Ordinary differential equations Asymptotic and perturbation methods Theorem

Levinson's theorem on asymptotic integration

Builds on System of linear differential equations Fundamental matrix Diagonal matrix Integrable function

Statement

Let $t_0 \in \R$ and let $n \ge 1$ . Let $\Lambda(t) = \operatorname{diag}(\lambda_1(t), \dots, \lambda_n(t))$ be a continuous diagonal matrix function on $[t_0, \infty)$ with complex entries. Let $R(t)$ be a continuous complex $n \times n$ matrix function on $[t_0, \infty)$ with $\int_{t_0}^{\infty} \norm{R(t)} \, dt < \infty.$ Assume the dichotomy condition: there is a constant $K$ such that for each pair $i \ne j$ one of the following holds, where $D_{ij}(s, t) = \int_s^t \operatorname{Re}\bigl(\lambda_i(u) - \lambda_j(u)\bigr) \, du$ :

$D_{ij}(t_0, t) \to \infty$ as $t \to \infty$ , and $D_{ij}(s, t) \ge -K$ for all $t \ge s \ge t_0$ ;
$D_{ij}(s, t) \le K$ for all $t \ge s \ge t_0$ .

Then the linear system $x' = \bigl(\Lambda(t) + R(t)\bigr) x$ has a fundamental matrix of the form $X(t) = \bigl(I + E(t)\bigr) \exp\Bigl(\int_{t_0}^{t} \Lambda(s) \, ds\Bigr), \qquad E(t) \to 0 \text{ as } t \to \infty.$ Equivalently, for each $k$ there is a solution $x_k(t) = \bigl(e_k + o(1)\bigr) \exp\bigl(\int_{t_0}^t \lambda_k(s) \, ds\bigr)$ , where $e_k$ is the $k$ -th standard basis vector.

Remarks

The proof writes the $k$ -th solution as $x_k = \exp\bigl(\int \lambda_k\bigr) z$ and turns the equation for $z$ into an integral equation. The dichotomy condition splits the indices into those whose integrals are taken from $t_0$ to $t$ and those taken from $t$ to $\infty$ . With this split the integral operator is a contraction on a far half-line $[T, \infty)$ , because $\int_T^\infty \norm{R}$ is small. Its fixed point gives $z(t) \to e_k$ .

Example: if $\Lambda$ is a constant diagonal matrix, then each $D_{ij}(s, t)$ is $(t - s)$ times a constant, so the dichotomy condition holds automatically. Hence for a constant diagonal $\Lambda$ with distinct or equal eigenvalues and an integrable perturbation $R$ , each solution of $x' = (\Lambda + R) x$ behaves like $e^{\lambda_k t}$ times a vector close to $e_k$ .

The integrability of $R$ cannot be dropped. For $n = 1$ , $\Lambda = 0$ and $R(t) = 1/t$ on $[1, \infty)$ , the solutions are $c t$ , which do not tend to a constant multiple of $e^0 = 1$ .

Going deeper

For a student

The theorem is the basic tool of asymptotic integration of linear systems. One first brings a system into the form diagonal part plus integrable part, often by a change of variables that diagonalises the leading term. Then the theorem gives the leading behaviour of a fundamental matrix. The WKB, or Liouville–Green, approximation for $y'' = Q(t) y$ can be justified in this way after a suitable substitution. Hartman and Wintner proved a related result in which $R$ is only square integrable, at the cost of a stronger dichotomy condition and a correction term in the exponent.

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Diagonal matrix concept
Fundamental matrix concept
Integrable function concept
System of linear differential equations concept
Levinson's theorem on asymptotic integration statement

Tag	`5ZFRRMBFV0`
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About	System of linear differential equations, Fundamental matrix
Uses	System of linear differential equations, Fundamental matrix, Diagonal matrix, Integrable function