arXiv subjects

Jihed Hedhly

Publications and source records attributed to Jihed Hedhly.

3 recordsLinked to original sources

arXiv2021

Eigenvalue ratios for vibrating String equations with concave densities

In this paper, we prove the optimal lower bound $\frac{λ_n}{λ_m}\geq(\frac{n}{m})^2$ of vibrating string $$-y''=λρ(x) y,$$ with Dirichlet boundary conditions for concave densities. Our aproach is based on the method of Huang [Proc. AMS., 1999]. The main argument is to restrict the two consecutive eigenfunction $y_{n-1}$ and $y_n$ between two successive zeros of $y_{n-1}$. We also prove the same result for the Dirichlet Sturm-Liouville problems.

math.SP↗

arXiv2021

Eigenvalue Ratios for vibrating string equations with single-well densities

In this paper, we prove the optimal upper bound $\frac{λ_n}{λ_m}\leq(\frac{n}{m})^2$ of vibrating string $$-y''=λρ(x) y,$$ with Dirichlet boundary conditions for single-well densities. The proof is based on the inequality $\frac{λ_n(ρ)}{λ_{m}(ρ)}\leq \frac{λ_n(L)}{λ_{m}(L)} ,$ with $L$ must be a stepfunction. We also prove the same result for the Dirichlet Sturm-Liouville problems.

math.AP↗

arXiv2018

Upper Bound For The Ratios Of Eigenvalues Of Schrodinger Operators With Nonnegative Single-Barrier Potentials

In this paper we prove the optimal upper bound $\frac{λ_{n}}{λ_{m}}\leq\frac{n^{2}}{m^{2}}$ $\Big(λ_{n}>λ_{m}\geq 11\sup\limits_{x\in[0,1]}q(x)\Big)$ for one-dimensional Schrodinger operators with a nonnegative differentiable and single-barrier potential $q(x)$, such that $\mid q'(x) \mid\leq q^{*},$ where $q^{*}=\frac{2}{15}\min\{q(0) , q(1)\}$. In particular, if $q(x)$ satisfies the additional condition $\sup\limits_{x\in[0,1]}q(x)\leq \frac{π^{2}}{11}$, then $\frac{λ_{n}}{λ_{m}}\leq \frac{n^{2}% }{m^{2}}$ for $n>m\geq 1.$ For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.

math.SP↗