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

TL;DR

This paper provides quantitative upper bounds on the number of eigenvalues below the essential spectrum for Schroedinger operators with Ahlfors regular potentials in a strip, considering Robin and Dirichlet boundary conditions, using weighted norms.

Contribution

It introduces new upper estimates for eigenvalues of Schroedinger operators with Ahlfors regular potentials, expressed via weighted L^1 and Orlicz norms, for different boundary conditions.

Findings

Eigenvalue bounds depend on weighted L^1 norms of potentials.

Orlicz norms provide alternative estimates.

Results apply to operators with Robin and Dirichlet boundary conditions.

Abstract

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.

Equations

$$H_{\mu }:=-\Delta -V\mu ,V\ge 0,\text{on}{L}^2(S),$$

$$u_{x_2}(x_1,0)+\alpha u(x_1,0)=u_{x_2}(x_1,a)+\beta u(x_1,a)=0,$$

$$N_{-}(\mathbf{q}):=\sup \left\{\dim \mathcal{L}|\mathbf{q}[u]<0,\forall u\in \mathcal{L}\setminus \{0\}\right\},$$

$$H_{{\lambda }_1,\mu }=-\Delta -{\lambda }_1-V\mu \text{mbox}on{L}^2(S)$$

$${\mathcal{E}}_{{\lambda }_1,\mu ,S}[u]$$

$$\text{Dom}\left({\mathcal{E}}_{{\lambda }_1,\mu ,S}\right)$$

$$N_{-}\left({\mathcal{E}}_{{\lambda }_1,\mu ,S}\right)\le \sum _{n\in \mathbb{Z}}{N}_{-}\left({\mathcal{E}}_{{\lambda }_1,\mu ,S_n}\right),$$

$${\mathcal{E}}_{{\lambda }_1,\mu ,S_n}[u]=-{\int }_{\overline{S_n}}V(x)|u_1(x_2){|}^{2}d\mu (x)$$

$$\mu\Big{(}\mbox{supp}\big{(}V|u_{1}|^{2}\big{)}\cap\overline{S_{n}}\Big{)}=0.$$

$$\Vert f{\Vert }_{\Psi ,\mu }=\Vert f{\Vert }_{\Psi ,\Omega ,\mu }=\sup \left\{\left|{\int }_{\Omega }fgd\mu \right|:{\int }_{\Omega }\Phi (|g|)d\mu \le 1\right\}$$

$$\Vert f{\Vert }_{(\Psi ,\mu )}=\Vert f{\Vert }_{(\Psi ,\Omega ,\mu )}=\inf \left\{\kappa >0:{\int }_{\Omega }\Psi \left(\frac{|f|}{\kappa }\right)d\mu \le 1\right\}.$$

$$\Vert f{\Vert }_{(\Psi ,\mu )}\le \Vert f{\Vert }_{\Psi ,\mu }\le 2\Vert f{\Vert }_{(\Psi ,\mu )},\forall f\in L_{\Psi }(\Omega ),$$

$${\int }_{\Omega }\Psi \left(\frac{|f|}{{\kappa }_0}\right)d\mu \le C_0,C_0\ge 1⟹\Vert f{\Vert }_{(\Psi )}\le C_0{\kappa }_0.$$

$${\int }_{\Omega }\Psi \left(\frac{|f|}{\kappa }\right)d\mu \le {\int }_{\Omega }\Psi \left(\frac{|f|}{C_0{\kappa }_0}\right)d\mu \le \frac{1}{C_0}{\int }_{\Omega }\Psi \left(\frac{|f|}{{\kappa }_0}\right)d\mu \le 1$$

$$\Vert f{\Vert }_{(\Psi ,\mu )}\le \max \left\{1,{\int }_{\Omega }\Psi (|f|)d\mu \right\}.$$

