Property of being an Abelian basis for the system of eigen- and adjoint functions of multipoint boundary value problems for differential equations with variable coefficients (Q1902839)

The author studies the existence of an Abelian basis for a system of eigenfunctions and adjoint eigenfunctions of a second order operator differential equation with mixed boundary conditions. The operator studied is \[ Lu(x)= u''(x)+A(x)u(x)=0, \] where \(A\) is a strongly positive operator in the Hilbert space \(H\) (this holds uniformly with respect to \(x\); that is, the domain \(D(A(x))\) is independent of \(x\) and is dense in \(H\)). It is assumed further that the resolvent set contains a cone \(|\arg (\lambda)|>\delta\) for some \(\delta\in (0,\pi)\) and \[ |R(\lambda, A(x))|=|(A(x)-\lambda I)^{-1}|\leq M/(1+|\lambda|), \] with \(M\) independent of \(x\). The operator is considered on the space \[ W^2_2(0,1;H(A),H)= \{u\mid A(0)u\in L_2(0,1;H),\;u''\in L_2(0,1;H)\}, \] equipped with the obvious norm \(|u|^2=|A(0)u|^2_{L_2}+|u''|^2_{L_2}\), and for a strongly positive operator \(A\), \(H(A^\alpha)\) is the Hilbert space \[ H(A^\alpha)= \{u\mid u\in D(A^\alpha),\;|u|=|A^\alpha u|_H,\;\alpha>0\}. \] The author considers \(L\) on the subspace \[ D(L)= \{u\in W^2_2(0,1;H(A),H),\;L_\nu u=0,\;\nu=1,2\}, \] where \[ L_\nu u=\gamma_\nu u^{(m_\nu)}(0)+ \delta_\nu u^{(m_\nu)}(1)+ \sum^n_{i=1} N_{\nu i}u^{(m_\nu)}(c_i)+ \sum^n_{i=1} Q_{\nu i}u(c_i)+ K_\nu u(0)+M_\nu u(1), \] \(\nu=1,2\); \(m_\nu=0,1\). Here \(\gamma_\nu\) and \(\delta_\nu\) are complex constants, \(K_\nu\), \(M_\nu\), \(N_{\nu i}\), and \(Q_{\nu i}\) are linear operators in \(H\), and \(0<c_i<1\). The definition of an Abelian summation method of order \(\gamma\) is as follows. If \(K\) is an operator with discrete spectrum in \(H\) and if \(\{f_i\}\) are eigenvectors of \(K\) which do not contain adjoint vectors corresponding to the associated eigenvalues \(\lambda_i\), then we say that \(\{f_i\}\) is a basis for Abelian summation method of order \(\gamma\) if there are \(0<m_1< m_2<\cdots< m_\ell<\dots\) such that the series \[ f(t)= \sum^\infty_{\ell=0} \sum^{m_{\ell+1}}_{i=m_\ell+1} c_ie^{-\lambda_i^\gamma t}f_i \] converges for every \(f\in H\), \(t>0\), and \(f(t)\to f\) as \(t\to 0+\). If adjoint vectors exist, the definition is modified: for example, if \(f_p,\dots,f_q\) is a basis for the root subspace, we replace the corresponding term \[ c_p e^{-\lambda_p^\gamma t}f_p+\cdots+ c_q e^{-\lambda_q^\gamma t}f_q \] by the integral \[ -1/2\pi i\int_{|\lambda- \lambda'|=\varepsilon} e^{-\lambda^\gamma t} (K-\lambda I)^{-1}f d\lambda, \] where the integration path lies in a certain angle containing all of the eigenvalues. The author gives a complicated sufficient condition (on \(A(x)\), the boundary constants and boundary operators) for the resolvent operator of \(L\) to exist in an angle \(|\arg (\lambda)|>\delta\) and satisfy the standard estimate \[ |R(\lambda,L)|\leq C|\lambda|^{-1}, \] which is used to prove the following result, where \(\sigma_p(H)\) denotes the von Neumann-Schatten class of a complete continuous operator \(A\). Theorem. If \(L\) satisfies the conditions mentioned above on \(A\), the boundary constants and boundary operators, and further \(A^{-1}(0)\in \sigma_p(H)\), \(p>0\), and \(\delta<\pi/ (1+2p)\), then the eigenfunctions and adjoint vector functions of \(L\) form a basis of Abelian summation of order \(\gamma\) in \(L_2(0,1;H)\) if \(\gamma\in ((2p+1)/2, \pi/2\delta)\). He remarks that a similar result holds for eigenvalue problems of \(Lu+Pu=pu\), where \(Pu(x)= B(x)u'(x)+ C(x)u(x)\) and the same boundary conditions are imposed.