# Eigenvalue Bounds for a Class of Schrödinger Operators in a Strip

Authors: Martin Karuhanga
Journal: Journal of Mathematics
 (2018)

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## 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 weightedL1 norms and Lln⁡L norms of the potential. Estimates involving the norms of the potential supported by a curve embedded in a strip are also presented.


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## Body

## 1. Introduction

The celebrated Cwikel-Lieb-Rozenblum (CLR) inequality [1] gives an upper bound for the number of negative eigenvalues of the Schrödinger operator -Δ-V,V≥0 on L2(Rd),d>2. It is known that the CLR inequality does not hold for d=2 and one of the reasons is that the Sobolev space H1(R2) is not continuously embedded in L∞(R2). However, H1(R2)↪Lq(R2) for all q∈[2,+∞) and there are estimates involving ∫R2|V(x)|rdx,∀r>1 (see, e.g., [2–4]). More precisely, H1(R2) is embedded in a space of exponentially integrable functions which lies between L1(R2) and Lp(R2),p>1 (see, e.g., [5]). This gives rise to estimates involving a norm of V weaker than VLr,r>1, namely, the Orlicz Lln⁡L 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 -Δ-V on L2(S) whose domain is characterized by the Neumann boundary conditions, where S 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 VLp,p>1 in the estimates of [3, Section 7] by the Lln⁡L norms of V. 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.LetS≔{(x1,x2)∈R2:x1∈R,0<x2<a},a>0 and V:R2→R be a function integrable on bounded subsets of S. Consider the following self-adjoint operator on L2(S):(1)HV=-Δ-V,V≥0,with homogeneous Neumann boundary conditions both at x2=0 and x2=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 operator (1) to the subspace of functions of the form w(x1)u1(x2), where u1 is the first eigenfunction of the one-dimensional differential operator on L2((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 L2((0,a)) inner product.


## 2. Preliminaries

Let(Ω,Σ,μ) be a measure space and let Ψ:[0,+∞)→[0,+∞) be a nondecreasing function. The Orlicz class KΨ(Ω) is the set of all of measurable functions f:Ω→C(orR) such that(2)∫ΩΨfxdμx<∞.If Ψ(t)=tp,1≤p<∞, this is just the Lp(Ω) space. TheOrlicz spaceLΨ(Ω) is the linear span of the Orlicz class KΨ(Ω), that is, the smallest vector space containing KΨ(Ω).Definition 1.
A continuous nondecreasing convex functionΨ:[0,+∞)→[0,+∞) is called an N-function if (3)limt→0+⁡Ψtt=0andlimt→∞⁡Ψtt=∞.The function Φ:[0,+∞)→[0,+∞) defined by (4)Φt≔sups≥0⁡st-Ψsis called complementary to Ψ.Definition 2.
AnN-function Ψ is said to satisfy a global Δ2-condition if there exists a positive constant k such that, for every t≥0,(5)Ψ2t≤kΨt.Similarly Ψ is said to satisfy a Δ2-condition near infinity if there exists t0>0 such that (5) holds for all t≥t0.LetΦ and Ψ be mutually complementary N-functions, and let LΦ(Ω) and LΨ(Ω) be the corresponding Orlicz spaces. We will use the following norms on LΨ(Ω):(6)fΨ=fΨ,Ω=sup⁡∫Ωfgdμ:∫ΩΦgdμ≤1and(7)fΨ=fΨ,Ω=inf⁡κ>0:∫ΩΨfκdμ≤1.These two norms are equivalent:(8)fΨ≤fΨ≤2fΨ,∀f∈LΨΩ(see [5]).Note that(9)∫ΩΨfκ0dμ≤C0,C0≥1⇒fΨ≤C0κ0.It follows from (9) with κ0=1 that(10)fΨ≤max⁡1,∫ΩΨfdμ.We will need the following equivalent norm on LΨ(Ω) with μ(Ω)<∞ which was introduced in [7]:(11)fΨav=fΨ,Ωav=sup⁡∫Ωfgdμ:∫ΩΦgdμ≤μΩ.We will use the following pair of pairwise complementary N-functions:(12)As=es-1-s,Bs=1+sln⁡1+s-s,s∈R.Let I1,I2⊆R be nonempty open intervals. We denote by L1I1,LB(I2) the space of measurable functions f:I1×I2→C such that(13)fL1I1,LBI2≔∫I1fx1,·B,I2dx1<+∞.Let us recall that a sequence {an} belongs to the “weak l1-space” (Lorentz space) l1,w if the quasinorm(14)an1,w=sups>0⁡scardn:an>sis finite. It is a quasinorm in the sense that it satisfies the weak version of the triangle inequality: (15)an+bn1,w≤2an1,w+bn1,w.Quasinorm (14) induces a topology on l1,w in which this space is nonseparable. The closure of the set of elements an with only finite number of nonzero terms is a separable subspace in l1,w. It is well known that l1⊂l1,w and (16)an1,w≤anl1(see, e.g., [10] for more details).


