Singular Perturbations of Abstract Wave equations

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dc.creatorPosilicano, Andrea
dc.date2004-03-23
dc.date2005-03-10
dc.date.accessioned2026-07-07T05:06:40Z
dc.date.available2026-07-07T05:06:40Z
dc.descriptionGiven, 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 $\ddotϕ-(C_1+C_2)\dotϕ=-(B^2+C_1C_2)ϕ$. Given then an auxiliary Hilbert space $\fh$ and a linear map $τ:D(B^2)\to\fh$ with a kernel $\K$ dense in $\H_0$, we explicitly construct skew-adjoint operators $W_Θ$ on a Hilbert space $\H_Θ\simeq D(B)\oplus\H_0\oplus \fh$ which coincide with $W$ on $\N\simeq\K\oplus D(B)$. The extension parameter $Θ$ 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 $A_Θ$ of $A:=-B^2$ such that the diagram $$ \begin{CD} W @>>> W_Θ@AAA @VVV A@>>> A_Θ\end{CD} $$ is commutative.
dc.descriptionRevised version. Misprints corrected. New examples and a digression on a possible application to the electrodynamics of a point particle added. Accepted for publication in Journal of Functional Analysis
dc.identifierhttps://arxiv.org/abs/math/0403386
dc.identifierhttp://arxiv.org/abs/math/0403386
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70557
dc.subjectFunctional Analysis
dc.subjectMathematical Physics
dc.titleSingular Perturbations of Abstract Wave equations
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