Eigenvalue Bounds for a Class of Schrödinger Operators in a Strip

Abstract

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

1. Introduction

The celebrated Cwikel-Lieb-Rozenblum (CLR) inequality [1] gives an upper bound for the number of negative eigenvalues of the Schrödinger operator on . It is known that the CLR inequality does not hold for and one of the reasons is that the Sobolev space is not continuously embedded in . However, for all and there are estimates involving (see, e.g., [2–4]). More precisely, is embedded in a space of exponentially integrable functions which lies between and (see, e.g., [5]). This gives rise to estimates involving a norm of weaker than , namely, the Orlicz norm (see, e.g., [6, 7]). The strongest known estimates have been obtained in [6]. For more information regarding upper estimates for the number of negative eigenvalues of two-dimensional Schrödinger operators refer to [2–4, 6–9] and the references therein. This paper provides estimates for the number of negative eigenvalues of the Schrödinger operator on whose domain is characterized by the Neumann boundary conditions, where is an infinite straight strip. We use the results of Shargorodsky [6] to obtain improved versions of the estimates by Grigor’yan and Nadirashvili [3]. This improvement is achieved by replacing in the estimates of [3, Section 7] by the norms of . In addition, these estimates are extended to the case of strongly singular potentials (see Section 4). The precise description of the operator here studied is as follows.

Let and be a function integrable on bounded subsets of . Consider the following self-adjoint operator on :with homogeneous Neumann boundary conditions both at and . The main objective of this paper is to obtain estimates for the number of negative eigenvalues of (1) in terms of the norms of .

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 operator (1) to the subspace of functions of the form , where is the first eigenfunction of the one-dimensional differential operator on and, hence, is reduced to a well studied one-dimensional Schrödinger operator with the potential equal to a weighted mean value of over . The second problem is defined by a class of functions orthogonal to constant functions in the inner product.

2. Preliminaries

Let be a measure space and let be a nondecreasing function. The Orlicz class is the set of all of measurable functions such thatIf , this is just the space. The Orlicz space is the linear span of the Orlicz class , that is, the smallest vector space containing .

Definition 1. A continuous nondecreasing convex function is called an -function if The function defined by is called complementary to .

Definition 2. An -function is said to satisfy a global -condition if there exists a positive constant such that, for every ,Similarly is said to satisfy a -condition near infinity if there exists such that (5) holds for all .

Let and be mutually complementary -functions, and let and be the corresponding Orlicz spaces. We will use the following norms on :andThese two norms are equivalent:(see [5]).

Note thatIt follows from (9) with thatWe will need the following equivalent norm on with which was introduced in [7]:We will use the following pair of pairwise complementary -functions:Let be nonempty open intervals. We denote by the space of measurable functions such thatLet us recall that a sequence belongs to the “weak -space” (Lorentz space) if the quasinormis finite. It is a quasinorm in the sense that it satisfies the weak version of the triangle inequality: Quasinorm (14) induces a topology on in which this space is nonseparable. The closure of the set of elements with only finite number of nonzero terms is a separable subspace in . It is well known that and (see, e.g., [10] for more details).

3. Estimating the Number of Negative Eigenvalues in a Strip

Define (1) via its quadratic form where denotes the standard Sobolev space . Let denote the number of negative eigenvalues of (1) repeated according to their multiplicities. Then is given bywhere denotes a linear subspace of (see, e.g., [10, Theorem 10.2.3]).

Let andFurthermore, let Then we have the following result.

Theorem 3. There exist constants such that

Let and be a projection defined by Indeed, is a projection since . Let , and then one can show that . Here and below denotes a direct orthogonal sum.

Indeed for all we haveNow pick and , then, Similarly, letand then Indeed, for all and all we have This is so because , and . To see this note that and do not depend on implying that . Also, . So, . Hence .

Now for all , one has and where So Hencewhere and denote the restrictions of the form to the spaces and , respectively.

LetThen similarly to the estimate before in [6] one hasIn terms of the original potential and It now remains to find an estimate for in (34).

Let , where and are as defined in (21). Then the variational principle (see, e.g., [11, Ch.6, § 2.1, Theorem 4]) implieswhere

Lemma 4 (cf. [6, Lemma 7.8]). There exists such that(see (13)).

