In this paper, we derive an asymptotic approximation to the eigenvalues of the linear differential equation $$ -y"(x)+q(x)y(x)=λy(x), x\in (a,b) $$ with boundary conditions of general form, when $q$ is a measurable function which has a singularity in $(a,b)$ and which is integrable on subsets of $(a,b)$ which exclude the singularity.
A conjecture from a paper by J. Bierkens and A.C.M. Ran concerning the location of eigenvalues of rank one perturbations of singular M-matrices is shown to be false in dimension four and higher, but true for dimension two, as well as for dimension three with an additional condition on the perturbation.
We provide an answer to a question raised by Levine and Weinberger in their $1986$ paper concerning the difference between Dirichlet and Neumann eigenvalues of the Laplacian on bounded domains in $\mathbb{R}^{n}$. More precisely, we show that for a certain class of domains there exists a sequence $p(k)$ such that $λ_{k}\geq μ_{k+ p(k)}$ for sufficiently large $k$. This sequence, which is given explicitly and is independent of the domain, grows with $k^{1-1/n}$ as $k$ goes to infinity, which we conjecture to be optimal. We also prove the existence of a sequence, now not given explicitly and onl…
Eigenvalue asymptotics of Müller minimizers for atoms and molecules
Ludwig Maximilian University Munich（慕尼黑路德维希-马克西米利安大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；Caltech（加州理工学院）；Zhejiang University（浙江大学）
This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic …
For fixed $W\in \big(0,\frac{1}{2}\big)$ and positive integer $N\geq 1,$ the discrete prolate spheroidal wave functions (DPSWFs), denoted by $U_{k,W}^N,$ $0\leq k\leq N-1$ form the set of the eigenfunctions of the positive and finite rank integral operator $\widetilde Q_{N,W},$ defined on $L^2(-1/2,1/2),$ with kernel $K_N(x,y)=\frac{\sin(Nπ(x-y))}{\sin(π(x-y))}\, \mathbf 1_{[-W,W]}(y).$ It is well known that the DPSWF's have a wide range of classical as well as recent signal processing applications. These applications rely heavily on the properties of the DPSWFs as well as the behaviour of the…
