Topics
- Introduction to the course.
- Review of Chapters 1 and 2.
Readings
- Make sure you know the content of those two chapters.
Topics
- Review of Chapters 3 and 4.
- Topology induced by a family of seminorms.
- Nets.
Readings
- Make sure you know the content of those two chapters.
- Section 5.7
Topics
- Review of Chapter 5.
- Topology induced by a family of seminorms.
- Nets.
Readings
Topics
- Duality between $L^p_{loc}$ and $L^q_c$.
Readings
Jan 27
To be rescheduled
Topics
Readings
Topics
Readings
Topics
- The $W^{k,\infty}$ spaces.
- Complete proof of the extension theorem.
- The last item of the proof of the existence of the trace map.
Readings
Topics
Readings
Topics
Readings
Homework #1
Denote by $Lip(\Omega)$ the set of continuous functions on $\Omega$ that are globally Lipschitz. Namely, if $f\in Lip(\Omega)$,
there is a $L>0$ such that $|f(x)-f(y)|\leq L\|x-y\|$
for every $x,y\in\Omega$.
- Prove that $\|f\|_{Lip}=\|f\|_{C^0}+\sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|}$ is a norm on $Lip([0,1])$ and show that, with this norm, $Lip([0,1])$ is a Banach space.
- Prove that $W^{1,\infty}([0,1])=Lip([0,1])$.
- Let $f\in L^p(\mathbb R^2_+)$. Prove that $\|f\|_{L^p(\mathbb R\times(0,\varepsilon))}\to0$ as $\varepsilon\to0^+$ without using Poincare' inequality.
- Let $\eta\in C^\infty([0,\infty))$ be such that $\eta=0$ in $[0,1]$, $\eta=1$ in $[2,\infty)$ and $\eta\in(0,1)$ in $(1,2)$.
Set $\eta_\varepsilon(x)=\eta(x/\varepsilon)$. Show that there is a constant $c>0$ such that $|\eta'_\varepsilon(x)||\leq c/\varepsilon$ for $x\in(\varepsilon,2\varepsilon)$.
- Verify using Theorem 6.2.6 that $f(x)=0$ for $\in[-1,0)$ and $f(x)=1$ for $\in(0,1]$ does not belong to $W^{1,p}([-1,1])$ for $p\in[1,\infty)$ while $g(x)=x+1/2$ for $x\in(-1/2,0)$,
$g(x)=1/2-x$ for $x\in(0,1/2)$ and $g(x)=0$ otherwise does belong to $W^{1,p}([-1,1])$.
Topics
- An experience of interaction with AI.
- Embedding theorem for Soboleve spaces.
Topics
- The dual Sobolev spaces $H^{-k}$.
- The rainbow of functional spaces.
- Resolvant of an operator.
- $\rho(A)$ is an open set.
Topics
- Relation between spectral theory and quantum physics.
- Spectrum of the multiplication operator in $L^1([0,1])$.
Topics
- Relation between spectral theory and quantum physics.
- Spectrum of the derivtive operator in $L^1([0,\infty))$.
Topics
- An unbounded bijective linear operator with an unbouded inverse.
- The adjoint operator.
Topics
- Why is it important to find the spectrum of an operator.
- The multiplication operator by a real function on $L^2(\bR)$ is self-adjoint.
Topics
- Why is it important to find the spectrum of an operator.
- The multiplication operator by a real function on $L^2(\bR)$ is self-adjoint.
Topics
- Characterization of self-adjoint operators.
Homework #2
-
Prove Theorem 2.2.36
-
Prove Theorem 2.2.50
-
Let $X$ be a Banach space, $A$ a closed operator on $X$ and let $\lambda\in\mathbb C$.
Assume that there is a sequence $x_n\in X$ with $\|x_n\|=1$ and such that $Ax_n-\lambda x_n\to0$.
Prove that $\lambda\in\sigma(A)$.
-
Let $A$ be a bounded operator on a Hilbert space $H$.
Use the fact that $\ker A^*=(Im A)^\perp$, and so that $H=\ker A^*\oplus \overline{Im A}$, to prove that
$$
\sigma_r(A) = \{\lambda\in\mathbb C : \lambda\not\in\sigma_p(A)\text{ and }\bar\lambda\in\sigma_p(A^*)\}.
$$
-
Define the left and right shifts operators on $\ell^2$ by
$$
(Sx)_n = x_{n+1},\; (Tx)_1=0,\; (Tx)_n=x_{n-1},\;n\geq1.
$$
Show that $T$ and $S$ are bounded and evaluate $\|S\|$ and $\|T\|$;
then, determine $S^*$ and $T^*$ and find $\sigma_p(S)$, $\sigma_c(S)$,
$\sigma_r(S)$, $\sigma_p(T)$, $\sigma_c(T)$, $\sigma_r(T)$.
