# Spectral Bounds for Polydiagonal Jacobi Matrix Operators

Authors: Arman Sahovic
Journal: International Journal of Partial Differential Equations
 (2014)

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## Abstract

The research on spectral inequalities for discrete Schrödinger operators has proved fruitful in the last decade. Indeed, several authors analysed the operator’s canonical relation to a tridiagonal Jacobi matrix operator. In this paper, we consider a generalisation of this relation with regard to connecting higher order Schrödinger-type operators with symmetric matrix operators with arbitrarily many nonzero diagonals above and below the main diagonal. We thus obtain spectral bounds for such matrices, similar in nature to the Lieb-Thirring inequalities.


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## Body

## 1. Background

LetW be the self-adjoint Jacobi matrix operator acting on l2(Z) as follows:
(1)W=(⋱⋮⋮⋮⋮⋰…b-1a-100……a-1b0a00……0a0b1a1……00a1b2…⋰⋮⋮⋮⋮⋱),
via
(2)(Wφ)(n)=an-1φ(n-1)+bnφ(n)+anφ(n+1),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhforn∈Z,
where an>0 and bn∈R. This operator can be viewed as the one-dimensional discrete Schrödinger operator if an=1 for all n. A variety of papers examined such operators; for example, we quote the work by Killip and Simon in [1], where they obtained sum rules for such Jacobi matrices. Additionally, Hundertmark and Simon in [2] were able to find spectral bounds for these operators. We thus state their result.Ifan→1, bn→0 rapidly enough, as n→±∞, the essential spectrum σess(W) of W is absolutely continuous and coincides with the interval [-2,2] (see, e.g., [3]). Besides, W may have simple eigenvalues {Ej±}j=1N± where N±∈N¯(∶=N∪{∞}), and
(3)E1+>E2+>⋯>2>-2>⋯>E2->E1-.
Indeed, in [2] the authors found the following.Theorem 1.
If{bn}n∈Z, {an-1}n∈Z∈lγ+1/2(Z), γ≥1/2, then
(4)∑j=1N+‍|Ej+-2|γ+∑j=1N-‍|Ej-+2|γ≤kγ[∑n=-∞∞‍|bn|γ+1/2+4∑n=-∞∞‍|an-1|γ+1/2],
where
(5)kγ=2(3γ-1/2)Lγ,1cl,Lγ,1cl=Γ(γ+1)2πΓ(γ+3/2).The author (see [4]) then improved their result, achieving the smaller constant: kγ=3γ-1πLγ,1cl, by translating a well-known method employed by Dolbeaut et al. in [5] to the discrete scenario. They, in turn, used a simple argument by Eden and Foias (see [6]) to obtain improved constants for Lieb-Thirring inequalities in one dimension.The aim of this paper is to answer the natural question of whether these methods can be generalised to give bounds for higher order Schrödinger-type operators and thus “polydiagonal” Jacobi-type matrix operators, which we will define below.


## 2. Notation and Preliminary Material

For a sequence{φ(n)}n∈Z, let D and D* be the difference operator and its adjoint, respectively, denoted by Dφ(n)=φ(n+1)-φ(n) and D*φ(n)=φ(n-1)-φ(n). We then denote the discrete one-dimensional Laplacian by ΔD:=(D*Dφ)(n)=-φ(n+1)+2φ(n)-φ(n-1). For σ∈N, n∈Z, and a sequence φ∈l2(Z), with ΔD1=ΔD, we define ΔDσ by
(6)(ΔDσφ)(n):=(ΔD(ΔDσ-1φ))(n).
We note that ΔD being self-adjoint immediately implies that ΔDσ is also self-adjoint.Finding an explicit formula forΔDσ requires a few combinatorial techniques, all of which are standard. Let aCb:=(ab):=a!/((a-b)!b!), for a,b∈N. Then we have the following: (i) Cba+Cb+1a=Cb+1a+1, (ii) 2C0a+C1a=C1a+2, and (iii) 2Caa+Ca-1a=Ca+1a+2.A simple induction argument then delivers our formula for the σth order discrete Laplacian operator as follows:(7)(ΔDσφ)(n)=∑k=02σ‍Ck2σ(-1)k+σφ(n-σ+k).
Furthermore, in order to identify our essential spectrum, we apply the discrete fourier transform as follows:
(8)F(ΔDσφ)(x)=∑n∈Z‍einx(∑k=02σ‍Ck2σ(-1)k+σφ(n-σ+k)),
which, after some rearrangement, yields
(9)F(ΔDσφ)(x)=[Cσ2σ+2∑k=0σ-1‍2σCk(-1)k+σcos⁡((σ-k)x)]×(Fφ)(x).
The essential spectrum of the operator ΔDσ will thus be the range of the above symbol, which can be found to be ςess(ΔDσ)=[0,4σ].


