Let <$E GAMMA> be a closed subset of <$E bold roman R sup d> with strictly positive <$E c sub 2>-capacity. We provide a method to construct a large class of selfadjoint operators <$E H sub alpha sup mu> in <$E L sup 2 bold roman (R sup d)> which satisfy <$E H sub alpha sup mu f~=~- DELTA f> for every smooth function with compact support away from <$E GAMMA>. The operators are parametrized via a positive real number <$E alpha> and a measure <$E mu> supported by <$E GAMMA>. We shall show that the negative number <$E - beta> is an eigenvalue of <$E H sub alpha sup mu> if and only if 0 is an eigenvalue of the operator <$E ( - DELTA~+~alpha ) ( - DELTA~+~beta )~-~( alpha~-~beta ) mu> and give a linear bijection between the corresponding eigenspaces. Moreover we shall derive estimates for the number, counting multiplicities, of negative eigenvalues of the operator <$E H sub alpha sup mu>.

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