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

In this paper, quantitative upper estimates for the number of eigenvalues lying below the essential spectrum of Schroedinger operators with potentials generated by Ahlfors regular measures in a strip subject to two different types of boundary conditions (Robin and Dirichlet respectively) are presented. The estimates are presented in terms of weighted L^1 norms and Orlicz norms of the potential.


Introduction
Let V ∈ L 1 loc (R d ). According to the Cwikel-Lieb-Rozenblum (CLR) inequality (see, e.g., [3,25]), the number of negative eigenvalues of the Schrödinger operator −∆ − V, V ≥ 0 on L 2 (R d ) with d ≥ 3 is estimated above by V . In the case d = 2, this estimate fails, but there has been significant recent progress in obtaining estimates of the CLR-type in twodimensions and the best known estimates have been obtained in [26]. Estimates for the number of negative eigenvalues of Schrödinger operators with potential of the form V µ, where µ is a Radon measure and V is an appropriate function, were obtained in [6], and results from [26] were extended to this setting in [13,16]. In the present paper, we obtain estimates for the number of eigenvalues below the essential spectrum of a Schrödinger operator with potential of the form V µ similar to those in [16] in a strip subject to boundary conditions of the Robin type. Similar estimates are also obtained when the domain of the operator is characterized by Dirichlet boundary conditions (see remark 6.4). Below is a precise description of the operator studied herein.
Let S := R × (0, a), a > 0 be a strip, µ a σ-finite positive Radon measure on R 2 and V : R 2 −→ R a non-negative function integrable on bounded subsets of S with respect to µ. We study the following Schrödinger operator where ∆ := 2 k=1 ∂ 2 ∂x 2 k , subject to the following Robin boundary conditions where α, β ∈ R. Here u x i (i = 1, 2) denotes the partial derivative of u with respect to x i and note that µ does not have to be the two-dimensional Lebesgue measure. A physical motivation to this problem is closely related to the study of spectral properties of quantum waveguides (see, e.g., [8,9,18,21,24]).
Under certain assumptions about V and µ, (1) is well defined and self-adjoint on L 2 (S) and its essential spectrum is the interval [λ 1 , +∞), where λ 1 is the first eigenvalue of −∆ = −∂ 2 x 2 considered along the width of the strip with boundary conditions (2) (see, e.g., [2,12,18,29]). 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 . Below, we prove stronger results in our more general setting.
Let H be a Hilbert space and let q be a Hermitian form with a domain Dom (q) ⊆ H. Set where L denotes a linear subspace of Dom (q). The number N − (q) is called the Morse index of q in Dom (q). If q is the quadratic form of a self-adjoint operator A with no essential spectrum in (−∞, 0), then N − (q) is the number of negative eigenvalues of A repeated according to their multiplicity (see, e.g., [4, S1.3] or [5,Theorem 10.2.3]). Estimating the number of eigenvalues of (1) below its essential spectrum is equivalent to estimating the number of negative eigenvalues of the operator subject to boundary conditions in (2). Now, defining (4) via its quadratic form we have Note that we take the closure S of the open strip S in the terms involving µ as this measure might charge subsets of the horizontal lines x 2 = 0 and We denote by N − (E λ 1 ,µ,S ) the number of negative eigenvalues of (4) counting multiplicities.
Let S n := (n, n + 1) × (0, a), n ∈ Z. Then it follows from the variational principle [13, Lemma 1.6.2] (see also [7, Ch.6, §2.1, Theorem 4] for the case when µ is absolutely continuous with respect to the Lebesgue measure) that where N − (E λ 1 ,µ,Sn ) are the restrictions of the form E λ 1 ,µ,S to S n . Let u(x) = u 1 (x 2 ), where u 1 is an eigenfunction of −∆ = −∂ 2 x 2 considered along the width of the strip with boundary conditions (2) corresponding to the first eigenvalue λ 1 . Then it is easy to see that and since V ≥ 0, the right-hand side is strictly negative unless So, one usually has N − (E λ 1 ,µ,Sn ) ≥ 1 and thus the right-hand side of (6) diverges. To avoid this, we shall split the problem into two problems. The first will be defined by the restriction of the form to the subspace of functions obtained by multiplying u 1 (x 2 ) by functions depending only on x 1 , and is thus reduced to a well studied one-dimensional Schrödinger operator. The second problem will be defined by a class of functions orthogonal to u 1 in the L 2 ((0, a)) inner product.

