= Solution
Splitting the <free Schrodinger equation> into real and imaginary parts gives
$$
\boxed{u_t=-v_{xx},\qquad v_t=u_{xx}.}
$$
Differentiating the first relation in time and using the second yields the <Euler-Bernoulli beam equation> $u_{tt}+u_{xxxx}=0$. This is the <Schrodinger factorization of the elastic beam equation>.
Assume the initial velocity has an integrable first spatial moment, as allowed by sufficient decay, and define
$$
v_0(x)=-\int_x^\infty(s-x)u_1(s)ds,\qquad q_0(x)=u_0(x)+iv_0(x).
$$
Then $v_0''=-u_1$ and $v_0$ decays at infinity. To encode the second boundary datum, define
$$
v_b(t)=v_0(0)+\int_0^t\widetilde u_1(s)ds,\qquad g_0(t)=\widetilde u_0(t)+iv_b(t).
$$
The <Dirichlet boundary condition> for the resulting <free Schrodinger equation> is compatible at the corner, since $q_0(0)=g_0(0)$. Insert these explicit $q_0,g_0$ into the data-only complex integral in part (b), with $\widehat q_0$ and $G_0$ defined as in part (a). \b[The required displacement is the real part of that integral.] Equivalently, the uniformly convergent lifted integral in part (b) may be used with the same complex data.
The <Schrodinger factorization of the elastic beam equation> verifies every condition: $u(x,0)=u_0(x)$, $u_t(x,0)=-v_0''(x)=u_1(x)$, $u(0,t)=\widetilde u_0(t)$, and
$$
u_{xx}(0,t)=v_t(0,t)=v_b'(t)=\widetilde u_1(t).
$$
The corner requirements on $u_0''$ and $\widetilde u_0'$ ensure consistency of these derivative traces; the natural interpretation of the last printed compatibility is $\widetilde u_0'(0)=u_1(0)$. If its prime were instead imposed for every $t$, that would simply be an extra restriction on the data, and the same construction would still solve them.
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