## Abstract

We consider a differential equation with parameter λ, (equation presented) where λ ∈ G = {λ ∈ ℂ: |λ| ≥ 1}, q_{k}(x) ∈ L_{1}(- 1, 1), and k = 0, 2n - 2. Let T = {y_{λ}(x)}_{λ ∈ G} be some set of solutions of (1). We show that the inequalities (equation presented) (2) can hold only if λ ∈ H. Here and below we denote by c(q) different positive constants, depending only on L_{1} norms ∥q_{k}∥L_{1}(-1, 1), k = 0, 2n - 2, of coefficients of (1), (equation presented) In addition, if (2) holds, then for any λ ∈ H, |λ| ≫ c(q), there is an integer s such that λ = |sπ|^{2n}[1 + O(1/|sπ|)]. Here the constant in O(·) depends only on L_{1} norms of coefficients of (1), and the exponent of (1 + |Rc λ|) in the definition of H cannot be lessened. As an example, we study the problem of localization of the spectrum of a class of semiregular boundary value problems.

Original language | English |
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Pages (from-to) | 363-375 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 229 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jan 1999 |

### Bibliographical note

Funding Information:* Supported by the Israel Academy of Sciences under Grant 505r95.