On the behaviour for large values of the time of the solution of the Cauchy problem for the equation $\frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}+\alpha(x)u=0$

We obtain an asymptotic expansion as $t\to\infty$ for the solution $u(t,x)$ of the Cauchy problem with initial functions of compact support for the equation
$$ u_{tt}-u_{xx}+(\alpha_0+\varphi(x))u=0,\qquad t>0,\quad-\infty<x<\infty, $$
where $\alpha_0=\text{const}$ and $\varphi(x)$ satisfies the following condition for some $k\geqslant1$:
$$ \int_{-\infty}^\infty|x|^k|\varphi(x)|\,dx<\infty. $$

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