$$\Vert f{\Vert }_{\Psi ,\Omega }^{(av)}:=\sup \left\{\left|{\int }_{\Omega }fgd\mu \right|:{\int }_{\Omega }\Phi (|g|)d\mu \le \mu (\Omega )\right\}.$$

$$\mathcal{A}(s)=e^{|s|}-1-|s|,\mathcal{B}(s)=(1+|s|)\ln (1+|s|)-|s|,s\in \mathbb{R}.$$

$$c_0{r}^d\le \mu (B(x,r))\le c_1{r}^d$$

$$c_0(R)r^d\le \mu (B(x,r))\le c_1(R)r^d$$

$$c_2\mu \left(\overline{S_{n\pm 1}}\right)\le \mu \left(\overline{S_n}\right)\le c_3\mu \left(\overline{S_{n\pm 1}}\right),\forall n\in \mathbb{Z}.$$

$$c_2^k\mu \left(\overline{S_{n\pm k}}\right)\le \mu \left(\overline{S_n}\right)\le c_3^k\mu \left(\overline{S_{n\pm k}}\right),\forall k\in \mathbb{N}.$$

$$I_n:=[2^{n-1},2^n],n>0,I_0:=[-1,1],I_n:=[-2^{|n|},-2^{|n|-1}],n<0,$$

$${\mathcal{F}}_n$$

$${\mathcal{F}}_0$$

$$M_n$$

$$N_{-}\left({\mathcal{E}}_{{\lambda }_1,\mu ,S}\right)\le 1+C\left(\sum _{\{{\mathcal{F}}_n>c,n\in \mathbb{Z}\}}\sqrt{{\mathcal{F}}_n}+\sum _{\{M_n>c,n\in \mathbb{Z}\}}{M}_n\right).$$

$$\Sigma :=\left\{\theta \in [0,\pi ):\exists l_{\theta }\text{mbox}suchthat\mu (l_{\theta })>0\right\},$$

$${\Sigma }_N:=\left\{\theta \in [0,\pi ):\exists {\text{l}}_{\theta }\text{mbox}suchthat\mu (l_{\theta }\cap B(0,N))>0\right\},$$

$$\Sigma =\underset{N\in \mathbb{N}}{\cup }{\Sigma }_N.$$

$${\Sigma }_{N,\delta }:=\left\{\theta \in [0,\pi ):\exists {\text{l}}_{\theta }\text{mbox}suchthat\mu (l_{\theta }\cap B(0,N))>\delta \right\}$$

$$\mu \left(l_{{\theta }_k}\cap B(0,N)\right)>\delta ,\forall k\in \mathbb{N}.$$

$$\mu \left(\underset{j\ne k}{\cup }(l_{{\theta }_j}\cap l_{{\theta }_k})\right)=0.$$

Full text

Keywords: Discrete spectrum; Schrödinger operators; Ahlfors regular potential; strip.

Mathematics Subject Classification (2010): 35P05, 35P15

1 Introduction

Let $V\in L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$. According to the Cwikel-Lieb-Rozenblum (CLR) inequality (see, e.g., [3, 25]), the number of negative eigenvalues of the Schrödinger operator $-\Delta-V,V\geq 0$ on $L^{2}(\mathbb{R}^{d})$ with $d\geq 3$ is estimated above by $\|V\|_{L^{\frac{d}{2}}(\mathbb{R}^{d})}$. In the case $d=2$, this estimate fails, but there has been significant recent progress in obtaining estimates of the CLR-type in two-dimensions 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 $V\mu$, where $\mu$ 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 $V\mu$ 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 $S:=\mathbb{R}\times(0,a),\,a>0$ be a strip, $\mu$ a $\sigma$-finite positive Radon measure on $\mathbb{R}^{2}$ and $V:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ a non-negative function integrable on bounded subsets of $S$ with respect to $\mu$. We study the following Schrödinger operator

