Variational proof of the existence of eigenvalues for star graphs

Abstract

We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schrödinger operator with an attractive δ-potential supported by a star graph, i.e. by a finite union of rays emanating from the same point. In contrast to the previous works, the construction is valid without any additional assumption on the number or the relative position of the rays. The approach is used to obtain an upper bound for the lowest eigenvalue.

On Schrödinger operators with δ′-potentials supported on star graphs

“…We already know from the first part of the proof that E n (H θ ) = Λ n (H θ ) for small θ. Hence, the last inequality gives the sought lower bound for E n (H θ ) in (16) and completes the proof.…”

“…For small θ the right-hand side is clearly smaller than inf spec ess H θ ≡ −4, which implies E n (H θ ) = Λ n (H θ ) and shows the sought upper bound for E n (H θ ) in (16).…”

Bound states asymptotics in the system with quantum wires in ℝ3

“…Let us consider equation (20). As we have already mentioned operator D κ,θ can induce only discrete spectrum points for θ ∈ 0, π 2 .…”

Strong coupling asymptotics for $δ$-interactions supported by curves with cusps

“…To our knowledge, no sufficiently detailed analysis for non-smooth Γ was carried out so far. Being based on the general machinery for problems with corners [5,8,16] one might expect that if Γ is piecewise smooth with non-zero angles, then at least several lowest eigenvalues behave as E n (H α ) ≃ −µ n α 2 as α → +∞, where µ n ∈ ( 1 4 , 1) are spectral quantities associated with some model operators (so-called star leaky graphs) whose exact values are not known: we refer to [7,9,11,18,21] for a number of estimates.…”