PhD Preliminary Qualifying Examination Applied Mathematics August 17, 2015

Assume a bounded, self-adjoint operator \(T:H\to H\), where \(H\) is a Hilbert space, satisfies \[ \langle Tx,x\rangle \ge \beta \lVert x\rVert ^2, \tag {1} \] for some \(\beta >0\). The goal of this problem is to show that \(T\) is one-to-one and onto.

1. Show that the nullspace of \(T\) is trivial, i.e. \(\mathcal {N}(T)=\{0\}\).
2. Show that the range \(\mathcal {R}(T)\) is closed.
3. Show that \(\mathcal {R}(T)^\perp =\{0\}\).
4. Why do we have \(\mathcal {R}(T)=H\)?

Let \(X\) be a Banach space, \(f:X\to \mathbb {R}\) be a bounded linear functional, \(y\in X\) fixed and \(\alpha \in \mathbb {R}\).

1. Prove that there exists a constant \(C>0\) such that if \(\lvert \alpha \rvert <C\), then the non-linear equation \[ x+\alpha f(x)x=y, \tag {2} \] has a unique solution \(x\) in the ball \(B=\{x\in X\mid \lVert x-y\rVert \le 1\}\).
2. Suggest an iterative procedure for approximating the unique solution \(x\) to equation (2).

Let \((e_n)\) be an orthonormal sequence of a Hilbert space \(H\). Show that \(e_n\to 0\) weakly.

Let \((\lambda _j)\) be a sequence real numbers with \(\lambda _j\ne 1\) for all \(j\) and \(\lambda _j\to 1\). Consider the operator \(T:\ell ^2\to \ell ^2\) defined for \((\xi _j)\in \ell ^2\) by \[ T(\xi _j)=(\lambda _j\xi _j). \]

1. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent set \(\rho (T)\).
2. Give a condition on the \(\lambda _j\) for \(T\) to be invertible.

Consider the distribution \(u\in \mathcal {D}'(\mathbb {R})\) defined for \(\phi \in \mathcal {D}(\mathbb {R})\) by \[ \langle u,\phi \rangle =\int _{-\infty }^{\infty }\lvert x\rvert \phi (x)\,dx. \] Find its derivative \(\partial u\).

The following functions \(f(x)\) [\(x\) is a real variable] can be represented by series (Taylor or Laurent) in powers of \((x-2)\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{x^2},\\[0.6ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x-3}{\sin x-2},\\[1.2ex] \text {(c)}\quad f(x)&=&\dfrac {e^x}{(x-2)(1+e^x)}. \end {array} \]

Integrate \[ \int _{-\infty }^{\infty }\frac {\sin \alpha x}{\sinh \pi x}\,dx. \]

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{(x,y)\in \mathbb {R}^2:\ x^2+y^2>1,\ x^2+(y+1)^2<4\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }x^2+y^2=1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }x^2+(y+1)^2=4 \] (\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in terms of the original real variables \(x\) ans \(y\)).

1. Calculate the Fourier transform of a Gaussian \(f(x)=e^{-x^2}\) (\(x\in \mathbb {R}\)).
2. Is it possible that a function \(f(x)\) (\(x\in \mathbb {R}\)) and its Fourier transform \(F(\mu )\) (\(\mu \in \mathbb {R}\)) both have finite support?

Find the three-term asymtotic expansion of the integral \[ I(s)=\int _0^1 \ln (t)\,e^{ist}\,dt \] with large real parameter \(s\to +\infty \).