User asghar ghorbanpour - MathOverflow

What are the invariant Pseudo-differential operators on a Lie group?

2013-04-18T19:21:11Z

It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group.

On the other hand $\Psi$Do are generalization of the differential operators on general smooth manifold.

My question is that: Does any algebraic description for the $G$-invariant $\Psi$DOs on the Lie groups (or more general on homogeneous spaces $G/H$) exist ? Thanks

When is a Pseudo-differential operator trace class or in Dixmier ideal?

2013-01-25T23:46:59Z

Let's denote the set of all Pseudo-differential operators with symbol of “order” $d$ by $\Psi_d(M)$ and Sobolev space on $M$ by $H_s(M)$. It is known that

If $P\in\Psi_d(M)$ Then $P$ extends to a continuous map $P:H_{s}(M)\to H_{s-d}(M)$ for all $s$. Moreover, since the natural inclusion $H_s\to H_t$, for $s>t$ is compact, $P:H_{s}(M)\to H_{t}(M)$ is compact operator if $t< s-d$.

See for example Lemma 1.3.4, Gilkey's book Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem.

In special case, when $s=0$, $L^2(M)=H_0(M)$, $P:L^2(M)\to L^2(M)$ is continuous if $d\leq 0$. and it is compact if $d<0$.

Now my question is

when is $P:L^2(M)\to L^2(M)$ trace class? and when is it in Dixmier ideal $\mathcal{L}^{1,\infty}(L^2(M))$ or in general $\mathcal{L}^{(p,q)}(L^2(M))$?

Thanks

Eigenfunctions restricted on closed geodesics

2013-01-20T21:49:01Z

Consider the flat torus $T^2=\frac{\mathbb{R}^2}{l_1\mathbb{Z}\oplus l_2\mathbb{Z}}$. It is easy to see that the eigenvalues of the Laplacian on torus, $-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}$, are $\lambda_{m_1,m_2}=(2\pi)^2(\frac{m_1^2}{l_1^2}+\frac{m_2^2}{l_2^2})$ with the associated eigenfunction $$f_{(m_1,m_2)}(x,y)=e^{2\pi i(\frac{m_1}{l_1}x+\frac{m_2}{l_2}y)}.$$ where $m_1,m_2\in \mathbb{Z}$. Furthermore, The closed geodesics of $T^2$ parametrized by the arc length, are $$ \gamma_{(n_1,n_2)}(t)=\frac{1}{l}(n_1l_1t,n_2l_2t)$$ where $n_1,n_2\in \mathbb{Z}$ and $l=\sqrt{n_1^2l_1^2+n_2^2l_2^2}$. A simple computation shows that an eigenfunction, say $f_{(m_1,m_2)}$, restricted on a closed geodesic, $\gamma_{(n_1,n_2)}$, gives
$$f_{(m_1,m_2)}\circ \gamma_{(n_1,n_2)}(t)=e^{2\pi i(\frac{m_1n_1+m_2n_2}{l})t}$$ Which is an eigenfunction on the circle $\mathbb{R}/l\mathbb{Z}$ with the eigenvalue $\tilde{\lambda}=\left( \frac{2\pi}{l}(m_1n_1+m_2n_2)\right)^2$.

Now my question is: Is this true in the general cases? More precisely;

Let $\gamma:[0,l]\to M$ be a closed geodesics on the Riemannian manifold $(M,g)$ which is parametrized by the arc length. If $f\in C^\infty(M)$ is an eigenfunction for the Laplacian on $M$, i.e. $$\Delta(f)=\lambda f$$ Then

Question 1) Is $f\circ \gamma$ an eigenfunction on the circle $S^1=\mathbb{R}/l\mathbb{Z}$? Or, Is it in the form of $$f\circ \gamma(t)=c e^{2\pi i \tilde{\lambda}t}.$$

Question 2) If so, how does $\tilde{\lambda}$ depend on $\gamma$ and $\lambda$?

Thanks.

Comment by Asghar Ghorbanpour

2013-01-28T17:24:14Z

Thanks for the useful references. However, I still wondering if there is a bound like $l$ such that pseudo differential operator of the order $d$ is in the Dixmier ideal (not necessary measurable) when $d<l$. of course $l$ should be in $[-k,0)$ where $k=dim M$.

Comment by Asghar Ghorbanpour

2013-01-21T15:26:27Z

Thanks. You are right.