[TABLE]

where $\Delta:=\sum^{2}_{k=1}\frac{\partial^{2}}{\partial x^{2}_{k}}$, subject to the following Robin boundary conditions

[TABLE]

where $\alpha,\beta\in\mathbb{R}$. Here $u_{x_{i}}(i=1,2)$ denotes the partial derivative of $u$ with respect to $x_{i}$ and note that $\mu$ 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 $\mu$, (1) is well defined and self-adjoint on $L^{2}(S)$ and its essential spectrum is the interval $[\lambda_{1},+\infty)$, where $\lambda_{1}$ is the first eigenvalue of $-\Delta=-\partial^{2}_{x_{2}}$ considered along the width of the strip with boundary conditions (2) (see, e.g., [2, 12, 18, 29]). For a detailed discussion about what $\lambda_{1}$ is, see e.g. [14]. The case where both $\alpha$ and $\beta$ are equal to zero with $\mu$ 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 $\mathcal{H}$ be a Hilbert space and let $\mathbf{q}$ be a Hermitian form with a domain $\mbox{Dom}\,(\mathbf{q})\subseteq\mathcal{H}$. Set

[TABLE]

where $\mathcal{L}$ denotes a linear subspace of $\mbox{Dom}\,(\mathbf{q})$. The number $N_{-}(\mathbf{q})$ is called the Morse index of $\mathbf{q}$ in $\mbox{Dom}\,(\mathbf{q})$. If $\mathbf{q}$ is the quadratic form of a self-adjoint operator $A$ with no essential spectrum in $(-\infty,0)$, then $N_{-}(\mathbf{q})$ 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

[TABLE]

subject to boundary conditions in (2). Now, defining (4) via its quadratic form we have

[TABLE]

Note that we take the closure $\overline{S}$ of the open strip $S$ in the terms involving $\mu$ as this measure might charge subsets of the horizontal lines $x_{2}=0$ and $x_{2}=a$.

We denote by $N_{-}\left(\mathcal{E}_{\lambda_{1},\mu,S}\right)$ the number of negative eigenvalues of (4) counting multiplicities.

Let $S_{n}:=(n,n+1)\times(0,a),\,\,n\in\mathbb{Z}.$ Then it follows from the variational principle [13, Lemma 1.6.2] (see also [7, Ch.6, $\S$2.1, Theorem 4] for the case when $\mu$ is absolutely continuous with respect to the Lebesgue measure) that

[TABLE]

where $N_{-}\left(\mathcal{E}_{\lambda_{1},\mu,S_{n}}\right)$ are the restrictions of the form $\mathcal{E}_{\lambda_{1},\mu,S}$ to $\overline{S_{n}}$. Let $u(x)=u_{1}(x_{2})$, where $u_{1}$ is an eigenfunction of $-\Delta=-\partial^{2}_{x_{2}}$ considered along the width of the strip with boundary conditions (2) corresponding to the first eigenvalue $\lambda_{1}$. Then it is easy to see that

[TABLE]

and since $V\geq 0$, the right-hand side is strictly negative unless

[TABLE]

So, one usually has $N_{-}\left(\mathcal{E}_{\lambda_{1},\mu,S_{n}}\right)\geq 1$ 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.

2 Notation

In order to state the estimate for $N_{-}\left(\mathcal{E}_{\lambda_{1},\mu,S}\right)$, we need some notation from the theory of Orlicz spaces (see, e.g., [1, 17, 22]). Let $\Phi$ and $\Psi$ be mutually complementary $N$-functions, and let $L_{\Phi}(\Omega,\mu)$, $L_{\Psi}(\Omega,\mu)$ be the corresponding Orlicz spaces. We will use the following norms on $L_{\Psi}(\Omega,\mu)$

[TABLE]

and

[TABLE]

These two norms are equivalent

[TABLE]

(see [1]).