## 3. Estimating the Number of Negative Eigenvalues in a Strip

Define (1) via its quadratic form (17)qV,Su≔∫S∇ux2dx-∫SVxux2dx,dom⁡qV,S=W21S∩L2S,Vdx,where W21(S) denotes the standard Sobolev space H1(S). Let N-(qV,S) denote the number of negative eigenvalues of (1) repeated according to their multiplicities. Then N-(qV,S) is given by(18)N-qV,S=supL⁡dimL:qV,Su<0,∀u∈L∖0,where L denotes a linear subspace of dom⁡(qV,S) (see, e.g., [10, Theorem 10.2.3]).Let(19)In≔2n-1,2n,n>0,I0≔-1,1,In≔-2n,-2n-1,n<0,and(20)Gn≔∫In∫0ax1Vxdx,n≠0,G0≔∫I0∫0aVxdx.Furthermore, let (21)Jn≔n,n+1,n∈Z,I≔0,aandDn≔VL1Jn,LBI.Then we have the following result.Theorem 3.
There exist constantsc,C>0 such that(22)N-qV,S≤1+C∑n∈Z,Gn>cGn+∑n∈Z,Dn>cDn,∀V≥0.Let(23)L1≔u∈L2S:ux=ux1and P:L2(S)→L1 be a projection defined by (24)Pvx≔1a∫0avxdx2=Pvx1.Indeed, P is a projection since P2=P. Let L2≔(I-P)L2(S), and then one can show that L2(S)=L1⊕L2. Here and below ⊕ denotes a direct orthogonal sum.Indeed for allv∈L2 we have(25)∫0avxdx2=∫0aI-Pvxdx2=∫0avxdx2-∫0aPvxdx2=∫0avxdx2-∫0aPvx1dx2=∫0avxdx2-∫0avxdx2=0.Now pick w∈L1 and v∈L2, then, (26)v,wL2S=∫Svxwx1¯dx=∫R∫0avxdx2wx1¯dx1=0.Similarly, let(27)H1≔u∈W21S:ux=ux1andH2≔I-PW21S,and then (28)W21S=H1⊕H2.Indeed, for all v∈H1 and all w∈H2 we have (29)v,wW21S=∫Svx1wx¯+vx1x1wx1x¯+vx2x1wx2x¯dx=0.This is so because v,vx1∈L1,w¯,wx1¯∈L2, and vx2=0. To see this note that v(x1) and vx1(x1) do not depend on x2 implying that vx1∈L1. Also, w∈L2⇔∫0aw(x)dx2=0. So, (d/dx1)∫0aw(x)dx2=0⇒∫0awx1(x)dx2=0⇒wx1∈L2. Hence ∫Svx1(x1)wx1(x)¯dx=0.Now for allu∈W21(S), u=v+w,v∈H1,w∈H2 one has (30)∫S∇ux2dx=a∫Rv′x12dx1+∫S∇wx2dx+∫S∇vx1.∇wx¯dx+∫S∇wx.∇vx1¯dx︸=0=a∫Rv′x12dx1+∫S∇wx2dxand (31)∫SVxux2dx=∫SVxvx12dx+∫SVxwx2dx+∫S2VxRev.wdx≤2∫RV~x1vx12dx1+2∫SVxwx2dx,where (32)V~x1=1a∫0aVxdx2.So (33)∫S∇ux2dx-∫SVxux2dx≥∫Rv′x12dx1-2∫RV~x1vx12dx1+∫S∇wx2dx-2∫SVxwx2dx.Hence(34)N-qV,S≤N-q1,2V~,R+N-q2,2V,S,where q1,2V~,R and q2,2V,S denote the restrictions of the form q2V,S to the spaces H1 and H2, respectively.Let(35)Gn≔∫Inx1V~x1dx1,n≠0,G0≔∫I0V~x1dx1.Then