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Various qualitative methods for analyzing the discrete spectrum of SchrOdinger operators have been elaborated during the last three decades-for a review see Eef. 1, Section XIII. 3. They are of a great physical interest, especially because they can provide us with an information about -)*the few-body and many-body Hamlltonians, for which the quantitative methods are usually difficult to be applied. lt is not strange, therefore, that new re3ulta of this type continue to appear.

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In this paper, we study gravitational instantons (i.e., complete hyperkähler 4‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑D_k case, it must have an O(4)‑multiplet.

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