Notation
In order to state the estimate for N − (E λ 1 ,µ,S ), we need some notation from the theory of Orlicz spaces (see, e.g., [1,17,22]). Let Φ and Ψ be mutually complementary N-functions, and let L Φ (Ω, µ), L Ψ (Ω, µ) be the corresponding Orlicz spaces. We will use the following norms on L Ψ (Ω, µ) and These two norms are equivalent (see [1]). Note that Indeed, since Ψ is convex and increasing on [0, +∞), and Ψ(0) = 0, we get for any κ ≥ C 0 κ 0 , (see [26]). It follows from (10) with κ 0 = 1 that We will also need the following equivalent norm on L Ψ (Ω) with µ(Ω) < ∞, which was introduced in [27]: We will use the following pair of mutually complementary N-functions Definition 2.1. Let µ be a positive Radon measure on R 2 . We say the measure µ is Ahlfors regular of dimension d > 0 if there exist positive constants c 0 and c 1 such that for all 0 < r ≤ diam(supp µ) and all x ∈ supp µ, where B(x, r) is the ball of radius r centred at x, and the constants c 0 and c 1 are independent of the balls.

Definition 2.2. (Local Ahlfors regularity)
We say that a measure µ is locally Ahlfors regular on a bounded set G ⊂ R 2 if for every R < ∞ there exist d > 0 and positive constants c 0 (R) and c 1 (R) such that for all 0 < r ≤ R and all x ∈ supp µ ∩ G. We say that µ is locally Ahlfors regular on the strip S if (15) holds for all 0 < r ≤ R and all x ∈ supp µ ∩ S, and there exist constants c 2 , c 3 > 0 such that Thus for each n ∈ Z, From now onwards, it will be assumed that µ is a σ-finite positive Radon measure that is locally Ahlfors regular on S.

Statement of the main result
where u 1 is a normalized eigenfunction of −∆ = −∂ 2 x 2 on (0, a) with boundary conditions (2) corresponding to the first eigenvalue λ 1 (here the normalization is with respect to the Lebesgue measure). Theorem 3.1. Let µ be a σ-finite positive Radon measure on R 2 that is locally Ahlfors regular on S and V ∈ L B (S n , µ) for every n ∈ Z. Then there exist constants C, c > 0 such that

Auxiliary results
We start with a result that was obtained in [16] (see also [13, Lemma 3.1.1]). For the reader's convenience, we reproduce the proof here.
where l θ is a line in R 2 in the direction of the vector (cos θ, sin θ). Then Σ is at most countable.
Proof. Let where B(0, N) is the ball of radius N ∈ N centred at 0. Then It is now enough to show that Σ N is at most countable for ∀N ∈ N. Suppose that Σ N is uncountable. Then there exists a δ > 0 such that n would have been finite or countable.
Since l θ j ∩ l θ k , j = k contains at most one point, then This contradiction means that Σ N is at most countable for each N ∈ N.
Hence Σ is at most countable.
Let G ⊂ R 2 be a bounded set with Lipschitz boundary such that 0 < µ(G) < ∞. Let G * be the smallest square containing G with sides chosen in the directions θ 0 and θ 0 + π 2 from Corollary 4.2, and let G * be the closed square with the same centre as G * and sides in the same direction but of length 3 times that of G * . Let There exists a bounded linear operator [28, Ch.VI, Theorem 5]). We will use the following notation: where E ⊆ R 2 is a set of a finite two dimensional Lebesgue measure |E|.
The following result is similar to [13, Lemma 3.2.13] and follows directly from the proof of the latter. Lemma 4.3. Let G be as above and µ be a σ-finite positive Radon measure on R 2 that is locally Ahlfors regular on G. Choose and fix a direction satisfying Corollary 4.2. Further, for all x ∈ G and for all r > 0, let ∆ x (r) be a square with edges of length r in the chosen direction centred at x ∈ supp µ ∩G. Then for any V ∈ L B G, µ , V ≥ 0 and any m ∈ N there exists a finite cover of for all u ∈ W 1 2 (S n ). Let λ 1 ≤ λ 2 ≤ · · · ≤ λ j ≤ · · · be the eigenvalues of the above boundary value problem. By the Min-Max principle, we have 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 which in turn implies Since λ 1 < λ 2 , one gets Lemma 4.4. [Ehrling's Lemma] Let X 0 , X 1 and X 2 be Banach spaces such that X 2 ֒→ X 1 is compact and X 1 ֒→ X 0 . Then for every ε > 0, there exists a constant C(ε) > 0 such that See, e.g., [23] for details and proof. Let S(R 2 ) be the class of all functions ϕ ∈ C ∞ (R 2 ) such that for any multiindex γ and any k ∈ N, Denote by S ′ (R 2 ) the dual space of S(R 2 ). For s > 0, let Here, u(ξ) is the Fourier image of u(x) defined by Now, let X 0 = L 2 (S n ), X 1 = H s (S n ) for 1 2 < s < 1 and X 2 = W 1 2 (S n ) in Lemma 4.4. That X 2 ֒→ X 1 is compact follows from the Sobolev compact embedding theorem (see, e.g., [1,Ch. VII] or [20, § 1.4.6]). Thus we have the following lemma: Lemma 4.5. Let ε > 0 be given. Then there exists a constant C 1 > 0 such that Proof. In this proof, we make use of (22) (ε(λ 1 + 1) + C(ε)) .

