Complex Scaling

Complex scaling converts outgoing exponential growth into decay along a deformed spatial contour. Resonances can then appear as discrete eigenvalues of a nonself-adjoint operator. This page gives an explicit one-dimensional exterior-scaling construction, including the interface domain. It does not identify every complex eigenvalue of an arbitrary absorbing numerical model with a physical resonance.

Required background. Resonances defines poles of a continued resolvent and distinguishes momentum from energy. Helpful background. Domains of Hamiltonians explains why the differential expression and its domain must be specified together.

Exterior scaling leaves the interaction region real

Let

$$H=-\frac{d^2}{dx^2}+V(x),\qquad V\in L^\infty(\mathbb R;\mathbb R),\qquad \operatorname{supp}V\subset(-R,R),$$

with physical domain $H^2(\mathbb R)$ and units $\hbar=2m=1$. Choose $0<\theta<\pi/2$ and deform the exterior rays:

$$z_\theta(x)= \begin{cases} -R+e^{i\theta}(x+R),&x<-R,\\ x,&|x|\leq R,\\ R+e^{i\theta}(x-R),&x>R. \end{cases}$$

The potential is never evaluated at a complex argument inside its support. Outside, it is zero. Thus bounded compactly supported potentials need not be analytic to use this exterior construction.

On $L^2(\mathbb R,dx)$, representing functions on the contour by their parameter $x$, the differential expression is

$$H_\theta u= \begin{cases} -e^{-2i\theta}u'',&|x|>R,\\ -u''+Vu,&|x|<R. \end{cases}$$

Its domain consists of functions in $H^2$ on each of the three intervals whose values are continuous at $\pm R$ and whose derivative with respect to the contour coordinate is continuous:

$$\begin{aligned} e^{-i\theta}u'(-R-)&=u'(-R+),\\ u'(R-)&=e^{-i\theta}u'(R+). \end{aligned}$$

These are transmission conditions, not Dirichlet conditions at artificial walls. They follow from continuity of $u$ and $\partial_z u$. A smooth contour avoids corners; the piecewise straight version makes the matching conditions explicit.

The resulting closed operator is nonself-adjoint. Its exterior continuous spectrum lies on the rotated energy ray $e^{-2i\theta}[0,\infty)$. The interaction produces discrete spectral points away from that ray.

Outgoing waves become square-integrable in an exposed sector

For nonzero complex momentum $k$, a one-dimensional outgoing resonant solution obeys

$$u(z)=a_+e^{ikz}\quad\text{on the right},\qquad u(z)=a_-e^{-ikz}\quad\text{on the left}.$$

On either scaled exterior ray, its modulus decays like

$$\exp[-\operatorname{Im}(ke^{i\theta})|x|]$$

up to an $x$-independent factor. If

$$-\theta<\arg k<0,$$

then $\operatorname{Im}(ke^{i\theta})>0$. A fourth-quadrant resonance that grew on the real axis therefore becomes square-integrable on the contour.

The corresponding energy $z=k^2$ lies in the wedge $-2\theta<\arg z<0$, between the physical positive axis and the rotated continuum. The factor of two belongs to the energy map; the outgoing-wave decay condition concerns momentum.

Why scaled eigenvalues identify outgoing poles

Away from the support, a solution of $(H_\theta-k^2)u=0$ is a linear combination of outgoing and incoming exponentials. In the exposed sector, the outgoing term decays and the incoming term grows.

Square integrability forces the incoming coefficient to vanish on each exterior ray. The transmission conditions transport the remaining solution through the real interaction region. Hence a scaled eigenfunction yields a nonzero physical differential-equation solution with purely outgoing exterior behavior, which characterizes a resonance away from threshold.

Conversely, an outgoing resonant solution analytically continues along the exterior free rays, decays there, and satisfies the transmission conditions. It becomes an eigenfunction of $H_\theta$ at $k^2$.

This proves the basic eigenfunction correspondence in this one-dimensional setting. Fredholm theory establishes discreteness and the matching algebraic multiplicities; those additional operator-theoretic steps are supplied by the cited complex-scaling theorem. The local calculation does not by itself prove a general multidimensional scaling theorem.

Global dilation has stronger analytic requirements

For real $s$, the unitary dilation in $d$ dimensions is

$$(D(s)f)(x)=e^{ds/2}f(e^s x).$$

Conjugation gives

$$D(s)HD(s)^{-1}=e^{-2s}(-\Delta)+V(e^s x).$$

Replacing $s$ by $i\theta$ is an analytic continuation, not a unitary coordinate change on arbitrary $L^2$ functions. A global construction requires a suitable analytic family of potentials and operators. A discontinuous square well does not acquire a meaning at complex arguments merely by writing $V(e^{i\theta}x)$.

Exterior scaling avoids that issue when the potential is supported inside the unchanged real region. Long-range potentials require additional analyticity or a more general distortion framework.

Numerical eigenvalues need stability and convergence checks

A true exposed resonance is independent of the scaling angle while it remains in the permitted wedge. Discretized continuum points usually move with the rotated ray. This gives a useful diagnostic, but apparent angle stability alone is not a proof: mesh errors, finite exterior length, and nonnormal spectral sensitivity can also create misleading points.

A calculation should check the interface conditions, refinement, truncation, and residual together with the analytic hypotheses. The deformed operator is a spectral tool; its evolution is not the physical norm-preserving dynamics.

Exercises

1. For $k=2-0.2i$, which angles expose the outgoing mode?

Solution

One needs $\theta>\arctan(0.1)$, with $\theta<\pi/2$ in this construction. At equality the exterior exponential has constant modulus and is not square-integrable. Strict inclusion in the sector is required.

2. What goes wrong if ordinary derivative continuity is imposed at $R$?

Solution

The contour derivative is $e^{-i\theta}u'$ outside and $u'$ inside. Equating the unscaled parameter derivatives generally introduces an artificial interface interaction. The computed eigenvalues would then belong to a different operator rather than the intended exterior deformation.

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. — Section 2.6 and Theorem 6 give the exterior-contour construction and Fredholm correspondence.