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The Dirichlet eigenvalues $${\{\lambda_{n}\}_{n=1}^{\infty}}$${λn}n=1∞ and Neumann eigenvalues $${\{\mu_{n}\}_{n=1}^{\infty}}$${μn}n=1∞ of the string equation $${\varphi'' (x) +\lambda \rho (x) \varphi(x) =0}$$φ′′(x)+λρ(x)φ(x)=0 are considered. It is known that $${ \mu_{n} < \lambda_{n} <… Click to show full abstract
The Dirichlet eigenvalues $${\{\lambda_{n}\}_{n=1}^{\infty}}$${λn}n=1∞ and Neumann eigenvalues $${\{\mu_{n}\}_{n=1}^{\infty}}$${μn}n=1∞ of the string equation $${\varphi'' (x) +\lambda \rho (x) \varphi(x) =0}$$φ′′(x)+λρ(x)φ(x)=0 are considered. It is known that $${ \mu_{n} < \lambda_{n} < \mu_{n+2}}$$μn<λn<μn+2 for all n. The purpose of this paper is to provide conditions on the mass density $${\rho(x)}$$ρ(x) under which $${\lambda_{n} < \mu_{n+1}}$$λn<μn+1 or $${\mu_{n+1} < \lambda_{n}.}$$μn+1<λn.
Journal Title: Acta Mathematica Hungarica
Year Published: 2018
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