Page 74 - Textos de Matemática Vol. 34

64 SUSANA D. MOURA
where trΓb := idΓ ◦ b ◦ trΓ and t′ denotes the conjugate exponent of t. We will be concerned with the negative spectrum of
Hβ :=id−∆+βtrΓb
and on its behaviour as β → ∞. This will be based on the entropy version of the Birman-Schwinger principle, cf. [15, 5.2.1 & 5.4.1] or [19].
In the sequel we consider the Sobolev space W21(Rn) endowed with the scalar product
(6.1) (f, g) 1 n := (id − ∆)1/2f, (id − ∆)1/2g ; W2(R ) L2(Rn)
then the counterpart of Theorem 31.3 in [28, p. 245] reads as follows:
Theorem 6.1. Let Γ be a (d,Ψ)-set in Rn with (n−2)+ < d ≤ n. Let b ∈ Lr(Γ)
be real with
(6.2) 0≤ 1 <1−n−2 rd+
and let trΓb :=idΓ ◦b◦trΓ. Then
B := (id − ∆)−1 ◦ trΓb
is a compact self-adjoint operator in W21(Rn) whose null space satisfy N ( B ) ⊃ { f ∈ W 21 ( R n ) : t r Γ f = 0 } ,
where both sides coincide if b(γ) ̸= 0 μ-a.e. on Γ. Moreover, B is generated by the sesquilinear form

b(γ) (trΓf)(γ) (trΓg)(γ) dμ(γ) = (Bf, g)W21(Rn). Γ
For each β > 0, let
and
(6.3)
Gβ :=id+βB
Nβ :=#σ(Gβ)∩(−∞,0]
be the number of non-positive eigenvalues of Gβ (counting multiplicities). Then, there exists a number c > 0 such that
(6.4) Nβ ≤#k∈N:k−(2−n+d)/dΨ♯d(k−1)(n−2)/d ≥cβ∥b|Lr(Γ)∥−1
for all β > 0 and all b ∈ Lr(Γ), where Ψ♯d is the Bruijn conjugate of Ψ(r1/d)−1, according to (iv) of Proposition 2.1.