Non-self-adjoint fourth-order dissipative operators and the completeness of their eigenfunctions (Q2831910)

In this paper, the authors consider the following dissipative boundary value problem NEWLINE\[NEWLINEy^{(4)}+q(x)y=\lambda y,\quad x\in[a,b),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\gamma_1y(a)+\gamma_2y'(a)-y'''(a)=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\gamma_2y(a)+\gamma_3y'(a)+y''(a)=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\gamma_5[y,z_1]_b+\gamma_4[y,z_2]_b-[y,z_4]_b=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\gamma_4[y,z_1]_b+\gamma_6[y,z_2]_b+[y,z_3]_b=0,NEWLINE\]NEWLINE where \(a\) is the regular point and \(b\) is the singular point for the differential equation which is in limit-circle case at \(b\), \(\gamma_1,\gamma_2,\gamma_3,\gamma_4\) are real numbers with \(\gamma_1\gamma_3-\gamma_2^2>0\), \(\gamma_5,\gamma_6\) are complex numbers with \(\mathrm{Im}(\gamma_5+\gamma_6)\geq 0\) and \(\mathrm{Im }\gamma_5\mathrm{Im }\gamma_6\geq |((\gamma_2)/(\gamma_3))\gamma_5-((\gamma_2)/(\gamma_1))\gamma_6|^2\). Here NEWLINE\[NEWLINE[f,g]=-fg'''+f'g''-f''g'+f'''gNEWLINE\]NEWLINE and \(z_1,z_2,z_3,z_4\) are the linearly independent solutions of the differential equation when \(\lambda=0\). The authors construct a linear differential operator and show that the operator is dissipative in the Hilbert space. To prove the completeness of the root functions of that operator they pass to the inverse operator. Using Livsic's theorem, they complete the proof.