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Let ℒ($H$) denote the space of operators on a Hilbert space$H$. We show that the extreme points of the unit ball of the space of continuous functions$C(K, ℒ(H))$($K$-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dim$H$<∞ and card$K$<∞, (b) exposed if and only if$H$is separable and$K$carries a strictly positive measure.

We discuss operators of the type H=-\Delta+ V (x)-lpha\delta (x-\Sigma) with an attractive interaction, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , in H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , where H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is a potential bias being a positive constant H=-\Delta+ V (x)-lpha\delta (x-\Sigma) in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) . We show that H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is then empty if the bias is supported in theexterior'region, while in the opposite case isolated eigenvalues may exist.