##
**Sufficient and necessary conditions for the classification of Sturm-Liouville differential equations with complex coefficients.**
*(English)*
Zbl 1241.34031

The authors determine important new asymptotic behavior relative to the class of solutions to the spectral problem for non-self-adjoint Sturm-Liouville ordinary differential equations developed by B. M. Brown, D. K. R. McCormack, W. D. Evans and M. Plum [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1984, 1235–1257 (1999; Zbl 0944.34018)]. The equation is
\[
y= w^{-1}(-(py')'+ qy)= \lambda y,~ a\leq x< b,\tag{i}
\]
\(w> 0\), \(p\), \(q\) complex determine an unbounded operator \(T\) on \(L^2_w\) with domain \(D(T)\). Let \(\Omega\) be the smallest closed convex set in \(\mathbb C\) containing points
\[
\{p/w+ rq\}, ~r> 0,~ a\leq x<b.\tag{ii}
\]
\(\Omega\) is contained in a half plane whose support line is a rotation of angle \(\theta\) from \(\text{Re\,}\mathbb C= 0\) and touches \(\Omega\) at \(K\) on the boundary. Let \(S\) be the set of \((\theta,K)\) such that \(\text{Re}(\exp(i\theta)(\mu- K))\geq 0\), \(\mu\in\Omega\), \(K\) not an extreme point and let \(\Lambda(\theta,K)\) be the half plane \(\{\mu: \text{Re}(\exp(i\theta)(\mu- K))< 0\}\). Let \(\lambda\in\Lambda(\theta, K)\) be such that (i) has solutions \((\lambda,y)\). These are classified into three cases, I, II, III. Let \(E\) be the set of all \(\theta\) such that \((\theta,K)\in S\).

The following asymptotic results are proved when \(b=\infty\), \(0\leq x< b\):

(a) \(T\) is in case I if and only if for \(y_1,y_2\in D(T)\), \(\theta\in E\), \[ p(y_1' y- y_1 y_2')\to 0, ~x\to\infty.\tag{iii} \]

(b) If \(E\) has more than one point then \(T\) is in case I if and only if \[ py_1 y_2'\to 0,~ x\to\infty,~ y_1,y_2\in D(T).\tag{iv} \]

(c) \(T\) is in case II with respect to \((\theta_0,K_0)\in S\) if and only if \(D(T)\neq D_{\theta_0}(T)\neq 0\), \[ \overline py_1\overline y_1'+ \exp(2i\theta_0) p\overline y_1y_2'\to 0~, x\to\infty,\tag{v} \] where \(D_{\theta_0}(T)\) denotes elements of \(D(T)\) such that \[ \int^\infty_0 (\text{Re}(\exp(i\theta_0) p)|y'|^2+ \text{Re}(\exp(i\theta_0)q)|y|^2)\,dx<\infty.\tag{vi} \] The method of proof is to reduce the equation (i) to a \((2,2)\) matrix Hamiltonian system and apply the general results about such systems by the authors [J. Qi and H. Wu, Math. Nachr. 284, No. 5–6, 764–780 (2011; Zbl 1235.34084)]. Cases I and II are generalizations of the limit point case, \(p\), \(q\) real. Case III is a generalization of the limit circle case.

The following asymptotic results are proved when \(b=\infty\), \(0\leq x< b\):

(a) \(T\) is in case I if and only if for \(y_1,y_2\in D(T)\), \(\theta\in E\), \[ p(y_1' y- y_1 y_2')\to 0, ~x\to\infty.\tag{iii} \]

(b) If \(E\) has more than one point then \(T\) is in case I if and only if \[ py_1 y_2'\to 0,~ x\to\infty,~ y_1,y_2\in D(T).\tag{iv} \]

(c) \(T\) is in case II with respect to \((\theta_0,K_0)\in S\) if and only if \(D(T)\neq D_{\theta_0}(T)\neq 0\), \[ \overline py_1\overline y_1'+ \exp(2i\theta_0) p\overline y_1y_2'\to 0~, x\to\infty,\tag{v} \] where \(D_{\theta_0}(T)\) denotes elements of \(D(T)\) such that \[ \int^\infty_0 (\text{Re}(\exp(i\theta_0) p)|y'|^2+ \text{Re}(\exp(i\theta_0)q)|y|^2)\,dx<\infty.\tag{vi} \] The method of proof is to reduce the equation (i) to a \((2,2)\) matrix Hamiltonian system and apply the general results about such systems by the authors [J. Qi and H. Wu, Math. Nachr. 284, No. 5–6, 764–780 (2011; Zbl 1235.34084)]. Cases I and II are generalizations of the limit point case, \(p\), \(q\) real. Case III is a generalization of the limit circle case.

Reviewer: J. B. Butler jun. (Portland)

### MSC:

34B24 | Sturm-Liouville theory |

34C40 | Ordinary differential equations and systems on manifolds |

34B20 | Weyl theory and its generalizations for ordinary differential equations |

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\textit{B. Xie} and \textit{J. G. Qi}, Abstr. Appl. Anal. 2011, Article ID 340796, 15 p. (2011; Zbl 1241.34031)

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