# 3-Manifolds which are unions of three solid tori by Larranaga J. C. G. By Larranaga J. C. G.

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Extra info for 3-Manifolds which are unions of three solid tori

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186(1997), 481–493.  Rofe-Beketov, F. , Kneser constants and eﬀective masses for band potentials, Soviet Phys. Dokl. 29(5)(1985), 391–393.  Schmidt, K. , On the essential spectrum of Dirac operators with spherically symmetric potentials, Math. Ann. 297(1993), 117–131.  Schmidt, K. , Dense point spectrum and absolutely continuous spectrum in spherically symmetric Dirac operators, Forum Math. 7(1995), 459–475.  Schmidt, K. , Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential, Proc.

2 . < ∞ and assume that σe (T ) ∩ [λ−a, λ+a] = ∅. Then for every µ ∈ [λ−a, λ+a ] there is an ε > 0 with dim ((Eµ+ε − Eµ−ε ) H) < ∞. A compactness argument yields dim ((Eλ+a+ε − Eλ−a−ε ) H) < ∞ for some ε > 0. ,K be an orthonormal basis of E H. Then K K E un = k=1 because un un , E ek ek → 0, E un , ek ek = k=1 0. Moreover, |T − λ| un 2 = |µ − λ| d Eµ un 2 ≥ |a + ε| {1 − (λ − a − ε < µ ≤ λ + a + ε)} d Eµ un = |a + ε| un 2 − E un 2 2 . For n → ∞, the right-hand side tends to |a + ε|, in contradiction to the deﬁnition of a.

14 o. There exists a sequence with λn → λ and (λn )n∈N ⊂ σe (S) or (λn )n∈N ⊂ σd (S). In the former case we may select an appropriate mixed sequence from the singular sequences of the λn s. For λn ∈ σd (S) we chose a normalized eigenfunction vn ∈ D(T ). 11 and Pythagoras’s theorem they converge weakly to 0 and Rd obviously fulﬁl eS (vn ) = λn and sS (vn ) = 0. We approximate vn by vn ∈ C∞ 0 and proceed as in the proof of [28, Lemma 1] to obtain the desired sequence by cutting out inside balls of increasing radii.