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A computational investigation of an integro-differential inequality with periodic potential. (English) Zbl 0972.34021

Let \(M(f)\) be the Sturm-Liouville differential expression \(M(f)= -f''+q(x)f\) over \([a,b)\). The paper is devoted to the study of the inequality \[ \left(\int^\infty_0 \bigl(|f'|^2 +q|f|\bigr) dx\right)^2\leq K\int_0^\infty |f|dx\int^\infty_0 M[f]dx, \] where \(K\) is some constant, \(f\) is a locally absolutely continuous function on \([a,b)\) and is Lebesgue square integrable; \(q(x)\) is (i) \(\pm\sin x\), (ii) \(\pm\cos x\) and (iii) \(q(x)=-1(x\) on \([0,\Pi])\) and \(+1\) \((x\) on \([\Pi,2\Pi])\), and then extended periodically one \([0,\infty)\). For these cases the spectrum consists of bands which may have eigenvalues in the gaps. The authors provide strong numerical evidence that the inequality is valid when zero lies in one of the spectral band also when zero is a Neumann or Dirichlet eigenvalue (the problem is translated by a value equal to the eigenvalue). Finally, the authors compute the position of these bands using the Rayleigh-Ritz method and examine various translates to show how the values of best constant vary.

MSC:

34B24 Sturm-Liouville theory
34A45 Theoretical approximation of solutions to ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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