Asymptotic behaviour of the number of negative eigenvalues of a differential operator with operator coefficient

ABSTRACTLet L2 (0, ∞; H ) denote the set of all functions f , defined in the interval [0, ∞ ) with values inan infinite dimensional seperable Hilbert space H , which are measurable in the sense ofBochner and satisfy the condition∞2∫ f ( x) dx < ∞ .0The set L2 (0, ∞; H ) is a linear space. The space L2 (0, ∞; H ) becomes an infinite dimensionalseperable Hilbert space by defining the inner-product for any f and g in L2 (0, ∞; H ) as∞( f , g ) ( 0,∞ ) = ∫ ( f ( x), g ( x))dx .0In this thesis, entitled with ?Asymptotic Behaviour of the Number of Negative Eigenvalues ofa Differential Operator with Operator Coefficient?, we prove that the closure of a symmetricoperator formed by the differential expression′l ( y ) = −( p( x) y ′( x) ) − Q( x) y ( x)and the boundary conditioncos α . y (0) + sin α . y ′(0) = 0where α ∈ (−∞, ∞) is a constant, is self adjoint and this self adjoint operator is semi boundedbelow and negative part of its spectrum is discrete in the space L2 (0, ∞; H ) . Morover, fornumber N (ε ) of eigenvalues smaller than −ε (ε > 0 ) we find asymptotic formulas of theformα j ( x) − ε[ ]N (ε ) = π −1 1 + O(ε t0 ) ∑ ∫ dxp ( x)j α j ( x ) ≥εα j ( x) − ε[ ]N (ε ) = π −1 1 + O(e −ε ) ∑−β∫ dxp ( x)j α j ( x ) ≥εwhen ε → 0 . α1 ( x), α 2 ( x),… ,in the above formulas, denote the eigenvalues of the operatorQ ( x ) : H → H which is completly continuous and self adjoint. t 0 and β are positiveconstants.Key Words: Hilbert space, self adjoint operator, symmetric operator, closed operator,closable operator, Bochner integral, eigenvalue, discrete spectrum.vii