Note that

[TABLE]

Indeed, since $\Psi$ is convex and increasing on $[0,+\infty)$, and $\Psi(0)=0$, we get for any $\kappa\geq C_{0}\kappa_{0}$,

[TABLE]

(see [26]). It follows from (10) with $\kappa_{0}=1$ that

[TABLE]

We will also need the following equivalent norm on $L_{\Psi}(\Omega)$ with $\mu(\Omega)<\infty$, which was introduced in [27]:

[TABLE]

We will use the following pair of mutually complementary $N$-functions

[TABLE]

Definition 2.1**.**

Let $\mu$ be a positive Radon measure on $\mathbb{R}^{2}$. We say the measure $\mu$ is Ahlfors regular of dimension $d>0$ if there exist positive constants $c_{0}$ and $c_{1}$ such that

[TABLE]

for all $0<r\leq$ diam(supp$\,\mu$) and all $x\in\textrm{supp}\,\mu$, 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 $\mu$ is locally Ahlfors regular on a bounded set $G\subset\mathbb{R}^{2}$ if for every $R<\infty$ there exist $d>0$ and positive constants $c_{0}(R)$ and $c_{1}(R)$ such that*

[TABLE]

for all $0<r\leq R$ and all $x\in\textrm{supp}\,\mu\cap\overline{G}$. We say that $\mu$ is locally Ahlfors regular on the strip $S$ if (15) holds for all $0<r\leq R$ and all $x\in\textrm{supp}\,\mu\cap\overline{S}$, and there exist constants $c_{2},c_{3}>0$ such that

[TABLE]

Thus for each $n\in\mathbb{Z}$,

[TABLE]

From now onwards, it will be assumed that $\mu$ is a $\sigma$-finite positive Radon measure that is locally Ahlfors regular on $S$.

3 Statement of the main result

Let

[TABLE]

[TABLE]

where $u_{1}$ is a normalized eigenfunction of $-\Delta=-\partial^{2}_{x_{2}}$ on $(0,a)$ with boundary conditions (2) corresponding to the first eigenvalue $\lambda_{1}$ (here the normalization is with respect to the Lebesgue measure).

Theorem 3.1**.**

Let $\mu$ be a $\sigma$-finite positive Radon measure on $\mathbb{R}^{2}$ that is locally Ahlfors regular on $S$ and $V\in L_{\mathcal{B}}(S_{n},\mu)$ for every $n\in\mathbb{Z}$. Then there exist constants $C,c>0$ such that

[TABLE]

4 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.

Lemma 4.1**.**

Let $\mu$ be a $\sigma$-finite Radon measure on $\mathbb{R}^{2}$ such that $\mu(\{x\})=0$ for all $x\in\mathbb{R}^{2}$. Let

[TABLE]

where $l_{\theta}$ is a line in $\mathbb{R}^{2}$ in the direction of the vector $(\cos\theta,\sin\theta)$. Then $\Sigma$ is at most countable.

Proof.

Let

[TABLE]

where $B(0,N)$ is the ball of radius $N\in\mathbb{N}$ centred at [math]. Then

[TABLE]

It is now enough to show that $\Sigma_{N}$ is at most countable for $\forall N\in\mathbb{N}$. Suppose that $\Sigma_{N}$ is uncountable. Then there exists a $\delta>0$ such that

[TABLE]

is infinite. Otherwise, $\Sigma_{N}=\underset{n\in\mathbb{N}}{\cup}\Sigma_{N,\frac{1}{n}}$ would have been finite or countable. Now take $\theta_{1},...,\theta_{k},...\in\Sigma_{N,\delta}$. Then

[TABLE]

Since $l_{\theta_{j}}\cap l_{\theta_{k}},\;j\neq k$ contains at most one point, then

[TABLE]

Let

[TABLE]

Then $\tilde{l}_{\theta_{j}}\cap\tilde{l}_{\theta_{k}}=\emptyset,\;j\neq k$ and $\tilde{l}_{\theta_{k}}\cap B(0,N)\subset B(0,N)$. So

[TABLE]

But

[TABLE]

which implies

[TABLE]

This contradiction means that $\Sigma_{N}$ is at most countable for each $N\in\mathbb{N}$. Hence $\Sigma$ is at most countable. ∎

Corollary 4.2**.**

There exists $\theta_{0}\in[0,\pi)$ such that $\theta_{0}\notin\Sigma$ and $\theta_{0}+\frac{\pi}{2}\notin\Sigma$.

