Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 10

Expand the two operators before comparing coefficients:Equality $L=w\tilde{L}$ requires $a=\tilde{a}$, $c=\tilde{c}$, and $(aw{)}^{\prime }=w(a^{\prime }+\tilde{b})$. Thus $a{w}^{\prime }=\tilde{b}w$, giving the positive integrating factorWith $p=aw>0$, the result is the Sturm-Liouville form $L=-(pd/dx{)}^{\prime }-wc$. The arbitrary positive multiplier $C$ rescales both sides of the eigenvalue problem and does not change its eigenvalues. The coefficient ${\tilde{a}}^{\prime }$ is already present in the divergence term of $\tilde{L}$; it must not be counted a second time in the exponential.

Fix one self-adjoint choice of separated homogeneous Dirichlet boundary conditions or Neumann boundary conditions at each endpoint. The stated product condition then makes the boundary term $[awy{y}^{\prime }{]}_{\alpha }^{\beta }$ vanish. Multiplication of $Ly=\lambda wy$ by $y$ and integration by parts giveHence define the quadratic functionalsand $\Lambda [y]=F[y]/G[y]$ for nonzero admissible $y$. The first expression for $F$ is linear in the operator $L$, and $G$ is linear in the weight $w$.

To prove the minimum statement, expand an admissible function in a complete weighted orthonormal system of eigenfunctions of the Sturm-Liouville problem, $y={\sum }_n{c}_n{y}_n$, with ordered eigenvalues ${\lambda }_0\le {\lambda }_1\le \cdots$. Self-adjointness and orthogonality of Sturm-Liouville eigenfunctions giveEquality is attained by a lowest eigenfunction, soThis is the Rayleigh-Ritz variational principle. The endpoint product condition is shorthand for fixed choices of boundary type: allowing each trial function to choose different endpoint types would not define one fixed self-adjoint eigenvalue problem.

Move the differential terms to the left in the sign convention of part (a). Then $\tilde{a}=1+x$, $\tilde{c}=0$, and $\tilde{b}=(3+2x)-{\tilde{a}}^{\prime }=2(1+x)$. The integrating factor is therefore $w=e^{2x}$, up to a positive constant. The Sturm-Liouville problem isThe trial $y=x{e}^{-x}$ is admissible because $y(0)=0$ and $y^{\prime }=e^{-x}(1-x)$ vanishes at one. Its Rayleigh quotient has numeratorand denominatorConsequently the Rayleigh-Ritz variational principle givesThe numerator is positive for every nonzero admissible function, so the lowest eigenvalue is positive. The trial is not an exact eigenfunction: substituting it would require $x^2+2x-1=\lambda x$ for every $x$. Thus the upper inequality is actually strict. This example underlies the exponential-polynomial trial space for a mixed Sturm-Liouville problem.

For the added trial term $q_n=x^n{e}^{-Ax}$, the left Dirichlet boundary condition is automatic for a positive integer $n$. At the right endpoint,To preserve $y^{\prime }(1)=0$ for an arbitrary variational coefficient $C$, choose $\boxed{A=n}$. Then minimize the Rayleigh quotient of $x{e}^{-x}+C{x}^n{e}^{-nx}$ over $C$.

Since $C=0$ remains available, this optimization cannot worsen the original upper bound. For $n=1$, the added function is a multiple of the first trial, so the quotient does not change: an independent choice $n\ge 2$ is needed for extra freedom. More generally, several such functions form the exponential-polynomial trial space for a mixed Sturm-Liouville problem, and the Rayleigh-Ritz method reduces minimization to a small generalized eigenvalue problem.

The added sine vanishes at zero. Its derivative at one is $B\cos B$, so nontrivial admissible choices areNegative values give the same trial directions after changing the sign of $C$; $B=0$ adds nothing. For each choice, minimize the Rayleigh quotient of $x{e}^{-x}+C\sin (Bx)$ over $C$, or optimize over several coefficients simultaneously. The original trial is included, so the best bound is no larger than $5/4$. Enlarging the trial subspace successively is the Rayleigh-Ritz method; the boundary-compatible half-integer sines provide additional independent shapes.