Can a positive definite second order elliptic operator over an unbounded domain give rise to a compact semigroup? Let $A$ denote some positive definite second order elliptic operator which is defined over $L^2(\Omega)$ with domain $D(A) = H^{2m}(\Omega) \cap H^{m}_{0}(\Omega)$. Here $\Omega$ is a bounded domain in $R^m$. Assume that the coefficient functions of $A$ are nice enough, then it can be shown that the semigroup generated by $A$ is compact.
Now what if $\Omega$ is $[0, \infty)$? Is it possible that $A$ can still generate compact semigroup, if replacing the function spaces in the bounded case by some weighted function spaces? If so, are there any references please?
 A: Let $\Omega\subset \mathbb R^n$ be any domain, bounded or unbounded. If $(e^{-tA})$ is ultracontractive, i.e., $e^{-tA}$ maps $L^2(\Omega)$ into $L^\infty(\Omega)$ for $t>0$ and $V\in L^1_{\mathrm{loc}}(\Omega)$ is nonnegative and $\{x\in \Omega|V(x)\leq M\}$ has finite measure for every $M>0$, then $(e^{-t(A+V)})$ is compact in $L^2(\Omega)$. This is Theorem 1 in Simon. Schrödinger Operators with Purely Discrete Spectrum.
For example, the Dirichlet Laplacian on any domain generates an ultracontractive semigroup. On $\mathbb R^n$, this can be seen from the explicit formula for $(e^{t\Delta})$, and for general domains this follows from a comparison principle. Also, if $A$ satisfies the Gagliardo-Nirenberg-Sobolev inequality
$$
\|u\|_{p^\ast}\leq C\langle Au,u\rangle^{1/2}
$$
for some $p^\ast>2$, then $(e^{-tA})$ is ultracontractive. In particular, if $A$ is an elliptic operator in divergence form
$$
Au=\sum_{j,k}\partial_j(a_{jk}\partial_ku)
$$
with coefficients $a_{j,k}\in L^1_{\mathrm{loc}}(\Omega)$ with $a_{jk}=a_{kj}$ and $\sum_{j,k}a_{jk}\xi_j\xi_k\geq \lambda\sum_{j}\xi_j^2$ for some $\lambda>0$, then $(e^{-tA})$ is ultracontractive.
A note on the domains: The domain of such operators is best described by the form method. In the case of the Laplacian the quadratic form associated with $-\Delta+V$ is given by
\begin{align*}
D(q_{-\Delta+V})&=\left\{u\in H^1_0(\Omega):\int_\Omega V|u|^2\,dx<\infty\right\}\\
q_{-\Delta+V}(u,v)&=\int _\Omega(\nabla u\cdot\nabla v+Vuv)\,dx,\;u,v\in D(q_{-\Delta+V}).
\end{align*}
Then
$$
D(-\Delta+V)=\{u\in D(q_{-\Delta+V}):\exists v\in L^2(\Omega)\,\forall w\in D(q_{-\Delta+V})\colon q_{-\Delta+V}(u,w)=\langle v,w\rangle_2\}.
$$
In the case of general elliptic operators in divergence form as described above, one has to replace $H^1_0(\Omega)$ by the form domain of $A$ (with Dirichlet boundary conditions), everything else stays the same. I won't go into details here, suffice it to say that if the coefficient matrix is uniformly bounded, then the form domain of $A$ is simply $H^1_0(\Omega)$.
