Schr\"odinger operators with $\delta$-interactions supported on conical surfaces

We investigate the spectral properties of self-adjoint Schr\"odinger operators with attractive $\delta$-interactions of constant strength $\alpha>0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum is given by $[-\alpha^2/4,+\infty)$ and that the discrete spectrum is infinite and accumulates to $-\alpha^2/4$. Furthermore, an asymptotic estimate of these eigenvalues is obtained.

In addition, we obtain an asymptotic estimate of the eigenvalues of −∆ α,C θ lying below −α 2 /4, and the results also extend to local deformations of the conical surface C θ , see Theorem 3.2 and Theorem 3.3 for details. The proof of our main result is based on standard techniques in spectral theory of self-adjoint operators: we construct singular sequences and use Neumann bracketing in the spirit of [EN03] to show the assertion on the essential spectrum; for the infiniteness of the discrete spectrum we employ variational principles. The same approach was applied in [S70] in the context of Schrödinger operators with slowly decaying negative regular potentials, see also [RS-IV, §XIII.3]. Similar arguments were also used in [DEK01,ET10] for the closely related question of infiniteness of the discrete spectrum for the Dirichlet Laplacian in a conical layer, see also [CEK04,J13,KV08,LL07,LR12] for further progress in this problem. We also point out [BEW09,DR13,EK02] for related spectral problems for Schrödinger operators with δ-potentials.

Essential spectrum of −∆ α,C θ
In this section we show that the essential spectrum of the operator −∆ α,C θ is given by [−α 2 /4, +∞). The proof of the inclusion σ ess (−∆ α,C θ ) ⊇ [−α 2 /4, +∞) makes use of singular sequences and for the other inclusion a specially chosen Neumann bracketing is used. A similar type of argument was also used in [BEL13,EN03] for δ and δ ′ -interactions on broken lines in the two-dimensional setting. For completeness we mention that the theorem (and its proof) below is also valid for θ = π/2, in which case the conical surface is a half-plane, and the result is wellknown.
Define for all p ∈ R and n ∈ N the functions ω n,p : R 2 + → C as + with the coordinate system (r, z). The ray Γ θ emerges from the origin and constitutes the angle θ ∈ (0, π/2) with the z-axis. The coordinate system (s, t) is associated with Γ θ .
By dominated convergence, using (2.1) and (2.2), we get (2.4) We denote by ω n,p,± the restrictions of ω n,p onto the open subsets The partial derivatives of ω n,p,± with respect to s and t are given by Similarly as in (2.4), using dominated convergence, we get (2.5) Let us define the sequence of functions ψ n,p : R 3 → C as where the functions ω n,p : R 2 + → C are interpreted as rotationally invariant functions on R 3 in the cylindrical coordinate system (r, ϕ, z). The hypersurface C θ separates the Euclidean space R 3 into two unbounded Lipschitz domains Ω + and Ω − , where Ω + = (x, y, z) ∈ R 3 : z > cot(θ) x 2 + y 2 , Ω − = (x, y, z) ∈ R 3 : z < cot(θ) x 2 + y 2 .
The subsets π 1 n , π 2 n and π 3 n of the closed half-plane R 2 + .

Discrete spectrum of −∆ α,C θ
In this section we show that the discrete spectrum of the self-adjoint operator −∆ α,C θ below the bottom −α 2 /4 of the essential spectrum is infinite for all angles θ ∈ (0, π/2) and we estimate the rate of the convergence of these eigenvalues to −α 2 /4 with the help of variational principles. The following lemma will be useful.
Proof. First of all observe that and setting ρ := inf{r : (r, z) ∈ supp ω} > 0 we obtain (3.4) Hence (3.2) and (3.4) imply ψ ∈ H 1 (R 3 ). Next we substitute ψ in the form a α,C θ in (1.1). It follows from the form of ∂ z ψ in (3.3) and ψ| C θ where we integrated by parts and used the fact that supp ω is contained in the open half-plane R 2 + . Hence, (3.6) and (3.7) imply Substituting this expression for the first integral in (3.5) we obtain (3.1).
Now we are ready to formulate and prove our main result on the infiniteness of the discrete spectrum of −∆ α,C θ below the bottom of the essential spectrum for all α > 0 and θ ∈ (0, π/2). Recall that −∆ α,C θ is bounded from below, and hence it also follows that the discrete spectrum has a single accumulation point, namely −α 2 /4. Theorem 3.2. Let −∆ α,C θ be the self-adjoint operator in L 2 (R 3 ) associated to the form (1.1) and let α > 0 and θ ∈ (0, π/2). Then the discrete spectrum of −∆ α,C θ below −α 2 /4 is infinite, accumulates at −α 2 /4, and the eigenvalues λ k < −α 2 /4 (enumerated in non-decreasing order with multiplicities taken into account) satisfy the estimate holds, where γ(θ) > 0, n k+1 := n 2 k + n k for k ∈ N, and n 1 = N with N ∈ N sufficiently large.
For geometric reasons we have |r(s, 0) − r(s, t)| ≤ a √ n with some 0 < a ≤ ε and r(s, t) > bn with some b > 0 for all (s, t) ∈ supp ω n . We first conclude from (3.20) that for S n in (3.11). In view of the above asymptotics and according to (3.9) there exists N ∈ N such that for all n ≥ N we have (3.25) S n ≤ − 2γ(θ) αn 4 for some constant γ(θ) > 0. Let us consider a sequence {n k } k , where n 1 := N and n k+1 := n 2 k + n k for k ∈ N. Then the functions ψ n k in (3.10) have disjoint supports for all k ∈ N and hence are orthogonal in L 2 (R 3 ). The space F k := span ψ n1 , ψ n2 , . . . , ψ n k ⊂ H 1 (R 3 ), has dimension k and for an arbitrary ψ = k l=1 a l ψ n l ∈ F k , a l ∈ C, we get where we have also used the estimate ω n l 2 L 2 (R 2 + ) ≤ 2 α . Employing (3.25) we obtain where we have again used the disjointness of the supports of {ψ n l } k l=1 . Combining the above estimate with (3.26) we get Hence, according to [BS87, Theorem 10.2.3] the operator −∆ α,C θ has at least k eigenvalues below the bottom of the essential spectrum −α 2 /4. The above construction works for any k ∈ N, so that the operator −∆ α,C θ has infinitely many eigenvalues below −α 2 /4. The eigenvalue estimate (3.8) follows from [BS87, Theorem 10.2.3] and (3.27).
Let θ ∈ (0, π/2) and C θ be the conical surface as above. A hypersurface Σ ⊂ R 3 , which for some compact set K ⊂ R 3 satisfies the condition Σ \ K = C θ \ K and which splits the space R 3 into two unbounded Lipschitz domains, is called a local deformation of C θ ; cf. [BEL13, Section 4.2]. Below we consider the self-adjoint Schrödinger operator −∆ α,Σ with an attractive δ-interaction of constant strength α > 0 supported on the Lipschitz hypersurface Σ. This Schrödinger operator is defined via the quadratic form The assertion on the essential spectrum in the next theorem is a consequence of [BEL13, Theorem 4.7]; the infiniteness of the discrete spectrum can be shown as in the proof of Theorem 3.2 using the same functions ψ n in (3.10) and n ∈ N sufficiently large.