Transformation operators for partial differential equations (Q1087751)

The reconstruction by trace u(0,t) for the differential equation \[ (1)\quad u_{tt}-u_{yy}+(A+c(y))u(y,t)=I\delta (y)\delta (t) \] \[ (2)\quad u(y,0)=u_ t(y,0)=0,\quad u_ y(0,t)=0,\quad t\geq 0,\quad y\geq 0 \] where A is a self-adjoint, semi-bounded operator in a Hilbert space \(H_ 0\) is considered. For the problem (1)-(2) existence and uniqueness are proved. In terms of some conditions for the coefficients the correctness of the representation \[ u(y,t)=v_ 0(y,t)+\int^{y}_{0}K(y,s)v_ 0(s,t)ds \] \[ v_ 0(y,t)=u(y,t)-\int^{y}_{0}H(y,s)u(s,t)ds;\quad t\geq y\geq s\geq 0 \] is shown. The author calls these equations the transformation operators.