As a consequence of Lemma 4.5 and (22) we have the following Lemma
Lemma 4.6. There exists a constant C 2 > 0 such that Proof.
Integration by parts and Lemma 4.7 give Thus, this together with Lemma 4.8 yield This means that for all u ∈ W 1 2 (S) Then one has where E 1,2µ and E 2,2µ are the restrictions of the form E λ 1 ,2µ,S to the spaces H 1 and H 2 respectively. We start by estimating the first term in the right-hand side of (27).
Recall that for all u ∈ H 1 , u(x) = w(x 1 )u 1 (x 2 ) (see (26)). Let I be an arbitrary interval in R and let On the subspace H 1 , one has Hence, we have the following one-dimensional Schrödinger operator Then (see [13, (2.42)], see also the estimate before (39) in [26]). To write the above estimate in terms of the original measure, let Then F n = F n . Hence Next, we consider the subspace H 2 ⊂ W 1 2 (S). By (22) and (25), one has for all u ∈ W 1 2 (S n ), u ⊥ u 1 . Let S n := (n, n + 1) × (0, a), n ∈ Z with µ(S n ) > 0 be the set G in Lemma 4.3 and S * n be defined as above (see the paragraph after Corollary 4.2). For each n, S * n intersects not more than N 0 rectangles to the left of S n and N 0 rectangles to right of S n , where N 0 ∈ N depends only on a and θ 0 in Corollary 4.2. (It is not difficult to see that the side length of S * n is less than or equal to 3 √ a 2 + 1 and hence 3 √ 2 √ a 2 + 1 + 1 provides an upper estimate for N 0 .) Then (17) implies

Now it follows from Lemma 4.3 that for any
for all u ∈ W 1 2 (S n ) ∩ C(S n ) satisfying the m 0 orthogonality conditions in Lemma 4.3, where the constant C 0 is independent of V , m, and n. Hence (31) implies for all u ∈ W 1 2 (S n ) ∩ C(S n ), u ⊥ u 1 satisfying the m 0 orthogonality conditions, where (see (21)). Taking m = 2C 3 V B,Sn,µ + 1 in (32), one has where C 4 := 2C 3 (see [13,Lemma 3.2.14]). Again, taking m = 1 (and m 0 = 0; see Lemma 4.3) in (32), we get Let M n = V B,Sn,µ (see Section 3). Then for any c ≤ 1 C 4 , the variational principle (see (6)) implies Thus (27) 6 Concluding remarks Remark 6.1. Recall that a sequence {a n } belongs to the "weak l 1 -space" (Lorentz space) l 1,w if the following quasinorm is finite. It is a quasinorm in the sense that it satisfies the weak version of the triangle inequality: (see, e.g., [10] for more details).
Proof. Consider the function Then by a computation similar to the one leading to (28) we get It follows from the above that E λ 1 ,µ,S [v n ] < 0 if F n > 5, n > 0. One can define functions v n for n ≤ 0 similarly to the above and extend to them the previous estimate. The fact that v n and v k have disjoint supports if |m − k| ≥ 3 implies that (see [26,   This estimate is stronger than (18). Indeed, suppose that V (B,Sn) = 1.
By the same procedure as the one leading to (56) in [15], one has the following estimate where Estimates (37) and (39) are equivalent to each other but the advantage of the latter is that it separates the contribution to the eigenvalues of V (x) = V (x 1 ) from that of V (x) = V (x 2 ). The condition (F n ) 1,w < ∞ is necessary and sufficient for the semi-classical behaviour of the estimate coming from the subspace H 1 (see Theorem 6. Remark 6.4. One can think of the Dirichlet boundary conditions as the limit of the boundary conditions in (2) as α and β tend to infinity. In this case, λ 1 = π 2 a 2 , u 1 (x 2 ) = sin π a x 2 , and λ 2 = min 4 π 2 a 2 , π 2 a 2 + π 2 > λ 1 .
Then similarly to (27), we have where q 1,2µ and q 2,2µ are the restrictions of the form q 2µ,S to the subspaces X 1 and X 2 respectively. For an arbitrary interval I on R, let ν(I) := 2 a I a 0 V (x) sin 2 π a x 2 dµ(x).
Then on the subspace X 1 , a procedure similar to the one leading to (30) gives an estimate for the first term in (42), where in this case F n is given by On the subspace X 2 , it follows from (40) and (41) that there is a constant C ′ > 0 such that Sn |u(x)| 2 dx , ∀u ∈ X n , u ⊥ sin π a x 2 .
Thus, we obtain, similarly to (35), an estimate for the second term in (42).

Acknowledgments
The author acknowledges the funding from the Commonwealth Scholarship Commission in the UK, grant UGCA-2013-138. The author is also very grateful to Eugene Shargorodsky for his valuable guidance and discussions during the preparation of this work.