High frequency asymptotic analysis of a string with rapidly oscillating density (Q2709834)

The authors study the eigenfunctions \(\varphi_k^{\varepsilon}(x)\) and the eigenvalues \(\sqrt{\lambda_k^{\varepsilon}}\) of the problem NEWLINE\[NEWLINEu''+\lambda p(\tfrac{x}{\varepsilon})u=0, \qquad u(0)=0,\;u(1)=0, \tag{1}NEWLINE\]NEWLINE where \(p\in W^{N+1,\infty}(\mathbb{R})\) is a periodic function with \(0<p_m\leq p(x)\leq p_M\) and \(\varepsilon>0\) is a small parameter. The main result is an asymptotic expansion, as \(\varepsilon\) goes to zero, of the high frequencies eigenfunctions and eigenvalues i.e. the functions \(\varphi_k^{\varepsilon}(x)\) and \(\sqrt{\lambda_k^{\varepsilon}}\) so that \(k\geq C\varepsilon^{-(1+1/N)}\). All order correction formulae are provided in case the density \(p\) is \(C^{\infty}\) smooth. From such a result, some estimates for the gap \(\sqrt{\lambda_{k+1}^{\varepsilon}}-\sqrt{\lambda_k^{\varepsilon}}\) between two consecutive eigenvalues are given. Also a uniform observability of high frequency eigenfunctions NEWLINE\[NEWLINE\tfrac{1}{C} (|(\varphi_k^{\varepsilon})'(0)|^2+|(\varphi_k^{\varepsilon})'(1)|^2) \leq \int_0^1|(\varphi_k^{\varepsilon})'(x)|^2 dx \leq C(|(\varphi_k^{\varepsilon})'(0)|^2+|(\varphi_k^{\varepsilon})'(1)|^2) NEWLINE\]NEWLINE is worked out. At last, the authors consider the eigenvalue problem NEWLINE\[NEWLINE(a(\tfrac{x}{\varepsilon})u')'+\lambda u=0, \quad u(0)=0,\;u(1)=0, NEWLINE\]NEWLINE where \(a\in L^{\infty}(\mathbb{R})\) is a periodic function with \(0<a_m\leq a(x)\leq a_M\). Similar results are obtained reducing this problem to problem (1). The low frequencies can be obtained using multiple scales. To deal with the high frequencies, the authors use the WKB method.