Short-Range Potentials

A short-range potential becomes integrably small along freely escaping trajectories. This criterion concerns spatial decay together with the kinetic dynamics, not merely the boundedness of the interaction. Here a simple pointwise condition gives a precise existence and completeness theorem, and a momentum-localized argument explains the critical decay exponent.

Required background. Cook’s Method turns an integrable trajectory estimate into a wave operator. Asymptotic Completeness defines exhaustion of the interacting absolutely continuous sector.

A sufficient short-range theorem

Work in units $\hbar=2m=1$ on $L^2(\mathbb R^d)$, $d\geq1$. Let

$$H_0=-\Delta,\qquad D(H_0)=H^2(\mathbb R^d),$$

and assume $V$ is a real measurable multiplication operator satisfying

$$|V(x)|\leq C\langle x\rangle^{-\rho}, \qquad \rho>1,\qquad \langle x\rangle=(1+|x|^2)^{1/2}.$$

Then $H=H_0+V$ is self-adjoint on $H^2$, and the ordinary wave operators exist and are complete:

$$\operatorname{Ran}W_\pm=\mathcal H_{\rm ac}(H).$$

For this bounded pointwise-decay class, the standard spectral results also give no singular continuous spectrum and no positive eigenvalues. The essential and absolutely continuous spectra are $[0,\infty)$. Negative eigenvalues and a zero-energy obstruction remain possible.

The bounded hypothesis avoids local singularities. It is sufficient rather than necessary. Coulomb behavior at the origin can be compatible with self-adjointness, but a Coulomb tail at infinity lies outside the displayed short-range condition. These are different issues.

Yafaev’s Lecture 2 gives the stationary theorem for this class. Teschl’s Theorems 12.11–12.12 prove spectral and completeness conclusions under a more general operator-tail condition; that general condition alone should not be silently strengthened to every assertion about positive eigenvalues in the pointwise class.

Why decay faster than inverse distance appears

Take a packet with $\widehat\phi\in C_c^\infty(\mathbb R^d\setminus\{0\})$, supported where $a\leq|k|\leq b$ for some $a>0$. Such packets form a dense set in $L^2$. Its free evolution is

$$(U_0(t)\phi)(x) =(2\pi)^{-d/2}\int e^{i(k\cdot x-t|k|^2)} \widehat\phi(k)\,dk.$$

Inside $|x|<a|t|$, the phase gradient obeys

$$|x-2tk|\geq a|t|.$$

Repeated integration by parts in momentum therefore gives, for every sufficiently large integer $N$,

$$\|\mathbf1_{\{|x|<a|t|\}}U_0(t)\phi\|_2 \leq C_{\phi,N}|t|^{d/2-N}, \qquad |t|\geq1.$$

The volume factor explains the exponent $d/2$. This is a nonstationary-phase estimate; the momentum support stays away from zero so the packet cannot have a stationary phase point inside the slower cone.

Outside that cone, the potential bound gives

$$\|V\mathbf1_{\{|x|\geq a|t|\}}U_0(t)\phi\|_2 \leq C\langle a|t|\rangle^{-\rho}\|\phi\|_2.$$

Combining the two regions produces an integrable time bound when $\rho>1$ and $N>d/2+1$. The finite-time portion is controlled by boundedness of $V$. Cook’s method proves existence on this dense class and then on all of $L^2$.

No smallness of $C$ was assumed. Short range is a condition on the tail, and a strong interaction can support bound states while having complete scattering on its absolutely continuous sector.

An operator-tail formulation and completeness

A useful sufficient formulation allows a symmetric $H_0$-bounded $V$ of relative bound below one. Put $\chi_r=\mathbf1_{\{|x|\geq r\}}$. Teschl’s short-range condition is integrability in $r\geq0$ of

$$h(r)=\|\chi_r V(H_0+1)^{-1}\|.$$

An equivalent formulation puts the spatial cutoff on the other side of the free resolvent. Under the pointwise bound above,

$$h(r)\leq C\langle r\rangle^{-\rho}\|(H_0+1)^{-1}\|,$$

so the criterion holds immediately.

Existence uses free propagation estimates. Completeness needs an additional argument about interacting states. The Enss proof combines energy localization away from zero, incoming/outgoing decompositions, and compactness of localized resolvent differences. It shows that an interacting continuous state orthogonal to the wave-operator range must have zero norm. Teschl supplies the detailed compactness and propagation lemmas; this paragraph describes their role, not a substitute proof.

The same estimates exclude singular continuous spectrum in that setting. Bound states remain orthogonal to the scattering range, so completeness does not assert that every normalized vector escapes.

Thresholds are outside uniform positive-energy estimates

Energy localization to a compact interval in $(0,\infty)$ keeps free velocities away from zero. As that interval approaches zero, the constants in the argument need not remain bounded.

A zero-energy eigenfunction or resonance can change low-energy resolvent behavior without contradicting completeness of the wave operators. Thresholds are a separate question treated in Threshold Phenomena. In particular, an almost-everywhere spectral theorem cannot determine a pointwise threshold limit by itself.

Exercises

1. For $V(x)=g(1+|x|^2)^{-\rho/2}$ in three dimensions, compare this theorem with the square-integrable-potential Cook example.

Solution

The present theorem applies for $\rho>1$. Square integrability requires $\rho>3/2$ because the radial integral behaves like $\int^\infty r^{2-2\rho}dr$. Thus $1<\rho\leq3/2$ requires more information than the elementary global dispersive estimate used in that example.

2. Does large coupling $g$ invalidate the short-range condition?

Solution

No, provided it is finite and the stated decay and reality hold. The decay exponent remains the same. Large coupling can change bound levels and threshold behavior, but this sufficient theorem is not a perturbative expansion in coupling.

References

• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Chapter 12, author’s text.
• D. Yafaev, “Lectures on scattering theory,” 2004, arXiv:math/0403213.