Eigenvalue bounds for a class of Schroedinger operators in a strip

This paper is concerned with the estimation of the number of negative eigenvalues (bound states) of Schroedinger operators in a strip subject to Neumann boundary conditions. The estimates involve weighted L^1 norms and L ln L norms of the potential. Estimates involving the norms of the potential supported by a curve embedded in the strip are also presented.


Introduction
This paper provides estimates for the number of negative eigenvalues of the Schrödinger operator −∆−V on L 2 (S) subject to Neumann boundary conditions, where S is an infinite straight strip . We use the results of Shargorodsky [10] to obtain improved versions of the estimates by Grigor'yan and Nadirashvili [7]. This improvement is achieved by replacing V L p , p > 1 in the estimates of [7,Section 7] by the L log L norms of V . In addition, these estimates are extended to the case of strongly singular potentials (see Section 4). On R 2 , estimates of a similar nature have been presented in [8]. The precise description of the operator here studied is as follows: Let S := {(x 1 , x 2 ) ∈ R 2 : x 1 ∈ R, 0 < x 2 < a}, a > 0 and V : R 2 −→ R be a function integrable on bounded subsets of S. Consider the following self-adjoint operator on L 2 (S) with homogeneous Neumann boundary conditions both at x 2 = 0 and x 2 = a.
The main objective of this paper is to obtain estimates for the number of negative eigenvalues of (1) in terms of the norms of V .
The strategy used here is as follows: The problem is split into two problems. The first one is defined by the restriction of the quadratic form associated with the operator (1) to the subspace of functions of the form w(x 1 )u 1 (x 2 ), where u 1 is the first eigenfunction of the one-dimensional differential operator on L 2 ((0, a)) and hence, is reduced to a well studied one-dimensional Schrödinger operator with the potential equal to a weighted mean value V of V over (0, a). The second problem is defined by a class of functions orthogonal to constant functions in the L 2 ((0, a)) inner product.
is called complementary to Ψ.
2. An N-function Ψ is said to satisfy a global ∆ 2 -condition if there exists a positive constant k such that for every t ≥ 0, Similarly Ψ is said to satisfy a ∆ 2 -condition near infinity if there exists t 0 > 0 such that (3) holds for all t ≥ t 0 .
Let Φ and Ψ be mutually complementary N-functions, and let L Φ (Ω), L Ψ (Ω) be the corresponding Orlicz spaces. We will use the following norms on L Ψ (Ω) and These two norms are equivalent (see [1]). Note that It follows from (7) with κ 0 = 1 that We will need the following equivalent norm on L Ψ (Ω) with µ(Ω) < ∞ which was introduced in [11]: We will use the following pair of pairwise complementary N-functions Let I 1 , I 2 ⊆ R be nonempty open intervals. We denote by L 1 (I 1 , L B (I 2 )) the space of measurable functions f : I 1 × I 2 → C such that f L 1 (I 1 ,L B (I 2 )) := Let us 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: The quasinorm (12) induces a topology on l 1,w in which this space is nonseparable. The closure of the set of elements a n with only finite number of non-zero terms is a separable subspace in l 1,w . It is well known that l 1 ⊂ l 1,w and {a n } 1,w ≤ {a n } l 1 (see, e.g., [3] for more details).

Estimating the number of negative eigenvalues in a strip
Define (1) via its quadratic form We shall denote by N − (q V,S ) the number of negative eigenvalues of (1) repeated according to their multiplicities. Then N − (q V,S ) is given by where L denotes a linear subspace of Dom(q V,S ) (see, e.g., [3, Theorem 10.

