Euler differential operator ($D=xd/dx$)

The Euler differential operator differentiates in logarithmic position: if $r=\log x$ and $V(r)=v(e^r)$, then $Dv=V^{\prime }$. In particular $D(x^n)=n{x}^n$, so a polynomial $p(D)$ acts on the power law $x^n$ by multiplication by $p(n)$. This converts a Cauchy-Euler equation into a constant-coefficient linear differential equation. Repeated roots give additional logarithmic factors, as ordinary constant-coefficient equations give polynomial factors multiplying exponentials.

For real $\beta$, the displayed boundary value problem with $v(\xi )\to 0$ as $\xi \to \infty$ has a solution exactly when $\beta >0$. Put $a={\beta }^{1/3}>0$. Its unique real solution isIndeed the Euler differential operator has characteristic roots $0,a,-a/2\pm i\sqrt{3}a/2$. Decay retains only the conjugate pair, and the two endpoint conditions determine their sine combination. For negative $\beta$ only one root has negative real part, so the two boundary conditions are incompatible. At zero every solution is a cubic polynomial in $\log \xi$, excluding nonzero decaying solutions. Complex exponents thus produce an oscillatory but decaying power-law envelope.