Singular Perturbations of Abstract Wave equations

Abstract

Given, on the Hilbert space $\H_0$, the self-adjoint operator $B$ and the skew-adjoint operators $C_1$ and $C_2$, we consider, on the Hilbert space $\H\simeq D(B)\oplus\H_0$, the skew-adjoint operator $$W=[\begin{matrix} C_2&\uno -B^2&C_1\end{matrix}]$$ corresponding to the abstract wave equation $\ddot\phi-(C_1+C_2)\dot\phi=-(B^2+C_1C_2)\phi$. Given then an auxiliary Hilbert space $\fh$ and a linear map $\tau:D(B^2)\to\fh$ with a kernel $\K$ dense in $\H_0$, we explicitly construct skew-adjoint operators $W_\Theta$ on a Hilbert space $\H_\Theta\simeq D(B)\oplus\H_0\oplus \fh$ which coincide with $W$ on $\N\simeq\K\oplus D(B)$. The extension parameter $\Theta$ ranges over the set of positive, bounded and injective self-adjoint operators on $\fh$. In the case $C_1=C_2=0$ our construction allows a natural definition of negative (strongly) singular perturbations $A_\Theta$ of $A:=-B^2$ such that the diagram $$ \begin{CD} W @>>> W_\Theta @AAA @VVV A@>>> A_\Theta \end{CD} $$ is commutative.