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1 results

  • Minimal spectral functions of an ordinary differential operator

  • Part of
  • Vadim Mogilevskii
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 55 / Issue 3 / October 2012
    Published online by Cambridge University Press:
    30 August 2012, pp. 731-769
    • Article
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  • Let l[y] be a formally self-adjoint differential expression of an even order on the interval [0, b〉(b ≤ ∞) and let L0 be the corresponding minimal operator. By using the concept of a decomposing boundary triplet, we consider the boundary problem formed by the equation l[y] − λy = f, f ∈ L2[0, b〉, and the Nevanlinna λ-dependent boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the m-function, which in the case of self-adjoint separated boundary conditions coincides with the classical characteristic (Titchmarsh–Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e. all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices n±(L0) and non-separated boundary conditions) the known estimate of the spectral multiplicity of the (exit space) self-adjoint extension à ⊃ L0. Results are obtained for expressions l[y] with operator-valued coefficients and arbitrary (equal or unequal) deficiency indices n±(L0).