## Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions

##### Date
2009-09-14T08:40:01Z
Zinsou, Bertin
##### Abstract
The eigenvalue problem y(4)(¸; x) ¡ (gy0)0(¸; x) = ¸2y(¸; x) with boundary conditions y(¸; 0) = 0; y00(¸; 0) = 0; y(¸; a) = 0; y00(¸; a) + i®¸y0(¸; a) = 0; where g 2 C1[0; a] is a real valued function and ® > 0, has an operator pencil L(¸) = ¸2 ¡ i®¸K ¡ A realization with self-adjoint operators A, M and K. It was shown that the spectrum for the above boundary eigenvalue problem is located in the upper-half plane and on the imaginary axis. This is due to the fact that A, M and K are self-adjoint. We consider the eigenvalue problem y(4)(¸; x) ¡ (gy0)0(¸; x) = ¸2y(¸; x) with more general ¸-dependent separated boundary conditions Bj(¸)y = 0 for j = 1; ¢ ¢ ¢ ; 4 where Bj(¸)y = y[pj ](aj) or Bj(¸)y = y[pj ](aj) + i²j®¸y[qj ](aj), aj = 0 for j = 1; 2 and aj = a for j = 3; 4, ® > 0, ²j = ¡1 or ²j = 1. We assume that at least one of the B1(¸)y = 0, B2(¸)y = 0, B3(¸)y = 0, B4(¸)y = 0 is of the form y[p](0)+i²®¸y[q](0) = 0 or y[p](a)+i²®¸y[q](a) = 0 and we investigate classes of boundary conditions for which the corresponding operator A is self-adjoint.