Tagged Questions
2
votes
2answers
73 views
Composition of pseudo-differential operators
Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| ...
9
votes
1answer
211 views
Applications of Pseudodifferential Operators
I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
4
votes
1answer
89 views
differential operator on manifold
I am currently trying to understand the local expression of a (pseudo)differential operator
$$
\int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi
$$
on a manifold $M$ (compact and boundaryless, ...
1
vote
0answers
24 views
How to prove this equivalence?
Consider the general elliptic operator
$$M=\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j},$$
where $a_{ij}$ are continous functions. The function $u$ satisfies
$$|Mu|\leq A(|\nabla ...
8
votes
2answers
190 views
Are there n-th roots of differential operators?
In analogy to a Dirac operator, it seems to me that formally, the equation
$$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$
is solved by
$$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$
Is there a ...
0
votes
1answer
42 views
Infinite propagation speed for the Schrodinger operator
Question related to:
On the propagation of singularities in PDE and
Hypoellipticity and singular support.
in what sense is to interpret the sentence the schrodinger operator
has infinite propagation ...
0
votes
0answers
34 views
Hypoellipticity and singular support
There is a theorem that states that if $p(D)$ is a linear
partial differential operator with constant coefficients
and its fundamental solution $E$ is $C^{\infty}$ outside $\{0\}$ then
the operator ...
0
votes
2answers
186 views
Fundamental solution of the wave operator
What is the explicit formula for a fundamental solution of the wave operator, where the space variable is in $\mathbb{R}^n$ with $n>1$? Thanks.
The operator I'm talking about is
...
0
votes
1answer
83 views
On the propagation of singularities in PDE
This question might be a little generic, but i wanted to get some idea on the concept of propagation of singularities in PDE. Searching the internet i only found very complicated things about the ...
1
vote
0answers
38 views
Non-hypoelliptic operator
Consider the following operator $$\partial_t^2-\partial_x^2+\lambda(x,t)$$ where $\lambda$ is a $C^{\infty}$ function on $\mathbb{R}^2$ and the operators above are second partial derivatives. How can ...
1
vote
0answers
51 views
Cayley Transform (PDE)
I really need your help in solving the following problem:
Prove that a unitary operator $U$ acting on a Hilbert space $H$ is the Cayley transform of some self-adjoint operator if and only if $1$ is ...
2
votes
1answer
2k views
Transforming the Laplace operator from Polar to Cartesian coordinates
I'm trying to find the error in my logic here.
Let's say we are given the Laplace operator in polar coordinates:
$$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...
1
vote
1answer
267 views
Symbol of a (non linear) differential operator
I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear.
In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by ...
