-angular-momentum zero-energy solution be an eigenstate rather than a resonance?

Solution

Its decaying exterior radial wavefunction behaves like $r^{-\ell-1}$. The norm tail is proportional to $\int_R^\infty r^{-2\ell}dr$, which converges for $\ell\geq1$. If interior regularity and matching also hold, the resulting zero mode is square-integrable. The s-wave $1/r$ tail fails this test.

2. In the spectral model, compare amplitude and probability decay for $\alpha=1/2$.

Solution

The amplitude magnitude decays as $|t|^{-3/2}$ and the survival probability as $|t|^{-3}$. Confusing these exponents would also confuse a resonance amplitude width with a probability decay rate.

References

• A. Jensen and T. Kato, “Spectral properties of Schrödinger operators and time-decay of the wave functions,” Duke Mathematical Journal 46, 583–611, 1979, doi:10.1215/S0012-7094-79-04631-3 — threshold spectral analysis and weighted decay.
• A. Jensen and G. Nenciu, “A unified approach to resolvent expansions at thresholds,” Reviews in Mathematical Physics 13, 717–754, 2001; erratum 16, 675–677, 2004, doi:10.1142/S0129055X01000843, author preprint — dimension-dependent expansions and threshold obstructions.
• M. Zworski, “Mathematical study of scattering resonances,” Bulletin of Mathematical Sciences 7, 1–85, 2017, doi:10.1007/s13373-017-0099-4, author text.