2.3]). Let
Indeed, P is a projection since P 2 = P . Let L 2 := (I −P )L 2 (S), then one can show that L 2 (S) = L 1 ⊕ L 2 . Here and below ⊕ denotes a direct orthogonal Similarly, let then where q 1,2 V ,R and q 2,2V,S denote the restrictions of the form q 2V,S to the spaces H 1 and H 2 respectively. Let Then similarly to the estimate before (39) in [10] one has In terms of the original potential V It now remains to find an estimate for N − (q 2,2V,S ) in (15).
Proof. Let I = I, the unit interval. Then it follows from Lemma [10, Lemma 7.7] that there is a constant d 1 > 0 such that Similarly to (62) in [10], there is a constant d 2 > 0 such that Proof. If D n < 1 C 1 , then N − (q 2,2V,Sn ) = 0 and one can drop this term from the sum (18). Hence for any c < 1 C 1 , (18) and Lemma 3.1 imply that This together with (15) and (17) imply (21).
One can easily show that (21) is an improvement of the result by A. Grigor'yan and N. Nadirashvili [7, Theorem 7.9] with a different c and that (21) is strictly sharper. Indeed, let B n := V B,Sn . Then it follows from the embedding L p (S n ) ֒→ L B (S n ) that there is a constant C(p), p > 1 such that p (see [10, Remark 6.3]). Now it follows from the embedding of mixed-norm Orlicz spaces (see, e.g., [4,6]) that where C 2 := C 1 C(p). The scaling V −→ tV, t > 0, allows one to extend the above inequality to an arbitrary V ≥ 0. Thus for any c < 1 C 2 , (21) implies [7, Theorem 7.9].
Next we will discuss different forms of (21).

Equation (24) in turn implies the following
which is equivalent to Thus (25) and (26) are equivalent. Similarly, Hence (24) is equivalent to the following estimate Note the last term in right hand side of (27) (and (26)) drops out if V does not depend on x 2 .
Let α > 0 be given. It is well known that the lowest possible (semi-classical) rate of growth of N − (q αV,S ) is It turns out that the finiteness of the first term in (21) is necessary for N − (q αV,S ) = O(α) as α −→ +∞ to hold (see next Theorem).
Proof. Consider the function for n > 0. Let u n (x) = w n (x 1 )u 1 (x 2 ), where u 1 is the first eigenfunction of the one-dimensional second order differential operator on L 2 ((0, a)) which is identically equal to 1. Then we have If G n > C 7 , then q V,S [u n ] < 0. The auxiliary functions w n can be defined similarly for n ≤ 0. Since u n and u k have disjoint supports for |n − k| ≥ 3, then and so card n ∈ Z : G n > C 7 α ≤ 3C 8 α .

Estimates involving norms of the potential supported by a Lipschitz curve inside a strip
In this section we obtain estimates analogous to those in the previous section when the potential V is strongly singular, i.e., when V is supported by a Lipschitz curve ℓ embedded in S. When dealing with function spaces on ℓ, we will always assume that ℓ is equipped with the arc length measure. Before we introduce the estimates, let us first look at the following operator that we shall need in the sequel: Consider a one-dimensional Schrödinger operator H X,α , with point δ-interactions on a countable set X = {x k } ∞ k=1 of points, called points of interaction and intensities α = {α k } ∞ k=1 , defined by the differential expression − d 2 dx 2 on functions w(x) that belong to the Sobolev space W 2 2 (R \ X) satisfying, in the points of the set X, the following conjugation conditions: Since for each k, (x k , x k+1 ) is an open interval, then any function in W 2 2 ((x k , x k+1 )) and its derivative have well defined (one-sided) values at the end-points. The operator H X,α has the following representation where δ is the Dirac's delta function. We shall assume that H X,α is selfadjoint (see, e.g., [2]) in case the set X is finite. One can also define the operator (29) via its quadratic form as follows Lemma 4.1. Given an infinite sequence of positive numbers (α k ), there is a sequence of points (x k ) in X such that (29) has infinitely many negative eigenvalues.
Proof. Let ψ ∈ C ∞ 0 (R) such that Let L be a linear subspace of W 1 2 (R) defined by Since ϕ k and ϕ j for k = j have disjoint supports, then dimL = ∞. Thus for all w ∈ L \ {0}, it follows from (30) that then q[w] < 0 and the operator (29) has infinitely many negative eigenvalues.
Let us now return to the operator (1) with V supported by and locally integrable on a Lipschitz curve ℓ. Let Let {x  Theorem 4.2. Suppose that N is the cardinality of Σ. Then there exist constants c 1 , c 2 , C 10 , C 11 > 0 such that Proof. Let q 1,2V,ℓ and q 2,2V,ℓ be the restrictions of the form q V,ℓ to the subspaces H 1 and H 2 respectively (see (14)), then (cf. (15)). Let us start by estimating the first term in the right-hand side of (32). On the complement of Σ, ν({x 1 }) = 0 for all x 1 ∈ R. This implies where c k := γ k V (x 2 ) dx 2 < ∞. Hence where c ′ k := 2c k . Let Dom(q 2,c ′ k ) = W 1 2 (R).