On the Spectrum of the Laplacian on a Strip With Various Boundary Conditions

On the discrete spectrum of Schrödinger operators with Ahlfors regular potentials in a strip

“…, where u 1 is a normalized eigenfunction corresponding to λ 1 . The function u 1 does not depend on x 1 and, viewed as a function of one variable x 2 , it is a normalized eigenfunction of −∆ = −∂ 2 x 2 on (0, a) with boundary conditions (2) corresponding to the first eigenvalue λ 1 , moreover λ 1 < λ 2 (see [14] or [13, Section 1.5]). It follows from the above that for all u ∈ W 1 2 (S n ), u ⊥ u 1 , one has…”

“…For a detailed discussion about what λ 1 is, see e.g. [14]. The case where both α and β are equal to zero with µ being the two-dimensional Lebesgue measure has been previously studied by A. Grigor'yan and N. Nadirashvili [11] who obtained estimates in terms of weighted L 1 norms and L p , p > 1 norms of V .…”