Resonance Poles and Resolvent Continuation

A resonance is a pole of a specified analytic continuation of a scattering object. It is not a complex eigenvalue of the original self-adjoint Hamiltonian. This page constructs the continuation for bounded compactly supported potentials in three dimensions and distinguishes its poles from bound eigenvalues, threshold singularities, and approximate exponential decay.

Required background. Resolvent Methods distinguishes ordinary resolvents from continuation. Helpful background. Rigorous Lippmann–Schwinger Equation supplies the compact local operator used below. Physical phase motion and line shapes belong to Scattering Resonances.

A momentum-plane continuation in three dimensions

Set $\hbar=2m=1$ and let

$$H=-\Delta+V,\qquad V\in L^\infty(\mathbb R^3;\mathbb R), \qquad \operatorname{supp}V\subset B_R.$$

Write the spectral parameter as $z=k^2$. Initially, for $\operatorname{Im}k>0$ away from bound-state poles,

$$\mathcal R_V(k)=(H-k^2)^{-1}.$$

The free kernel is

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

For sources and observation regions in fixed compact sets, this kernel depends holomorphically on every $k\in\mathbb C$. It therefore defines an entire family from compactly supported $L^2$ sources to local responses. It does not remain bounded on global $L^2$ when $\operatorname{Im}k<0$: the outgoing exponential then grows at infinity.

Let $X=L^2(B_R)$ and $K(k)=\mathbf1_{B_R}\mathcal R_0(k)V$, with the input extended by zero. The compactness proof on the Lippmann–Schwinger page applies, and the compact-support kernel makes $K(k)$ holomorphic in Hilbert–Schmidt norm.

At $k=i\kappa$ with $\kappa$ sufficiently large,

$$\|K(i\kappa)\|\leq\frac{\|V\|_\infty}{\kappa^2}<1.$$

Thus $I+K(k)$ is invertible at one point. Analytic Fredholm theory makes its inverse meromorphic, with finite-rank principal parts.

For a compactly supported source $f$, solve the local equation and reconstruct:

$$\mathcal R_V(k)f =\mathcal R_0(k)f -\mathcal R_0(k)V(I+K(k))^{-1} \bigl(\mathcal R_0(k)f|_{B_R}\bigr).$$

This proves meromorphic continuation from $L^2_{\rm comp}$ to $H^2_{\rm loc}$. The elliptic local regularity follows from the differential equation. Analytic Fredholm theory is the non-elementary input; all spatial restrictions in its use have now been specified.

Poles and their parameter matter

A pole of this continued family is a resonance in the broad resolvent sense. For real $V$, poles with $\operatorname{Im}k>0$ occur at $k=i\kappa$ and correspond to negative bound eigenvalues $-\kappa^2$. In scattering terminology one often reserves “resonance” for nonphysical-sheet poles and lists bound states separately.

For a fourth-quadrant pole $k=a-ib$ with $a,b>0$,

$$z=k^2=(a^2-b^2)-2iab.$$

The associated energy has negative imaginary part. A pole on the negative imaginary $k$ axis instead has real negative energy but an exponentially growing outgoing spatial behavior; it is commonly called a virtual or antibound state. Equal values of $k^2$ do not identify equal sheets.

The threshold $k=0$ needs separate analysis. Moreover, the three-dimensional entire free continuation is not dimension-independent. In even dimensions the continuation generally lives on a logarithmic cover; in one dimension the free resolvent itself has a $1/k$ threshold singularity.

Resonant states satisfy an outgoing condition

At a simple nonzero pole, a nonzero vector in the range of the resolvent residue solves

$$(H-k_*^2)u=0$$

distributionally and obeys the continued outgoing condition. For a lower-half-plane pole it generally grows exponentially at infinity and is not in $L^2$.

This is consistent with self-adjointness: the equation is posed in a local or continued space, outside the ordinary operator domain. A generalized resonance vector is not a normalizable state of complex energy.

Higher-order poles can produce generalized resonant chains rather than a single eigenfunction. Multiplicity is a finite-dimensional analytic notion and should not be read off from the height of a measured peak.

Pole decay is a controlled approximation only with a remainder

If an energy pole is written $z_*=E_R-i\Gamma/2$, its formal time factor is

$$e^{-itz_*/\hbar} =e^{-itE_R/\hbar}e^{-\Gamma t/(2\hbar)}.$$

The corresponding squared amplitude decays at rate $\Gamma/\hbar$. Applying this to physical wave packets requires a justified contour deformation, localization, and a bound on the non-pole remainder.

The full self-adjoint evolution remains unitary. Threshold contributions and background spectral weight can compete with the pole term, especially at long times or near channel openings. For a normalized vector with finite energy variance, the short-time survival probability is quadratic, which also rules out an exact positive-rate exponential starting at $t=0$.

Complex Scaling realizes suitable resonance poles as discrete eigenvalues of a deformed nonself-adjoint operator. Threshold Phenomena explains why the energy origin changes the analysis.

Exercises

1. For $k_*=a-ib$ in the fourth quadrant, identify the energy width $\Gamma$.

Solution

Since $\operatorname{Im}(k_*^2)=-2ab$, comparison with $E_R-i\Gamma/2$ gives $\Gamma=4ab$ in the stated units. The momentum imaginary part $b$ is not itself the energy width.

2. Can a particular scalar matrix element fail to display a pole of the operator resolvent?

Solution

Yes. If $\mathcal R_V(k)=A_{-1}/(k-k_*)+\cdots$, a pairing $\langle f,\mathcal R_V(k)g\rangle$ has no residue when $\langle f,A_{-1}g\rangle=0$. A chosen source or detector may decouple from the resonant subspace. Absence of a peak in one response does not exclude an operator pole.

References

• M. Zworski, “Mathematical study of scattering resonances,” Bulletin of Mathematical Sciences 7, 1–85, 2017, doi:10.1007/s13373-017-0099-4, author text.