similarly to the estimate before (39) in [6] one has(36)N-q1,2V~,R≤1+7.61∑n∈Z,Gn>0.046Gn.In terms of the original potential V(37)Gn=∫Inx1V~x1dx1=∫Inx1∫0aVxdx2dx1=∫In×0,ax1Vxdxand (38)G0=∫I0V~x1dx1=∫I0∫0aVxdx2dx1=∫I0×0,aVxdx.It now remains to find an estimate for N-(q2,2V,S) in (34).LetSn≔Jn×I,n∈Z, where Jn and I are as defined in (21). Then the variational principle (see, e.g., [11, Ch.6, § 2.1, Theorem 4]) implies(39)N-q2,2V,S≤∑n∈ZN-q2,2V,Sn,where (40)q2,2V,Snw≔∫Sn∇wx2dx-2∫SnVxwx2dx,dom⁡q2,2V,Sn=w∈W21Sn∩L2Sn,Vxdx:∫Snwxdx=0.Lemma 4 (cf. [6, Lemma 7.8]).
There existsC1>0 such that(41)N-q2,2V,Sn≤C1VL1Jn,LBI,∀V≥0(see (13)).Proof.
LetI=I, the unit interval. Then it follows from [6, Lemma 7.7] that there is a constant d1>0 such that(42)N-q2,2V,Sn≤d1VL1Jn,LBI+1,∀V≥0.Similarly to (62) in [6] and using the Poincaré inequality (see, e.g., [12, 1.1.11]), there is a constant d2>0 such that (43)2∫SnVxwx2dx≤d2VL1Jn,LBI∫Sn∇wx2dxfor all w∈W21(Sn)∩L2Sn,V(x)dx such that ∫Snw(x)dx=0. If VL1Jn,LB(I)≤1/d2, then N-(q2,2V,Sn)=0. Otherwise (42) implies (44)N-q2,2V,Sn≤C1VL1Jn,LBI,∀V≥0,where C1≔d1+d2. Hence (41) follows by the scaling x2→ax2.Proof of Theorem3Proof.
IfDn<1/C1, then N-(q2,2V,Sn)=0 and one can drop this term from the sum (39). Hence, for any c<1/C1, (39) and Lemma 4 imply that (45)N-q2,2V,S≤C1∑n∈Z,Dn>cDn∀V≥0.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 c and that (22) is strictly sharper. Indeed, let Bn≔VB,Sn. Then it follows from the embedding Lp(Sn)↪LB(Sn) that there is a constant C(p),p>1 such that (46)Bn=VB,Sn≤Cp∫SnVxpdx1/p=CpbnV,where bn(V)≔∫SnV(x)pdx1/p (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 (47)Dn≤CpbnV.Hence(48)N-q2,2V,Sn≤C2bnV,∀V≥0,where C2≔C1C(p). The scaling V↦tV,t>0, allows one to extend the above inequality to an arbitrary V≥0. Thus, for any c<1/C2, (22) implies [3, Theorem 7.9].Next we will discuss different forms of (22).Remark 5.