Proof.

The set

[TABLE]

is at most countable. This implies that there exists a $\theta_{0}\notin\Sigma\cup(\Sigma-\frac{\pi}{2})$. Thus $\theta_{0}+\frac{\pi}{2}\notin\Sigma$. ∎

Let $G\subset\mathbb{R}^{2}$ be a bounded set with Lipschitz boundary such that $0<\mu(\overline{G})<\infty$. Let $G_{*}$ be the smallest square containing $G$ with sides chosen in the directions $\theta_{0}$ and $\theta_{0}+\frac{\pi}{2}$ 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

[TABLE]

There exists a bounded linear operator

[TABLE]

which satisfies

[TABLE]

(see, e.g., [28, Ch.VI, Theorem 5]). We will use the following notation:

[TABLE]

where $E\subseteq\mathbb{R}^{2}$ 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 $\mu$ be a $\sigma$-finite positive Radon measure on $\mathbb{R}^{2}$ that is locally Ahlfors regular on $G$. Choose and fix a direction satisfying Corollary 4.2. Further, for all $x\in\overline{G}$ and for all $r>0$, let $\Delta_{x}(r)$ be a square with edges of length $r$ in the chosen direction centred at $x\in\textrm{supp}\,\mu\cap\overline{G}$. Then for any $V\in L_{\mathcal{B}}\left(\overline{G},\mu\right),\;V\geq 0$ and any $m\in\mathbb{N}$ there exists a finite cover of $\textrm{supp}\,\mu\cap\overline{G}$ by squares $Q_{x_{k}}(r_{x_{k}}),r_{x_{k}}>0,k=1,2,...,m_{0}$, such that $m_{0}\leq m$ and

[TABLE]

for all $u\in W^{1}_{2}(G)\cap C(\overline{G})$ with $(Tu)_{Q_{x_{k}}(r_{x_{k}})}=0,k=1,...,m_{0}$, where the constant $C(G)$ depends only on $G$ and is invariant under parallel translations of $G$, $C(d)$ depends only on $d$ in (15), and $R$ is the diameter of $G^{*}$. If $m=1$, one can take $m_{0}=0$.

Let

[TABLE]

for all $u\in W^{1}_{2}(S_{n})$.

Let $\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{j}\leq\cdots$ be the eigenvalues of the above boundary value problem. By the Min-Max principle, we have

[TABLE]

where $u_{1}$ is a normalized eigenfunction corresponding to $\lambda_{1}$. 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 $-\Delta=-\partial^{2}_{x_{2}}$ on $(0,a)$ with boundary conditions (2) corresponding to the first eigenvalue $\lambda_{1}$, moreover $\lambda_{1}<\lambda_{2}$ (see [14] or [13, Section 1.5]).

It follows from the above that for all $u\in W^{1}_{2}(S_{n}),\;\;u\perp u_{1}$, one has

[TABLE]

which in turn implies

[TABLE]

Since $\lambda_{1}<\lambda_{2}$, one gets

[TABLE]

Lemma 4.4**.**

[Ehrling’s Lemma]* Let $X_{0},\;X_{1}\;\textrm{and}\;X_{2}$ be Banach spaces such that $X_{2}\hookrightarrow X_{1}$ is compact and $X_{1}\hookrightarrow X_{0}$. Then for every $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that*

[TABLE]

See, e.g., [23] for details and proof.

