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On the \(L_w^2\)-solutions to general second-order nonsymmetric differential equations. (English. Russian original) Zbl 0973.34027
Sib. Math. J. 42, No. 1, 181-189 (2001); translation from Sib. Mat. Zh. 42, No. 1, 202-211 (2001).
The authors extend the results by R. J. Amos [Questions Math. 3, 53-65 (1978; Zbl 0411.34041)] and F. V. Atkinson, W. D. Evans [Math. Z. 127, 323-332 (1972; Zbl 0226.34026)] to the case in which \(M\) is a general second-order nonsymmetric operator. R. Amos proved that all solutions to the second-order differential equation \(M[y] = \lambda wy\), \(\lambda\in\mathbb C\), belong to \(L_w^2(a,\infty)\) whenever \(M\) is a second-order symmetric differential expression of the form \(M[f] = - (pf')' + qf\) on \([a,\infty)\) under fulfillment of suitable conditions on the coefficients \(p\) and \(q\). F. Atkinson and W. Evans studied the case in which there are solutions that do not belong to \(L_w^2(a,\infty)\).
The aim of the article under review is to study the \(L_w^2\)-solutions to general second-order nonsymmetric differential equations \(M[f] = \lambda wf\) and \(M^+[g] = \overline\lambda wg\), with \[ \begin{gathered} M[f] = - (p(f' - rf))' + up(f' - rf) + qf, \\ M^+[g] = - (p(g' + uf))' + rp(p' + uf) + qg. \end{gathered} \] Here, \(p\), \(r\), \(u\), \(q\) are complex-valued Lebesgue-measurable functions on the interval \([a,b)\), \(-\infty < a < b\leq \infty\), satisfying the conditions \[ p(x)\neq 0\quad\text{for a.e. } x\in [a,b),\quad \frac{1}{p}, r, u, q\in L_{\text{loc}}(a,b) . \]
34C11 Growth and boundedness of solutions to ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B24 Sturm-Liouville theory
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