## 3. Main Results

We now let{ψj}j=1N, N∈N¯, be the orthonormal system of eigensequences in l2(Z) corresponding to the negative eigenvalues {ej}j=1N of the (2σ)th order discrete Schrödinger-type operator as follows:
(10)(HDσψj)(n):=(ΔDσψj)(n)-bnψj(n)=ejψj(n),
where j∈{1,…,N} and we assume that bn≥0 for all n∈Z. Our next result is concerned with estimating those negative eigenvalues.Theorem 2.
Letbn≥0, {bn}n∈Z∈lγ+1/2σ(Z), γ≥1. Then the negative eigenvalues {ej}j=1N of the operator HDσ satisfy the inequality
(11)∑j=1N‍|ej|γ≤ησγ∑n∈Z‍bnγ+1/2σ,
where
(12)ησγ:=2σ(2σ+1)(2σ+1)/2σΓ((4σ+1)/2σ)Γ(γ+1)Γ(γ+(2σ+1)/2σ).Remark 3.
As the discrete spectrum ofHDσ lies in [-∞,0] and [4σ,∞], we shift our operator to the left by 4σ and by analogy have an estimate for the positive eigenvalues of that operator, thus immediately obtaining Corollary 4.Corollary 4.
Letbn≥0, {bn}n∈Z∈lγ+1/2σ(Z), γ≥1. Then the positive eigenvalues {ej}j=1N of the operator ΔDσ-4σ+b satisfy the inequality:
(13)∑j=1N‍ejγ≤ησγ∑n∈Z‍bnγ+1/2σ,withησγgivenabove.Finally we will apply these results to obtain spectral bounds for the following operator.We letWσ be a polydiagonal self-adjoint Jacobi-type matrix operator as follows:
(14)Wσ:=(⋱⋱⋱⋱⋱⋱⋱⋱⋰⋱⋱⋱⋱⋱a-2σ00⋱⋱⋱b-1a-11⋱⋱a-1σ0⋱⋱⋱a-11b0a01⋱⋱a0σ⋱⋱⋱⋱a01b1a11⋱⋱⋱⋱a-2σ⋱⋱a11b2a21⋱⋱⋱0a-1σ⋱⋱a21b3⋱⋱⋱00a0σ⋱⋱⋱⋱⋱⋰⋱⋱⋱⋱⋱⋱⋱⋱),
viewed as an operator acting on l2(Z) as follows: for n∈Z, i∈{1,…,σ},
(15)(Wσφ)(n)=∑i=1σ‍an-iiφ(n-i)+bnφ(n)+∑i=1σ‍aniφ(n+i)=an-σσφ(n-σ)+⋯+an-11φ(n-1)+bnφ(n)+an1φ(n+1)+⋯+anσφ(n+σ),
where ani, bn∈R, for all i∈{1,…,σ}. We denote (Wσ({an1},…,{anσ},{bn})φ)(n):=(Wσφ)(n) where we understand {·} to mean {·}n∈Z. We are then interested in perturbations of the following special case:
(16)(Wσ0φ)(n):=(Wσ({an1≡ω1},…,{anσ≡ωσ},hhhhWσ({an1≡ω1},…,{anσ≡ωσ},{bn≡0})φ)(n),
where ωi≔Cσ+i2σ(-1)i, and explicitly
(17)(Wσ0φ)(n)=((ΔDσ-Cσ2σ)φ)(n)=∑k=0,k≠σ2σ‍Ck2σ(-1)k+σφ(n-σ+k),
called the free Jacobi-type matrix of order σ. In particular, we examine the case where Wσ-Wσ0 is compact. Thus in what follows we assume that our sequences tend to the operator coefficients rapidly enough; that is, ani→ωi, bn→0, as n→±∞. Then the essential spectrum ςess is given by ςess(Wσ)=ςess(Wσ0)=[-Cσ2σ,4σ-Cσ2σ] and Wσ may have simple eigenvalues {Ej±}j=1N± where N±∈N¯, and
(18)E1+>E2+>⋯>4σ-Cσ2σ>-Cσ2σ>⋯>E2->E1-.Theorem 5.
Letγ≥1, {bn}n∈Z, and {ani-ωi}n∈Z∈lγ+1/2σ(Z) for all i∈{1,…,σ}. Then for the eigenvalues {Ej±}j=1N± of the operator Wσ we have
(19)∑j=1N-‍|Ej-+Cσ2σ|γ+∑j=1N+‍|Ej+-(4σ-Cσ2σ)|γ≤νσγ(∑n∈Z‍|bn|γ+1/2σ+4∑n∈Z‍∑k=1σ‍|ank-ωk|γ+1/2σ),
where
(20)νσγ=2σ(2σ+1)γ-2Γ((4σ+1)/2σ)Γ(γ+1)Γ(γ+(2σ+1)/2σ).