Let $S(\mathbb{R}^{2})$ be the class of all functions $\varphi\in C^{\infty}(\mathbb{R}^{2})$ such that for any multi-index $\gamma$ and any $k\in\mathbb{N}$,

[TABLE]

Denote by $S^{\prime}(\mathbb{R}^{2})$ the dual space of $S(\mathbb{R}^{2})$. For $s>0$, let

[TABLE]

Here, $\widehat{u}(\xi)$ is the Fourier image of $u(x)$ defined by

[TABLE]

Let

[TABLE]

[TABLE]

Now, let $X_{0}=L^{2}(S_{n}),X_{1}=H^{s}(S_{n})\;\textrm{for}\;\frac{1}{2}<s<1\;\textrm{and}\;X_{2}=W^{1}_{2}(S_{n})$ in Lemma 4.4. That $X_{2}\hookrightarrow X_{1}$ is compact follows from the Sobolev compact embedding theorem (see, e.g., [1, Ch. VII] or [20, $\S$ 1.4.6]). Thus we have the following lemma:

Lemma 4.5**.**

Let $\varepsilon>0$ be given. Then there exists a constant $C_{1}>0$ such that

[TABLE]

Proof.

In this proof, we make use of (22) and Lemma 4.4. For $s>\frac{1}{2}$, the trace theorem and Lemma 4.4 imply

[TABLE]

Take $\varepsilon\leq\frac{1}{2C_{s}\max\{|\beta|,|\alpha|\}}$. Then

[TABLE]

Hence (22) yields

[TABLE]

where

[TABLE]

∎

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

[TABLE]

Proof.

[TABLE]

where

[TABLE]

∎

Let $\mathcal{H}_{1}:=PW^{1}_{2}(S)$ and $\mathcal{H}_{2}:=(I-P)W^{1}_{2}(S)$, where

[TABLE]

and

[TABLE]

Then $P$ is a projection since $P^{2}=P$.

Lemma 4.7**.**

For all $u\in W^{1}_{2}(S),\;\;\langle(I-P)u(x_{1},.),u_{1}\rangle_{L^{2}(0,a)}=0$ for almost all $x_{1}\in\mathbb{R}$.

Proof.

Since $Pu=\langle u(x_{1},.),u_{1}\rangle_{L^{2}(0,a)}u_{1}$, then

[TABLE]

∎

Lemma 4.8**.**

For all $v\in\mathcal{H}_{1},\;\tilde{v}\in\mathcal{H}_{2},\;\;\langle v,\tilde{v}\rangle_{L^{2}(S)}=0$ and $\langle v_{x_{1}},\tilde{v}_{x_{1}}\rangle_{L^{2}(S)}=0$.

Proof.

[TABLE]

Since $(Pw)_{x_{1}}=Pw_{x_{1}}$ for every $w\in W^{1}_{2}(S)$, one has for all $v\in\mathcal{H}_{1}\;\textrm{and}\;\tilde{v}\in\mathcal{H}_{2}$,

[TABLE]

and hence it follows from the above that

[TABLE]

∎

Lemma 4.9**.**

Let

[TABLE]

Then

[TABLE]

Proof.

[TABLE]

Integration by parts and Lemma 4.7 give

[TABLE]

Thus, this together with Lemma 4.8 yield

[TABLE]

This means that for all $u\in W^{1}_{2}(S)$

[TABLE]

∎

5 Proof of Threorem 3.1

Let

[TABLE]

and

[TABLE]

Then one has

[TABLE]

where $\mathcal{E}_{1,2\mu}$ and $\mathcal{E}_{2,2\mu}$ are the restrictions of the form $\mathcal{E}_{\lambda_{1},2\mu,S}$ to the spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ respectively. We start by estimating the first term in the right-hand side of (27).

