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Since highly frequency dependent delta potentials are out of reach of the more general techniques in previous work, we study the special case where $\\Omega=B(0,1)\\subset \\mathbb{R}^2$ and $V\\equiv h^{-\\alpha }V_0>0$ with $\\alpha\\leq 1$. Here $h^{-1}\\sim \\Re \\lambda$ is the frequency. 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