The spectral theorem for compact self-adjoint operators used here concerns a compact self-adjoint operator: its nonzero eigenvalues are real, have finite multiplicity and can accumulate only at zero; the orthogonal complement of its kernel is the closed span of its orthonormal eigenvectors. Self-adjointness is essential for this eigenbasis statement.

Let $Kf$ be the unique zero-parameter weak solution. Uniqueness gives linearity. Testing with $Kf$ and using coercivity gives $\Vert Kf{\Vert }_{H^1}\le {\nu }^{-1}\Vert f{\Vert }_2$. The Rellich-Kondrachov theorem makes the embedding $H_0^1(\Omega )\hookrightarrow L^2(\Omega )$ compact on this bounded $C^1$ domain, so $K$ is compact on $L^2$. Symmetry givesIt is self-adjoint and positive. It is injective: if $Kf=0$, the weak equation pairs $f$ to zero with all $H_0^1$ tests, dense in $L^2$, so $f=0$. Moreover $⟨Kf,f⟩=B[Kf,Kf]>0$ for $f\ne 0$.

The parameter equation is equivalent to $u=K(f+\lambda u)$, that isA solution of this operator equation in $L^2$ lies in $H_0^1$, because its right-side representation is $K(f+\lambda u)$, and the equivalence follows in both directions. For $\lambda \le 0$, the form $B[u,v]-\lambda ⟨u,v⟩$ remains coercive; the preceding minimization argument gives a unique weak solution.

The spectral theorem now supplies a complete orthonormal basis $({\varphi }_j)$ of $L^2$ with $K{\varphi }_j={\kappa }_j{\varphi }_j$, ${\kappa }_j>0$, and ${\kappa }_j\to 0$. The elliptic eigenvalues are ${\lambda }_j=1/{\kappa }_j>0$, with finite multiplicity and tending to infinity, and $L{\varphi }_j={\lambda }_j{\varphi }_j$ weakly. If $\lambda$ is not one of them, thenThis converges even in $H_0^1$: its squared energy norm is ${\sum }_j{\lambda }_j|⟨f,{\varphi }_j⟩{|}^2/|{\lambda }_j-\lambda {|}^2$, finite because the multiplier is bounded over $j$. Partial sums solve the weak equation with the corresponding $L^2$ projections of $f$. Passing to the limit proves the equation for every test; the same coefficients force uniqueness. Thus the expansion represents the unique weak solution away from the elliptic eigenvalues.

Bounded coefficients and Cauchy-Schwarz give $|B[u,v]|\le C\Vert u{\Vert }_{H^1}\Vert v{\Vert }_{H^1}$. Uniform ellipticity gives $B[u,u]\ge \theta \Vert Du{\Vert }_2^2$ because $c\ge 0$. Poincaré inequality $\Vert u{\Vert }_2\le C_P\Vert Du{\Vert }_2$ holds on $H_0^1$ of a bounded domain, soA weak solution is $u\in H_0^1$ satisfying $B[u,\varphi ]=\int f\varphi$ for every $\varphi \in H_0^1$. The energy is finite by boundedness of $B$ and the forcing functional, andIt is bounded below and a minimizing sequence is bounded in the Hilbert space $H_0^1$. Its closed linear span is separable, so part 1 supplies a weakly convergent subsequence with limit $u$. Symmetry and nonnegativity of $B$ showThe forcing term converges by weak convergence. Thus $I(u)\le \mathrm{liminf}I(v_j)$, proving attainment. First variation in every zero-trace direction gives the weak equation. If $u,w$ are two solutions, test their difference equation with $u-w$ and use coercivity to get $u=w$. Hence there is exactly one weak solution, and it is the unique energy minimizer. Indeed $I(v)-I(u)=\frac{1}{2}B[v-u,v-u]$ directly verifies the minimizing property.

The first weak derivative $D_{j}u=v$ is characterized by ${\int }_{\Omega }u{\partial }_j\varphi =-{\int }_{\Omega }v\varphi$ for every $\varphi \in C_c^{\infty }(\Omega )$. The fundamental lemma of the calculus of variations in the locally integrable form says that a function pairing to zero with every such test is zero almost everywhere. Applied to the difference of two candidate derivatives, it proves uniqueness. The Sobolev space consists of $u\in L^p$ with all first weak derivatives in $L^p$, with normFor the one-dimensional tent, integration by parts on each side of one has cancelling junction terms because the function is continuous. Thus $u^{\prime }=1$ on $(0,1)$ and $u^{\prime }=-1$ on $(1,2)$, almost everywhere. Both it and its derivative are bounded, so it belongs to every stated Sobolev space; its displayed norm to the power $p$ is $2/(p+1)+2$.

