Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEigenvalue gap theorems for a class of non symmetric elliptic operators on convex domains
177
0
0.0
(
0
)
Authors
Jon Wolfson
Ask ChatGPT about the research
No Arabic abstract
Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.
rate research
Read More
In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter $D$ of sphere $mathbb S^n$ is $geq 3 frac{pi^2}{D^2}$ when $n geq 3$. We prove the same result when $n=2$. In fact our proof works for all dimension. We also give an asymptotic expansion of the first and second Dirichlet eigenvalues of the model in [SWW16].
The lowest eigenvalue of the Schrodinger operator $-Delta+mathcal{V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.
In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely when $D$, the diameter of a convex domain in the unit $S^n$ sphere, is $le frac{pi}{2}$, the gap is greater than the gap of the corresponding $1$-dim sphere model. We also prove the gap is $ge 3frac{pi^2}{D^2}$ when $n ge 3$, giving a sharp bound. As in Andrews-Clutterbucks proof of the fundamental gap, the key is to prove a super log-concavity of the first eigenfunction.
We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.
Let $mathcal{A}_0$ and $mathcal{A}_1$ be two self-adjoint Fredholm Dirac-type operators defined on two non-compact manifolds. If they coincide at infinity so that the relative heat operator is trace-class, one can define their relative eta function as in the compact case. The regular value of this function at the zero point, which we call the relative eta invariant of $mathcal{A}_0$ and $mathcal{A}_1$, is a generalization of the eta invariant to non-compact situation. We study its variation formula and gluing law. In particular, under certain conditions, we show that this relative eta invariant coincides with the relative eta invariant that we previously defined using APS index of strongly Callias-type operators.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