Topics
- Self-adjoint extensions
- Diagonalizing linear operators
- Classification of linear operators on the plane
- Normal operators
- Spectral theorem for compact normal eigenvalues
- Spectral theorem for self-adjoint operators with compact resolvent
Topics
- Spectral theorem for self-adjoint operators with compact resolvent
- Functional calculus
- Fourier transform on $\mathbb R^n$
Topics
- Fourier transform on $\mathbb R^n$
- Spectral decomposition for general unbounded self-adjoint operators on a Hilbert space
Topics
- Spectral decomposition for general unbounded self-adjoint operators on a Hilbert space
- Spectral measure
- Flows and semiflows
- Solutions of the free Schrodinger equation on $\mathbb R$.
- Classical and quantum harmonic oscillator
Topics
- Using the spectral decomposition: defining nonlocal operators, evaluating norms of functions of operators, writing resolvents in integral form.
- Semigroups of linear bounded operators.
Topics
- Semigroups of linear bounded operators.
- Semiflows of non-linear PDEs.
Homework #3
-
Diagonalize the Laplacian on $L^2(\mathbb S^1)$ and use this diagonalization to write the general solution
of the Schrodinger equation
$$
\dot\psi = -i\Delta\psi,\;\;\psi(0)=\psi_0\in L^2(\mathbb S^1).
$$
Then use chatgpt to create a mp4 video of the solution for
$$
\psi_0(x) = \sum_{n\in\mathbb Z}e^{-\sigma^2(n-n_0)^2-inx_0}e^{inx}
$$
with $\sigma=0.18$, $x_0=\pi$, $n_0=10$.
Is the solution periodic in time?
-
Let $A$ be a self-adjoint operator with spectrum $\sigma(A)$.
Show that
$$
e^A = \int_{\sigma(A)}e^\lambda dE(\lambda)
$$
by discretizing the integral to transform it into a finite sum and then taking the limit
as shown in the
Spectral Decomposition section of the textbook.
-
The classical harmonic oscillator can be written as the following first order ODE on the plane $(x,p)$:
$$
\begin{pmatrix}\dot x\\\dot y\end{pmatrix}
=
\begin{pmatrix} 0&1\\ -1&0\\ \end{pmatrix}
\begin{pmatrix} x_0\\ y_0\\ \end{pmatrix}
$$
Write explicitly the flow $(x(t),y(t))=F^t(x_0,y_0)$ generated by this ODE and visualize a few orbits of this
flow on the $(x,p)$ plane.
Then, use chatgpt to visualize find the evolution of the wave function
$$
\psi_0(x) = \pi^{-1/4}\exp\left(-\frac{(x-x_0)^2}{2}+ip_0(x-\frac{x_0}{2})\right)
$$
under the harmonic oscillator $H=(-d^2/dx^2+x^2)/2$ for some numerical value of $x_0$ and $p_0$.
Now, denote by $\psi(t) = e^{-iHt}\psi_0$ the corresponding solution of the Schrodinger equation.
Either by hand or with the help of some AI, evaluate the mean values
$$
\begin{align}
x_c(t) &= \langle \psi(t), M_x\psi(t)\rangle\\
p_c(t) &= \langle \psi(t), (-i d/dx)\psi(t)\rangle
\end{align}
$$
Do you detect any similarity between the classical solutions and the evolution of these "mean values"?
-
Using the same diagonalization found in problem 1, consider now the semiflow $e^{-\Delta t}$ of the heat equation on the circle.
Recall that a function $f=\sum_{n\in\mathbb Z} \hat f_n e^{inx}$ is analytical on the circle
if and only if its Fourier coefficients $\hat f_n$ go to 0 exponentially fast.
Use this fact to show that, given any $f_0\in L^2(\mathbb S^1)$, the function $f(t)=e^{-\Delta t}f_0$ is analytical
for every $t>0$.
Finally, recall that $f\in C^\infty(\mathbb S^1)$ if and only if its Fourier coefficients $\hat f_n$ go to zero
faster than any negative power of $n$. Hence, the Fourier coefficients of any $\eta\in C^\infty(\mathbb S^1)^*$
must grow at most as some power of $n$, so that $\eta(f)$ is finite for every $f\in C^\infty(\mathbb S^1)$.
Then show that, even if $\eta\in C^\infty(\mathbb S^1)^*$, we have that the formal expression
$e^{-\Delta t}\eta$ is an analytical function (and so $-\Delta$ generates a flow even on the space of distributions).