## 4. Auxiliary Results

We require the following discrete Kolmogorov-type inequality.Lemma 6.
For a sequenceφ∈l2(Z), and for n>k≥1, we have the following inequality:
(21)∥Dkφ∥l2(Z)≤∥φ∥l2(Z)1-k/n∥Dnφ∥l2(Z)k/n.Proof.
We proceed by induction, where we note that the initial case,k=1, n=2, holds true as the inequality
(22)∥Dφ∥l2(Z)≤∥φ∥l2(Z)1/2∥D2φ∥l2(Z)1/2
is in fact the simple inequality found by Copson in [7]. This case in turn, if used repeatedly, shows that the inequality holds true for all k, if n=k+1. We then take the inductive step on the variable n. Hence we assume that we have the required inequality for k<n≤m, given a fixed k, and proceed to prove the statement for n=m+1. Thus
(23)∥Dmφ∥l2(Z)2=〈Dmφ,Dmφ〉=〈D*Dmφ,Dm-1φ〉≤∥Dm+1φ∥l2(Z)∥Dm-1φ∥l2(Z).
We thus apply our induction hypothesis and set k=m-1 and n=m as follows:
(24)∥Dmφ∥l2(Z)2≤∥Dm+1φ∥l2(Z)∥Dmφ∥l2(Z)(m-1)/m∥φ∥l2(Z)1/m⟹∥Dmφ∥l2(Z)≤∥Dm+1φ∥l2(Z)m/(m+1)∥φ∥l2(Z)1/(m+1).
We now return to the induction hypothesis as follows:
(25)∥Dkφ∥l2(Z)≤∥Dmφ∥l2(Z)k/m∥φ∥l2(Z)(m-k)/m≤∥Dm+1φ∥l2(Z)k/(m+1)∥φ∥l2(Z)k/m(m+1)∥φ∥l2(Z)(m-k)/m=∥Dm+1φ∥l2(Z)k/(m+1)∥φ∥l2(Z)1-k/(m+1).We are now equipped to prove an Agmon-Kolmogorov-type inequality.Proposition 7.
For a sequenceφ∈l2(Z), we have for any σ∈N(26)∥φ∥l∞(Z)≤∥φ∥l2(Z)1-1/2σ∥Dσφ∥l2(Z)1/2σ.Proof.
First we use Lemma6 with k=1, n=σ as follows:
(27)∥Dφ∥l2(Z)≤∥φ∥l2(Z)1-1/σ∥Dσφ∥l2(Z)1/σ,
and we apply this estimate to the well-known discrete Agmon inequality (see [4]):
(28)|φ(n)|2≤∥φ∥l2(Z)∥Dφ∥l2(Z)≤∥φ∥l2(Z)∥φ∥l2(Z)1-1/σ∥Dσφ∥l2(Z)1/σ=∥φ∥l2(Z)2-1/σ∥Dσφ∥l2(Z)1/σ.Proposition 8.
Let{ψj}j=1N be an orthonormal system of sequences in l2(Z); that is, 〈ψj,ψk〉=δjk, and let ρ(n):=∑j=1N‍|ψj(n)|2. Then
(29)∑n∈Z‍ρ2σ+1(n)≤∑j=1N‍∑n∈Z‍|Dσψj(n)|2.Proof.
Letξ=(ξ1,ξ2,…,ξN)∈CN. By Proposition 7, we have
(30)|∑j=1N‍ξjψj(n)|2≤∥∑j=1N‍ξjψj∥l2(Z)(2σ-1)/σ∥Dσ∑j=1N‍ξjψj∥l2(Z)1/σ=(∑j,k=1N‍ξjξ¯k〈ψj,ψk〉)(2σ-1)/2σ×(∑j,k=1N‍ξjξ¯k〈Dσψj,Dσψk〉)1/2σ≤(∑j=1N‍|ξj|2)(2σ-1)/2σ×(∑j,k=1N‍ξjξ¯k〈Dσψj,Dσψk〉)1/2σ.
Let ξj:=ψj(n)¯ and as ρ(n)=∑j=1N‍|ψj(n)|2,
(31)ρ2(n)≤ρ(2σ-1)/2σ(n)×(∑j,k=1N‍ψj(n)ψk(n)¯〈Dσψj,Dσψk〉)1/2σ⟹ρ2σ+1(n)≤∑j,k=1N‍ψj(n)ψk(n)¯〈Dσψj,Dσψk〉⟹∑n∈Z‍ρ2σ+1(n)≤∑j=1N‍(∑n∈Z‍|Dσψj(n)|2).


