Spectral Decay and Principal Eigenvalues

Consider the complex function \(f(z)=\ln ((z-1)(z-2))\). Find the branch points. Choose a branch cut and find the corresponding principal branch of the function.

You must justify the choice of angles in the dual polar coordinate representation.

Assume a compact self-adjoint operator \(T:X\to X\), where \(X\) is a separable Hilbert space, satisfies \(\langle Tx,x\rangle \ge 0\) for all \(x\in X\).

1. State the spectral theorem for compact self-adjoint operators.
2. Show that all the eigenvalues of \(T\) are non-negative.
3. Construct an operator \(S:X\to X\) such that \(T=S^2\).

Let \(A(x)\) be an \(n\times n\) matrix-valued function defined on the smooth bounded connected domain \(U\subset \mathbb {R}^n\). Assume \(A(x)=(a_{i,j}(x))\) is uniformly symmetric positive definite, and \(a_{i,j}\in C^\infty (\overline {U})\) for all \(1\le i,j\le n\). Consider the problem \[ \begin {aligned} -\operatorname {div}(ADu) &= \lambda u, && \text {in }U,\\ u &= 0, && \text {on }\partial U. \end {aligned} \]

1. Characterize the principal (smallest) eigenvalue \(\lambda _1\) in terms of the Rayleigh quotient for this problem.
2. Prove that \(\lambda _1>0\), and that the associated eigenfunction \(u_1\) is a minimizer of the Rayleigh quotient.

The Legendre polynomials are given by \[ P_n(x)=\frac {1}{2^n n!}\frac {d^n}{dx^n}(x^2-1)^n, \] and are eigenfunctions of the differential operator \(Lu=-((1-x^2)u')'\) on the interval \(-1<x<1\).

1. Find the eigenvalue \(\lambda _n\) corresponding to the eigenfunction \(P_n(x)\).
2. In what sense, if any, are the functions \(\{P_n(x)\}\) complete?
3. Prove or disprove that \(L\) is a positive definite operator.

The Chebyshev functions are given by \(T_n(x)=\cos (n\cos ^{-1}x)\), and are eigenfunctions of the differential operator \[ Lu=-\sqrt {1-x^2}\left (\sqrt {1-x^2}u'\right )' \] on the interval \(-1<x<1\).

1. Find the eigenvalue \(\lambda _n\) corresponding to the eigenfunction \(T_n(x)\).
2. In what sense, if any, are the functions \(\{T_n(x)\}\) complete?
3. Prove or disprove that \(L\) is a positive definite operator.

Consider the Sturm-Liouville problem \[ x^2(x^2u')'+\lambda u=0 \] on \(1/2\le x\le 1\) with boundary conditions \(u(1/2)=u(1)=0\).

1. State without proof the main theorem on eigenfunction expansions which applies to this example.
2. Use the change of variables \(w(t)=u(1/t)\) to transform the differential equation into \(d^2w/dt^2+\lambda w=0\). Then calculate the eigenvalues \(\lambda _n\) and eigenfunctions \(u_n\), by citing without proof a result for the Sturm-Liouville problem \(y''+\lambda y=0\), \(y(a)=0=y(b)\).
3. Sturm oscillation theory and the Prüfer transformation are used in the general theory to produce the candidate eigenvalues and eigenfunctions. Sketch briefly how this is accomplished, without proofs.

All eigenvalues of \(A\) are real.

Eigenvectors corresponding to distinct eigenvalues are orthogonal.