Search Results

How can I find Eigenvalues of the differential operator $-\frac{d^2}{dt^2} + \mathrm{sin} =\Delta + \mathrm{sin}$, defined on $M=S^1$, the 1-Sphere? It seems that the underlying ODE is pretty much un …

I hope someone can help me, although this question is rather specific. I am reading John Roe's chapter on Getzler symbols in "Elliptic operators, topology and asymptotic methods" to understand the pr …

I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following: I hav …

Differential operators are never bounded unless they are of order zero. Standard references are Berline, Getzler, Vergne, "Heat Kernels and Dirac Operators" and Gilkey, "Invariance Theory, The Heat E …

For appropriate choices of $i$, $j$ (e.g. $i+j \neq n$), $\Omega^i(M)$ and $\Omega^j(M)$ have different ranks as $C^\infty(M)$ module, so at least they cannot be isomorphic as modules. Maybe they coul …

I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter. To clarify, suppose I have a 1-parameter family $T_h$ of …

On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that $X^{\mathrm{hor}}$ is a horizon …

The linearization of the vector field $X$ at the singular point zero is $$DX|_0 = \begin{pmatrix} 1 & 1\\ -1 & 0\end{pmatrix},$$ the eigenvalues of which are $$ \lambda_{1, 2} = \frac{1}{2} \pm \frac{ …

Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm. The Laplace-Belrami-Operator $\Delta$ on $X$ with do …

Let $X$ be a vector field on a manifold $M$ that induces a complete flow $\Theta_t$. Then the operator family $\Theta_t^*$, $$\Theta_t^*u(x) = u(\Theta_t(x))$$ is a strongly continuous semigroup of op …