Note that(49)∑n∈Z,Gn>cGn≤2cGnn∈Z1,w(see (49) in [6]). Estimate (22) implies the following estimate:(50)N-qV,S≤1+C3Gnn∈Z1,w+VL1R,LBI,∀V≥0.This follows from (49) and (51)∑n∈Z,Dn>cDn≤∑n∈ZDn=∫RVx1,.B,Idx1=VL1R,LBI.Equation (50) in turn implies the following:(52)N-qV,S≤1+C4Gnn∈Z1,w+∫R∫IVxpdx21/pdx1,∀V≥0,which is equivalent to(53)N-qV,S≤1+C5Gnn∈Z1,w+∫R∫IV∗xpdx21/pdx1,∀V≥0,where V∗(x)=V(x)-V~(x1),V~(x1)=(1/a)∫0aV(x)dx2.
Indeed,(54)∫R∫IVxpdx21/pdx1-∫R∫IV∗xpdx21/pdx1≤∫R∫IV~x1pdx21/pdx1=a1/p∫RV~x1dx1=a1/p∑n∈Z∫In×0,aVxdx≤a1/p∑n∈Z2-n+1Gn≤constsupn∈ZGn≤constGnn∈Z1,w.Thus (52) and (53) are equivalent.
Similarly,(55)VL1R,LBI-V∗L1R,LBI=∫RVx1,.B,Idx1-∫RV∗x1,.B,Idx1≤∫RV~x1B,Idx1=const∫RV~x1dx1≤constsupn∈ZGn≤constGnn∈Z1,w.Hence (50) is equivalent to the following estimate:(56)N-qV,S≤1+C6Gnn∈Z1,w+V∗L1R,LBI,∀V≥0.Note the last term on the right-hand side of (56) (and (53)) drops out if V does not depend on x2.
Letα>0 be given. It is well known that the lowest possible (semiclassical) rate of growth of N-(qαV,S) is (57)N-qαV,S=Oαasα→+∞(see, e.g., [2, 4]). It turns out that the finiteness of the first term in (22) is necessary for N-(qαV,S)=O(α)asα→+∞ to hold (see the next theorem).Theorem 6.
LetV≥0. If N-(qαV,S)=O(α)asα→+∞, then Gn1,w<∞.Proof.
Consider the function(58)wnx1≔0,x1≤2n-1orx1≥2n+2,4x1-2n-1,2n-1<x1<2n,2n+1,2n≤x1≤2n+1,2n+2-x1,2n+1<x1<2n+2,for n>0. Let un(x)=wn(x1)u1(x2), where u1 is the first eigenfunction of the one-dimensional second-order differential operator on L2((0,a)) which is identically equal to 1. Then we have (59)∫S∇unx2dx=a∫Rwn′x12dx1=a∫2n-12nwn′x12dx1+∫2n2n+1wn′x12dx1+∫2n+12n+2wn′x12dx1=C72n+1,where (60)C7≔5a.Now (61)∫SVxunx2dx≥∫2n2n+122n+1dx1∫0aVxdx2≥∫2n2n+12n+1dx1∫0ax1Vxdx2=1C7∫S∇unx2dxGn.If Gn>C7, then qV,S[un]<0. The auxiliary functions wn can be defined similarly for n≤0. Since un and uk have disjoint supports for |n-k|≥3, then (62)N-qV,S≥13cardn∈Z:Gn>C7(see [6, Theorem 9.1]). If, for some constant C8>0, N-qαV,S≤C8α, then (63)13cardn∈Z:αGn>C7≤C8α,and so (64)cardn∈Z:Gn>C7α≤3C8α.With s=C7/α we have (65)cardn∈Z:Gn>s≤C9s-1,s>0,where C9≔3C7C8.