## 5. Proof of Theorem2

We take the inner product withψj(n) on (10) and sum both sides of the equation with respect to j. We obtain
(32)∑j=1N‍ej=∑j=1N‍(∑n∈Z‍|Dσψj(n)|2)-∑j=1N‍(∑n∈Z‍bn|ψj(n)|2).
We now use Proposition 8 and apply the appropriate Hölder’s inequality; that is,
(33)∑j=1N‍ej≥∑n∈Z‍(∑j=1N‍|ψj(n)|2)2σ+1-(∑n∈Z‍bn(2σ+1)/2σ)2σ/(2σ+1)×(∑n∈Z‍(∑j=1N‍|ψj(n)|2)2σ+1)1/(2σ+1).
We define
(34)χ:=(∑n∈Z‍(∑j=1N‍|ψj(n)|2)2σ+1)1/(2σ+1),ς:=(∑n∈Z‍bn(2σ+1)/2σ)2σ/(2σ+1).
The latter inequality can be written as
(35)χ2σ+1-ςχ≤∑j=1N‍ej.
The LHS is maximal when
(36)χ=1(2σ+1)1/2σ(∑n∈Z‍bn(2σ+1)/2σ)1/(2σ+1).
Substituting this into (33), we obtain
(37)∑j=1N‍ej≥(1(2σ+1)(2σ+1)/2σ∑n∈Z‍bn(2σ+1)/2σ)-1(2σ+1)1/2σ∑n∈Z‍bn(2σ+1)/2σ.=-2σ(2σ+1)(2σ+1)/2σ∑n∈Z‍bn(2σ+1)/2σ.
Therefore,
(38)∑j=1N‍|ej|≤2σ(2σ+1)(2σ+1)/2σ∑n∈Z‍bn(2σ+1)/2σ.
We lift this bound now with regard to moments by using the standard Aizenman-Lieb procedure (see [8]). We let {ej(τ)}j=1N be the negative eigenvalues of the operator ΔDσ-(bn-τ)+. By the variational principle for the negative eigenvalues {-(|ej|-τ)+}j=1N of the operator ΔDσ-(bn-τ) we have
(39)(|ej|-τ)+≤|ej(τ)|.
By this estimate, we find that
(40)∑j=1N‍|ej|γ=1B(γ-1,2)∫0∞‍τγ-2∑j=1N‍(|ej|-τ)+dτ≤1B(γ-1,2)∫0∞‍τγ-2∑j=1N‍|ej(τ)|dτ≤2σ(2σ+1)(2σ+1)/2σ1B(γ-1,2)×∫0∞‍τγ-2∑n∈Z‍(bn-τ)+(2σ+1)/2σdτ,
by (38) above, where B(x,y)=Γ(x)Γ(y)/Γ(x+y) is the well-known Beta function. Thus, after a change of variable,
(41)∑j=1N‍|ej|γ≤2σ(2σ+1)(2σ+1)/2σ×Γ((4σ+1)/2σ)Γ(γ+1)Γ(γ+(2σ+1)/2σ)∑n∈Z‍bnγ+1/2σ,
completing our proof.


