Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 3

Letbe the harmonic-oscillator energy space, and denote the minimized quadratic functional by $J(u)$. A minimizing sequence $(u_n)\subset \mathcal{A}(M)$ is bounded in $\Sigma$, because $J$ is its squared Hilbert norm with positive coefficients. After taking a subsequence, $u_n\rightharpoonup u$ weakly in $\Sigma$.

The embedding $\Sigma \hookrightarrow L^4({\mathbb{R}}^2)$ is compact. On any fixed ball this follows from the Rellich-Kondrachov compactness theorem. Outside a large ball, the moment bound makes the $L^2$ tail uniformly small, and interpolation with the uniform $H^1\hookrightarrow L^q$ bounds for some $q>4$ makes the $L^4$ tail uniformly small. Hence $u_n\to u$ strongly in $L^4$.

The constraint passes to the limit, so $\int |u{|}^4=M$. Weak lower semicontinuity gives $J(u)\le \mathrm{liminf}J(u_n)=I(M)$. Therefore $u$ attains the infimum. This is Fixed-L4 minimization in the harmonic-oscillator energy space.

Replacing a minimizer by its absolute value does not increase its gradient norm, so choose a nonnegative minimizer $P$. The Euler-Lagrange equation isfor a Lagrange multiplier $\mu$. Multiplication by $P$ and integration show thatso $\mu >0$. Set $P_{\eta }=\sqrt{\mu }P$. Then $P_{\eta }$ is nonzero, belongs to $H^1({\mathbb{R}}^2)$, and satisfies the trapped focusing cubic ground-state equationStandard elliptic regularity and the strong minimum principle for elliptic operators make the nonnegative solution positive.

From $b_s+b^2=-\eta$ one obtainsSince ${\lambda }_s=-b\lambda$, the ratio $\sqrt{b^2+\eta }/\lambda$ is constant. Its value at $t=-1$ is one, soUsing $s_t={\lambda }^{-2}$ and ${\lambda }_s=-b\lambda$ givesThe initial conditions now giveFinally,These are the explicit lens parameters.

Substitute the Focusing Schrödinger lens ansatz intoThe imaginary terms proportional to $P_{\eta }+y\cdot \nabla P_{\eta }$ cancel because $b=-\lambda {\lambda }_t$. After multiplying the remaining real equation by ${\lambda }^3$, one obtainsThe dynamical system gives ${\lambda }^2{b}_t+b^2=-\eta$. Comparison with the equation for $P_{\eta }$ therefore requiresThus, up to an arbitrary constant phase,

For $T_2>T_1\ge -1$, the dual Strichartz estimate for the free Schrödinger equation givesThe assumed finite global $L_{t,x}^4$ norm makes the right side tend to zero as $T_1,T_2\to \infty$. The displayed family is therefore a Cauchy sequence in the complete space $L^2$ and has a strong limit $F_{\infty }$.

The Duhamel principle givesDefineSince the free Schrödinger group is unitary,This is scattering from a finite Strichartz norm.