Recall that for all $u\in\mathcal{H}_{1}\,,\;u(x)=w(x_{1})u_{1}(x_{2})$ (see (26)). Let $I$ be an arbitrary interval in $\mathbb{R}$ and let

[TABLE]

Then

[TABLE]

On the subspace $\mathcal{H}_{1}$, one has

[TABLE]

But

[TABLE]

which implies

[TABLE]

Hence, we have the following one-dimensional Schrödinger operator

[TABLE]

Let

[TABLE]

[TABLE]

Then

[TABLE]

(see [13, (2.42)], see also the estimate before (39) in [26]). To write the above estimate in terms of the original measure, let

[TABLE]

Then $F_{n}=\mathcal{F}_{n}$. Hence

[TABLE]

Next, we consider the subspace $\mathcal{H}_{2}\subset W^{1}_{2}(S)$. By (22) and (25), one has

[TABLE]

for all $u\in W^{1}_{2}(S_{n}),\;u\perp u_{1}$.

Let $S_{n}:=(n,n+1)\times(0,a)$, $n\in\mathbb{Z}$ with $\mu(\overline{S_{n}})>0$ 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 $N_{0}\in\mathbb{N}$ depends only on $a$ and $\theta_{0}$ in Corollary 4.2. (It is not difficult to see that the side length of $S^{*}_{n}$ is less than or equal to $3\sqrt{a^{2}+1}$ and hence $\left[3\sqrt{2}\sqrt{a^{2}+1}\right]+1$ provides an upper estimate for $N_{0}$.) Then (17) implies

[TABLE]

where

[TABLE]

Now it follows from Lemma 4.3 that for any $V\in L_{\mathcal{B}}(\overline{S_{n}},\mu),\,V\geq 0$

[TABLE]

for all $u\in W^{1}_{2}(S_{n})\cap C(\overline{S_{n}})$ 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

[TABLE]

for all $u\in W^{1}_{2}(S_{n})\cap C(\overline{S_{n}}),\,u\perp u_{1}$ satisfying the $m_{0}$ orthogonality conditions, where

[TABLE]

Let

[TABLE]

(see (21)). Taking $m=\left[2C_{3}\|V\|_{\mathcal{B},\overline{S_{n}},\mu}\right]+1$ in (32), one has

[TABLE]

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

[TABLE]

for all $u\in W^{1}_{2}(S_{n})\cap C(\overline{S_{n}}),\,u\perp u_{1}$. If $\|V\|_{\mathcal{B},\overline{S_{n}},\mu}\leq\frac{1}{C_{4}}$, then

[TABLE]

Otherwise, (34) implies

[TABLE]

where $C_{5}:=3C_{4}$.

Let $M_{n}=\|V\|_{\mathcal{B},\overline{S_{n}},\mu}$ (see Section 3). Then for any $c\leq\frac{1}{C_{4}}$, the variational principle (see (6)) implies

[TABLE]

Thus (27), (30) and (35) imply (18).

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

[TABLE]

is finite. It is a quasinorm in the sense that it satisfies the weak version of the triangle inequality:

[TABLE]

(see, e.g., [10] for more details).**

Theorem 6.2**.**

(cf. [26, Theorem 9.2])* Let $V\geq 0$. If $N_{-}\left(\mathcal{E}_{\lambda_{1},\gamma\mu,S}\right)=O(\gamma)\mbox{ as }\gamma\longrightarrow+\infty$, then $\|\mathcal{F}_{n}\|_{1,w}<\infty$.*

Proof.

Consider the function

[TABLE]

$n>0$. Let $v_{n}(x)=w_{n}(x_{1})u_{1}(x_{2})$. Then by a computation similar to the one leading to (28) we get

[TABLE]

Now

[TABLE]

It follows from the above that $\mathcal{E}_{\lambda_{1},\mu,S}[v_{n}]<0$ if $\mathcal{F}_{n}>5$, $n>0$.

