Quantum theory Scattering theory Theorem

Levinson's theorem

Builds on Scattering theory S-matrix Bound state Schrödinger equation

Statement

Let $V \colon (0, \infty) \to \R$ be a spherically symmetric potential on $\R^3$ , written as a function of $r = \abs{x}$ , with $\int_0^\infty r (1 + r) \abs{V(r)}\, dr < \infty$ . Use units in which the Schrödinger equation is $-\Delta \psi + V \psi = E \psi$ . Fix an integer $l \ge 0$ and $k > 0$ , and consider the radial equation $-u''(r) + \Big( \frac{l(l+1)}{r^2} + V(r) \Big) u(r) = k^2 u(r).$ Let $u$ be its solution that vanishes at $r = 0$ . Define the phase shift $\delta_l(k)$ by $u(r) \sim A \sin(kr - l\pi/2 + \delta_l(k))$ as $r \to \infty$ , with $A \ne 0$ . Normalize $\delta_l$ so that it is continuous in $k$ and $\delta_l(k) \to 0$ as $k \to \infty$ . Let $n_l$ be the number of bound states of angular momentum $l$ . These are the energies $-\kappa^2 < 0$ for which the radial equation, with $k^2$ replaced by $-\kappa^2$ , has a nonzero square-integrable solution vanishing at $r = 0$ . Assume that there is no zero-energy bound state or zero-energy resonance in the channel $l$ . Then $\delta_l(0) = n_l \pi.$ For $l = 0$ , if instead there is a zero-energy resonance, then $\delta_0(0) = \big(n_0 + \tfrac{1}{2}\big)\pi$ .

Remarks

The proof applies the argument principle to the Jost function $f_l(k)$ of the channel. It is analytic in the upper half-plane, tends to $1$ at infinity, and satisfies $\delta_l(k) = -\arg f_l(k)$ for $k > 0$ . Its zeros in the upper half-plane lie at the points $k = i\kappa$ with $-\kappa^2$ a bound state energy, so the change of phase along the real axis counts the bound states. Example: for the attractive square well $V = -V_0$ on $r < a$ and $V = 0$ outside, with $\sqrt{V_0}\, a$ slightly larger than $\pi/2$ , there is exactly one bound $s$ -state, and $\delta_0(0) = \pi$ . Without the exclusion of zero-energy resonances, the formula fails for $l = 0$ : the extra term $\pi/2$ appears. In terms of the S-matrix element $S_l(k) = e^{2i\delta_l(k)}$ , the theorem relates the winding of $S_l$ along the positive real axis to the number of bound states.

Going deeper

For a student

The theorem is a first instance of the link between the continuous spectrum and the discrete spectrum of a Schrödinger operator. It is used to count bound states from scattering data. It is also a key step in the inverse scattering problem on the half-line, where the phase shifts determine the potential only once the bound state data are added. Its analytic content is the argument principle for the Jost function, the same tool that locates resonances. Versions exist in one dimension, in two dimensions and for Dirac operators, with different half-integer corrections at zero energy.

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Bound state concept
S-matrix concept
Scattering theory concept
Schrödinger equation concept
Levinson's theorem statement

Tag	`AFJ4WFNKXK`
URI	`m2://AFJ4WFNKXK`
Kind	statement, theorem
Writing	W0 stub
MSC	81U05, 81U20
Status	This copy of the site has no build data. Run `m2 status AFJ4WFNKXK`.
About	S-matrix, Bound state
Uses	Scattering theory, S-matrix, Bound state, Schrödinger equation
Same as	Wikidata Q25110385 Wikipedia Levinson's theorem
Changed	3 November 2026