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Selma Yildirim Yolcu

Publications and source records attributed to Selma Yildirim Yolcu.

4 recordsLinked to original sources
arXiv2015

Refined Eigenvalue Bounds on the Dirichlet Fractional Laplacian

The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-Δ)^{α/2}|_Ω$ restricted to a bounded domain $Ω\subset{\mathbb R}^d$ with $d=2,$ $1\leq α\leq 2$ and $d\geq 3,$ $0< α\le 2$. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result in [36]. Furthermore, we give a result improving the bounds for analogous elliptic operators in [19].

math.AP↗
arXiv2009

An Improvement to a Berezin-Li-Yau type inequality for the Klein-Gordon Operator

In this article we improve a lower bound for $\sum_{j=1}^kβ_j$ (a Berezin-Li-Yau type inequality) in [E. M. Harrell II and S. Yildirim Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, J. Funct. Analysis, 256(12) (2009) 3977-3995]. Here $β_j$ denotes the $j$th eigenvalue of the Klein Gordon Hamiltonian $H_{0,Ω}=|p|$ when restricted to a bounded set $Ω\subset {\mathbb R}^n$. $H_{0,Ω}$ can also be described as the generator of the Cauchy stochastic process with a killing condition on $\partial Ω$. (cf. [R. Banuelos, T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincare inequality, J. Funct. Anal. 211 (2) (2004) 355-423]; [R. Banuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 234 (2006) 199-225].) To do this, we adapt the proof of Melas ([ A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proceedings of the American Mathematical Society, 131(2) (2002) 631-636]), who improved the estimate for the bound of $\sum_{j=1}^kλ_j$, where $λ_j$ denotes the $j$th eigenvalue of the Dirichlet Laplacian on a bounded domain in ${\mathbb R}^d$.

math.SP↗
arXiv2008

Eigenvalue inequalities for Klein-Gordon Operators

We consider the pseudodifferential operators $H_{m,Ω}$ associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian $\sqrt{|{\bf P}|^2+m^2}$ when restricted to a compact domain $Ω$ in ${\mathbb R}^d$. When the mass $m$ is 0 the operator $H_{0,Ω}$ coincides with the generator of the Cauchy stochastic process with a killing condition on $\partial Ω$. (The operator $H_{0,Ω}$ is sometimes called the {\it fractional Laplacian} with power 1/2, cf. \cite{Gie}.) We prove several universal inequalities for the eigenvalues $0 < β_1 < β_2 \le >...$ of $H_{m,Ω}$ and their means $\overline{β_k} := \frac{1}{k} \sum_{\ell=1}^k{β_\ell}$. Among the inequalities proved are: {\overline{β_k}} \ge {\rm cst.} (\frac{k}{|Ω|})^{1/d} for an explicit, optimal "semiclassical" constant, and, for any dimension $d \ge 2$ and any $k$: β_{k+1} \le \frac{d+1}{d-1} \overline{β_k}. Furthermore, when $d \ge 2$ and $k \ge 2j$, \frac{\overlineβ_{k}}{\overlineβ_{j}} \leq \frac{d}{2^{1/d}(d-1)}(\frac{k}{j})^{\frac{1}{d}}. Finally, we present some analogous estimates allowing for an external potential energy field, i.e, $H_{m,Ω}+ V(\bf x)$, for $V(\bf x)$ in certain function classes.

math.SP↗