Proof. Let , the unit interval. Then it follows from [6, Lemma 7.7] that there is a constant such thatSimilarly to in [6] and using the Poincaré inequality (see, e.g., [12, 1.1.11]), there is a constant such that for all such that . If , then . Otherwise (42) implies where . Hence (41) follows by the scaling .

Proof of Theorem 3

Proof. If , then and one can drop this term from the sum (39). Hence, for any , (39) and Lemma 4 imply that This together with (34) and (36) imply (22).

One can easily show that (22) is an improvement of the estimates obtained by A. Grigor’yan and N. Nadirashvili [3, Theorem 7.9] with a different and that (22) is strictly sharper. Indeed, let . Then it follows from the embedding that there is a constant such that where (see [6, Remark 6.3]). Now it follows from the known theory of embeddings of mixed-norm Orlicz spaces (see, e.g., [13, 14]) that Hencewhere . The scaling , allows one to extend the above inequality to an arbitrary . Thus, for any , (22) implies [3, Theorem 7.9].

Next we will discuss different forms of (22).

Remark 5. Note that(see in [6]). Estimate (22) implies the following estimate:This follows from (49) and Equation (50) in turn implies the following:which is equivalent towhere .
Indeed,Thus (52) and (53) are equivalent.
Similarly,Hence (50) is equivalent to the following estimate:Note the last term on the right-hand side of (56) (and (53)) drops out if does not depend on .
Let be given. It is well known that the lowest possible (semiclassical) rate of growth of is (see, e.g., [2, 4]). It turns out that the finiteness of the first term in (22) is necessary for to hold (see the next theorem).

Theorem 6. Let . If , then .

Proof. Consider the function for . Let , where is the first eigenfunction of the one-dimensional second-order differential operator on which is identically equal to . Then we have where Now If , then . The auxiliary functions can be defined similarly for . Since and have disjoint supports for , then (see [6, Theorem 9.1]). If, for some constant , , then and so With we have where .

4. 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 is strongly singular, i.e., when is supported by a Lipschitz curve embedded in . When dealing with function spaces on , we will always assume that 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 , with point -interactions on a countable set of points, called points of interaction, and intensities , defined by the differential expression on functions that belong to the Sobolev space satisfying, in the points of the set , the following conjugation conditions:Since, for each , is an open interval, then any function in and its derivative have well defined (one-sided) values at the end-points. The operator has the following representation:where is Dirac’s delta distribution. We shall assume that is self-adjoint (see, e.g., [15] in case the set is finite). One can also define operator (67) via its quadratic form as follows:

Lemma 7. Given an infinite sequence of positive numbers , there is a sequence of points in such that (67) has infinitely many negative eigenvalues.

Proof. Let such that Assume that , for all and let . ThenLet be a linear subspace of defined by Since and for have disjoint supports, then . Thus, for all , it follows from (68) that where . Take such that then and operator (67) has infinitely many negative eigenvalues.

Let us now return to operator (1) with supported by and locally integrable on a Lipschitz curve embedded in . Let Let be a sequence of points on satisfying conditions in (66). Define andThen the set is at most countable. Let be an arbitrary interval in and let Furthermore, let

Theorem 8. Suppose that is the cardinality of . Then there exist constants such thatIf is infinite, then .

Proof. Let and be the restrictions of the form to the subspaces and , respectively (see (27)), and then(cf. (34)). Let us start by estimating the first term on the right-hand side of (80). On the complement of , for all . This implies where . Hence where . Let Then, it follows from [3, Lemma 3.6] thatSimilarly to (36) one hasIf is finite, then(see, e.g., [15]). Otherwise, Lemma 7 impliesNow, it remains to estimate the second term on the right-hand side of (80). Let For any and any , following a similar argument in the proof of [6, Lemma 7.6], one can show that there exists a finite cover of by rectangles such that andfor all with , where is a constant independent of . Now, using (89) instead of [6, Lemma 7.6] in the proof of [6, Lemma 7.7], one can easily show similarly to [6, Lemma 7.8] that there is a constant such that If , then . Thus, similarly to (39) one has for any thatHence, the statement of the theorem follows from (80), (84), (85), (86), (87), and (91).

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

This research was supported by the Commonwealth Scholarship Commission in the United Kingdom, grant UGCA-2013-138.