Spectral theory of two-point differential operators determined by \(-D^ 2\). I: Spectral properties (Q919506)

The authors develop the tools for the complete resolution of the spectral theory for differential operators L in \(L^ 2(0,1)\) of the form: \({\mathcal D}(L)=\{u\in H^ 2(0,1):\) \(B_ iu=0\), \(i=1,2\}\), \(LU=-u''\), where \(H^ 2(0,1)\) denotes the usual Sobolev space and \(B_ 1\) and \(B_ 2\) are linearly independent boundary operators of the form \[ B_ 1(u)=a_ 1u'(0)+b_ 1u'(1)+a_ 0u(0)+b_ 0u(1) \] \[ B_ 2(u)=c_ 1u'(0)+d_ 1u'(1)+c_ 0u(0)+d_ 0u(1). \] The work is based on six numerical parameters, defined in terms of the coefficients of \(B_ 1\) and \(B_ 2\), which intervene in the development of the characteristic determinant of L, in the Green's function and in other spectral quantities. At the end of the paper, a complete classification scheme of spectral properties of L is given. Proofs will appear in the second part.