For $u=x_1/r$ in the punctured ball, ordinary differentiation givesThe function is bounded and $|D_{j}u|\le 2/r$, so polar integration gives finite derivative $L^p$ norms when ${\int }_0^1{r}^{n-1-p}dr<\infty$, namely $p<n$. To verify these are weak derivatives across the origin, integrate by parts outside a ball of radius $\epsilon$ against a compactly supported smooth test. The inner boundary term is at most a fixed constant times ${\epsilon }^{n-1}$, since $u$ is bounded. It tends to zero for $n>1$. The volume terms converge by local integrability of the displayed derivative. Thus $u\in W^{1,p}(B)$ for $1\le p<n$ with exactly these derivatives. Its value at the origin can be chosen arbitrarily.

Weak convergence means $\ell (u_j)\to \ell (u)$ for every bounded linear functional $\ell \in X^{\ast }$. Norm convergence implies this by $|\ell (u_j-u)|\le \Vert \ell \Vert \Vert u_j-u\Vert$. The converse fails for coordinate unit vectors in ${\ell }^2$: their norms are one, but pairing with any square-summable sequence tends to zero, so they are weakly null.

For the subsequence assertion apply Gram-Schmidt process to a countable dense subset to choose a countable orthonormal basis $(e_m)$ of the separable Hilbert space and a bound $M$ for the sequence. Repeated subsequence selection and diagonalization make every coefficient $⟨u_{j_k},e_m⟩$ converge to a scalar $a_m$. Bessel inequality gives ${\sum }_{m=1}^N|a_m{|}^2\le M^2$ for every $N$, so $u={\sum }_m{a}_m{e}_m$ exists in $H$ with norm at most $M$. Pairings with finite linear combinations of the basis converge. Approximating any other test vector by such combinations and using the common norm bound proves $u_{j_k}\rightharpoonup u$. The finite-dimensional case uses the same argument with a finite basis.

Apply this to every subsequence of the bounded $L^2$ sequence. Any weakly convergent further subsequence has limit $v$ with ${\int }_a^{b}v=0$, since interval indicators are $L^2$ tests. The permitted fact gives $v=0$ almost everywhere. If the whole sequence failed to converge weakly to zero, some fixed test functional and subsequence would stay a positive distance from zero. Its weakly convergent further subsequence would then contradict the preceding conclusion. Thus the entire sequence converges weakly to zero in $L^2(0,1)$.

For the Poisson distribution, summing the exponential series givesThe series is absolutely convergent for every real or complex $s$; its derivative at one is $\lambda$, the expected offspring count.

Inside its radius of convergence a power series can be differentiated term by term. HenceEvery mass is therefore recovered uniquely from the function, proving that the generating function determines the entire distribution. No derivatives at the potentially delicate endpoint one are needed.

If $|s|\le 1$, then ${\sum }_k|p_k{s}^k|\le {\sum }_k{p}_k=1$. Thus the power series converges absolutely at every such real $s$, including both endpoints. ThereforeThe radius of convergence is at least one, though it can be larger for particular distributions.

The probability generating function iswhere the series is defined, with the usual power-series convention $s^0=1$ even at $s=0$. In particular $G_X(0)=p_0$ and $G_X(1)=1$.

The Gaussian integral gives ${\int }_{{\mathbb{R}}^3}{e}^{-\gamma |y{|}^2}{d}^{3}y=(\pi /\gamma {)}^{3/2}$. Thus the normalized ground state is, up to an overall phase,Its probability density is even in each shifted coordinate, so all three expectations $⟨y_i⟩$ vanish by oddness. ThereforeThe mean position is the minimum of the displaced potential, rather than the original coordinate origin.

Put $\gamma =m\omega /\hslash$. Substitution of $\psi =F(y)e^{-\gamma y^2/2}$ in the one-dimensional Schrödinger equation with energy $(n+1/2)\hslash \omega$ givesFor degrees zero, one and two, coefficient comparison yields, up to independent nonzero constants,For the shifted coordinates $y_i=x_i-(r_0{)}_i$, let $G(y)=e^{-\gamma |y{|}^2/2}$. A basis of wave functions for the first three energy levels is:

These are products of the one-dimensional polynomial solutions. Arbitrary linear combinations within each listed eigenspace are also stationary states. Their counts $1,3,6$ agree with part (a), so the degeneracies have all been included.