## 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 potentialV 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 operatorHX,α, with point δ-interactions on a countable set X={xk}k=1∞ of points, called points of interaction, and intensities α={αk}k=1∞, defined by the differential expression -d2/dx2 on functions w(x) that belong to the Sobolev space W22(R∖X) satisfying, in the points of the set X, the following conjugation conditions:(66)wxk+0=wxk-0,w′xk+0-w′xk-0=αkwxk.Since, for each k, (xk,xk+1) is an open interval, then any function in W22((xk,xk+1)) and its derivative have well defined (one-sided) values at the end-points. The operator HX,α has the following representation:(67)HX,α≔-d2dx2+∑k=1∞αkδx-xk,where δ is Dirac’s delta distribution. We shall assume that HX,α is self-adjoint (see, e.g., [15] in case the set X is finite). One can also define operator (67) via its quadratic form as follows:(68)qw≔∫Rw′x2dx-∑k=1∞αkwxk2,∀w∈W21R.Lemma 7.
Given an infinite sequence of positive numbers(αk), there is a sequence of points (xk) in X such that (67) has infinitely many negative eigenvalues.Proof.
Letψ∈C0∞(R) such that (69)ψx=1ifx<12,0ifx≥1.Assume that xk+1-xk>xk-xk-1, for all k=1,2,… and let φk(x)≔ψ(2/(xk-xk-1))(x-xk). Then(70)φkx=1ifx-xk<xk-xk-14,0ifx-xk≥xk-xk-12.Let L be a linear subspace of W21(R) defined by (71)L≔w∈W21R:w=∑k=1∞akφk,ak∈C.Since φk and φj for k≠j have disjoint supports, then dimL=∞. Thus, for all w∈L∖{0}, it follows from (68) that (72)qw=∑k=1∞ak2∫Rφk′x2dx-αk=∑k=1∞ak24xk-xk-12∫Rψ′2xk-xk-1x-xk2dx-αk=∑k=1∞ak22xk-xk-1∫Rψ′t2dt-αk,where t=(2/(xk-xk-1))(x-xk). Take xk such that (73)xk-xk-1>2αk∫Rψ′t2dt,then q[w]<0 and operator (67) has infinitely many negative eigenvalues.Let us now return to operator (1) with V supported by and locally integrable on a Lipschitz curve l embedded in S. Let (74)qV,lu≔∫S∇ux2dx-∫lVxux2dsxdom⁡qV,l=W21S∩L2l,Vds.Let {x1(k)},k∈N be a sequence of points on R satisfying conditions in (66). Define (75)γk≔l∩x1k×0,a,x1∈R,k∈Nand(76)Σ≔x1k:γk>0.Then the set Σ is at most countable. Let I be an arbitrary interval in R and let (77)νI≔∫l∩I×0,aVxdsx.Furthermore, let (78)Fn≔∫Inx1dνx1,n≠0,F0≔∫I0dνx1cf.35,ln≔l∩Sn,Sn≔n,n+1×0,a,n∈N,Cn≔VB,lnav.Theorem 8.
Suppose thatN is the cardinality of Σ. Then there exist constants c1,c2,C10,C11>0 such that(79)N-qV,l≤1+N+C9∑n∈Z,Fn>c1Fn+C10∑n∈Z,Cn>c2Cn,∀V≥0.If Σ is infinite, then N-(qV,l)=∞.Proof.
Letq1,2V,l and q2,2V,l be the restrictions of the form qV,l to the subspaces H1 and H2, respectively (see (27)), and then(80)N-qV,l≤N-q1,2V,l+N-q2,2V,l(cf. (34)). Let us start by estimating the first term on the right-hand side of (80). On the complement of Σ, ν({x1})=0 for all x1∈R. This implies (81)∫lVxux2dsx=∫Rwx12dνx1+wx12∑k∈N∫γkVx2dx2=∫Rwx12dνx1+∑k∈Nckwx12,where ck≔∫γkV(x2)dx2<∞. Hence (82)q1,2V,lu=∫S∇ux2dx-2∫lVxux2dx=∫Rw′x12dx1-2∫Rwx12dνx1-∑k∈Nck′wx12,where ck′≔2ck. Let (83)q1,2νw≔∫Rw′x12dx1-2∫Rwx12dνx1,dom⁡q1,2ν=W21R∩L2R,dν,q2,ck′w≔∫Rw′x12dx1-∑k∈Nck′wx12,dom⁡q2,ck′=W21R.Then, it follows from [3, Lemma 3.6] that(84)N-q1,2V,l≤N-q1,2ν+N-q2,ck′.Similarly to (36) one has(85)N-q1,2ν≤1+7.16∑n∈Z,Fn>0.046Fn.If Σ is finite, then(86)N-q2,ck′≤N(see, e.g., [15]). Otherwise, Lemma 7 implies(87)N-q2,ck′=∞.Now, it remains to estimate the second term on the right-hand side of (80). Let (88)q2,2V,lnu≔∫Sn∇ux2dx-∫lnVxux2dsxdom⁡q2,2V,ln=u∈W21Sn∩L2ln,Vds:∫Snuxdx=0.For any V∈LB(ln),V≥0 and any r∈N, following a similar argument in the proof of [6, Lemma 7.6], one can show that there exists a finite cover of ln by rectangles Snk,k=1,…,r0 such that r0≤r and(89)∫lnVxux2dsx≤C12r-1VB,lnav∫Sn∇ux2dxfor all u∈W21(Sn)∩L2(ln,Vds) with ∫Snku(x)dx=0, where C12 is a constant independent of V. 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 C11>0 such that (90)N-q2,2V,ln≤C11VB,lnav,∀V≥0.If VB,ln(av)<1/C11, then N-(q2,2V,ln)=0. Thus, similarly to (39) one has for any c2<1/C11 that(91)N-q2,2V,l≤C11∑Cn>c2,n∈ZCn,∀V≥0.Hence, the statement of the theorem follows from (80), (84), (85), (86), (87), and (91).



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