Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 1

Choose a test function $\chi$ that equals one on the unit ball and vanishes outside the ball of radius two. Sinceintegration by parts givesThe first term on the right is supported where $1\le |x|\le 2$, where $u^2\le |x{|}^{\alpha }{u}^2$. The Cauchy-Schwarz inequality and Young inequality bound the second term byAfter absorbing the first integral,

This estimate also proves completeness. Indeed, a Cauchy sequence in $\Sigma$ is Cauchy in $H^1$ locally, while its gradients and the functions $|x{|}^{\alpha /2}{u}_n$ converge in $L^2$. The limits agree locally with a function $u$, so $u\in \Sigma$ and $u_n\to u$ in the energy norm. Thus $\Sigma$ is a Hilbert space; it is the confining-potential energy space for $V(x)=|x{|}^{\alpha }$.

On the unit ball, part 1 controls the $L^2$ norm. Outside it, $|x{|}^{\alpha }\ge 1$, soHence $\Vert u{\Vert }_2\le C\Vert u{\Vert }_{\Sigma }$, proving that the embedding $\Sigma \hookrightarrow L^2$ is continuous.

For compactness, let $(u_n)$ be bounded in $\Sigma$. On each ball, it is bounded in $H^1$, so the Rellich-Kondrachov compactness theorem gives a subsequence convergent in local $L^2$. The tail estimateis uniform in $n$ and tends to zero as $R\to \infty$. A diagonal argument therefore gives convergence in all of $L^2$. This is the compact embedding of a confining-potential energy space.

For fixed $f\in L^2$, the functionalis bounded on $\Sigma$ by the continuous embedding from part 2:The Riesz representation theorem gives a unique $T(f)\in \Sigma$ such that $⟨v,T(f){⟩}_{\Sigma }={\ell }_f(v)$ for every $v\in \Sigma$. It also givesso the linear operator $T:L^2\to \Sigma$ is bounded.

As a map into $L^2$, $T$ factors asThe first arrow is bounded by part 3 and the second is the compact embedding from part 2; hence $T:L^2\to L^2$ is a compact operator.

Writing $u=T(f)$ in the defining identity givesIn particular this holds for every test function $v$, so the definition of a distributional derivative yields

Minimize $\Vert u{\Vert }_{\Sigma }^2$ subject to $\Vert u{\Vert }_2=1$. A minimizing sequence is bounded in $\Sigma$, and part 2 supplies a subsequence converging strongly in $L^2$ and weakly in $\Sigma$. The constraint survives the strong convergence, while weak lower semicontinuity of the norm shows that the limit $\psi$ attains the minimum. Since $\Vert |\psi |{\Vert }_2=\Vert \psi {\Vert }_2$ and $|\nabla |\psi ||\le |\nabla \psi |$ almost everywhere, we may take $\psi \ge 0$.

The Lagrange multiplier equation iswhere testing with $v=\psi$ shows that $\lambda =\Vert \psi {\Vert }_{\Sigma }^2>0$. ThusThis is the ground-state eigenfunction of a confining Schrödinger operator.