One can define functions $v_{n}$ for $n\leq 0$ similarly to the above and extend to them the previous estimate. The fact that $v_{n}$ and $v_{k}$ have disjoint supports if $|m-k|\geq 3$ implies that

[TABLE]

(see [26, Theorem 9.1]). If $N_{-}\left(\mathcal{E}_{\lambda_{1},\gamma\mu,S}\right)\leq C\gamma$, then

[TABLE]

which implies

[TABLE]

With $s=\frac{5}{\gamma}$, we have

[TABLE]

where $C_{7}:=15\,C$. ∎

Remark 6.3**.**

Suppose that $\mu=|\cdot|$, the Lebesgue measure. Then

[TABLE]

Let $J_{n}:=(n,n+1),\;\;I:=(0,a)$ and write $\parallel V\parallel_{\mathcal{B},S_{n}}$ and $N_{-}\left(\mathcal{E}_{\lambda_{1},S}\right)$ instead of $\parallel V\parallel_{\mathcal{B},S_{n},|\cdot|}$ and $N_{-}\left(\mathcal{E}_{\lambda_{1},|\cdot|,S}\right)$ respectively. Further, let

[TABLE]

Then, using [26, Lemma 7.6] in place of our Lemma 4.3, one gets

[TABLE]

This estimate is stronger than (18). Indeed, suppose that $\|V\|_{(\mathcal{B},S_{n})}=1$. Since $\mathcal{B}(V)$ satisfies the $\Delta_{2}$-condition, then $\int_{S_{n}}\mathcal{B}(V(x))\,dx=1$ (see (9.21) in [17]). Using (9) and (12), we have

[TABLE]

The scaling $V\longmapsto tV,\;t>0$, allows one to extend the above inequality to an arbitrary $V\geq 0$.

By the same procedure as the one leading to (56) in [15], one has the following estimate

[TABLE]

where

[TABLE]

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 $\|(\mathcal{F}_{n})\|_{{1,w}}<\infty$ is necessary and sufficient for the semi-classical behaviour of the estimate coming from the subspace $\mathcal{H}_{1}$ (see Theorem 6.2 above). In addition, if $V_{*}\in L_{1}\left(\mathbb{R},L_{\mathcal{B}}(I)\right)$, then one gets an analogue of [19, Theorem 1.1], i.e.,

[TABLE]

if and only if $\mathcal{F}_{n}\in l_{1,w}$. **

Remark 6.4**.**

One can think of the Dirichlet boundary conditions as the limit of the boundary conditions in (2) as $\alpha$ and $\beta$ tend to infinity. In this case,

[TABLE]

Let $X_{n}:=\{u\in W^{1}_{2}(S_{n})\,:\,u(x_{1},0)=u(x_{1},a)=0\}$. Then for all $u\in X_{n},u\perp\sin\frac{\pi}{a}x_{2}$, one has an analogue of (22)

[TABLE]

Also, similarly to Lemma 4.6, there is a constant $C>0$ such that

[TABLE]

Now, for all $u\in X:=\{u\in W^{1}_{2}(S)\,:\,u(x_{1},0)=u(x_{1},a)=0\}$, let

[TABLE]

Then $P:X\longrightarrow X$ an orthogonal projection (cf. Lemma 4.9). Let $X_{1}:=PX$ and $X_{2}:=(I-P)X$. Furthermore, let

[TABLE]

Then similarly to (27), we have

[TABLE]

where $q_{1,2\mu}$ and $q_{2,2\mu}$ are the restrictions of the form $q_{2\mu,S}$ to the subspaces $X_{1}$ and $X_{2}$ respectively. For an arbitrary interval $I$ on $\mathbb{R}$, let

[TABLE]

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 $\mathcal{F}_{n}$ is given by

[TABLE]

On the subspace $X_{2}$, it follows from (40) and (41) that there is a constant $C^{\prime}>0$ such that

[TABLE]

Thus, we obtain, similarly to (35), an estimate for the second term in (42). **

7 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.