## 6. Proof of Theorem5

We have the following matrix bounds for square,m×m matrices, as given in [2]. For anm, ωm∈R, we have
(42)(-|anm-ωm|0…0ωm00…00⋮⋮⋱⋮⋮00…00ωm0…0-|anm-ωm|)≤(00…0anm00…00⋮⋮⋱⋮⋮00…00anm0…00)≤(|anm-ωm|0…0ωm00…00⋮⋮⋱⋮⋮00…00ωm0…0|anm-ωm|).
We thus use this on each block of indices of Wσ as follows:
(43)Wσ({an1≡ω1},…,{anσ≡ωσ},{bn(-)})≤Wσ({an1},…,{anσ},{bn})≤Wσ({an1≡ω1},…,{anσ≡ωσ},{bn(+)}),
where bn(±) is given by
(44)bn(±)=bn±((|an-11-ω1|+|an1-ω1|)hhhhh(|an-11-ω1|+|an1-ω1|)+⋯+(|an-σσ-ωσ|+|anσ-ωσ|));
that is,
(45)bn(±)=bn±(∑k=1σ‍|an-kk-ωk|+|ank-ωk|).
Now we relate these to our Schrödinger-type operators:
(46)ΔDσ-4σ+bn=Wσ0-(4σ-Cσ2σ)+bn=Wσ({an1≡ω1},…,{anσ≡ωσ},hhhhh{bn-(4σ-Cσ2σ)}),(47)ΔDσ+bn=Wσ0+Cσ2σ+bn=Wσ({an1≡ω1},…,{anσ≡ωσ},{bn+Cσ2σ}hhhhh{bn+Cσ2σ}).
Now (Ej+-(4σ-2σCσ)) are positive eigenvalues of Wσ({an1},…,{anσ},{bn-(4σ-Cσ2σ)}). Thus by using (43) and the variational principle, we have
(48)Wσ({an1},…,{anσ},{bn-(4σ-Cσ2σ)})≤Wσ({an1≡ω1},…,{anσ≡ωσ},{bn(+)-(4σ-Cσ2σ)}),⟹|Ej+-(4σ-Cσ2σ)|≤ej+,
where ej+ are the positive eigenvalues of
(49)Wσ({an1≡ω1},…,{anσ≡ωσ},{bn(+)-(4σ-Cσ2σ)})=ΔDσ-4σ+bn(+).
Let us now define (bn)+:=max⁡(bn,0), (bn)-:=-min⁡(bn,0). Then, by Corollary 4 for the positive eigenvalues of our operator, we have
(50)∑j=1N+‍(ej+)γ≤ησγ∑n∈Z‍(bn(+))+γ+1/2σ.
Thus, applying (48),
(51)∑j=1N+‍|Ej+-(4σ-Cσ2σ)|γ≤ησγ∑n∈Z‍((bn)++∑k=1σ‍(|an-kk-ωk|+|ank-ωk|))γ+1/2σ,
where
(52)ησγ:=2σ(2σ+1)(2σ+1)/2σΓ((4σ+1)/2σ)Γ(γ+1)Γ(γ+(2σ+1)/2σ).
Similarly, using Theorem 2 on (47),
(53)∑j=1N-‍|Ej-+Cσ2σ|γ≤ησγ∑n∈Z‍((bn)-+∑k=1σ‍(|an-kk-ωk|+|ank-ωk|))γ+1/2σ.
Using the following application of Jensen’s inequality, that is, for i∈{1,…,2σ+1}, let αi,q∈R, with q≥1,
(54)(∑i=12σ+1‍αi)q≤(2σ+1)q-1(∑i=12σ+1‍αiq),
to each of (51) and (53), we have
(55)((bn)±+∑k=1σ‍(|an-kk-ωk|+|ank-ωk|))γ+1/2σ≤(2σ+1)γ-(2σ-1)/2σ×((bn)±γ+1/2σ+∑k=1σ‍(|an-kk-ωk|γ+1/2σ(bn)±γ+1/2σ+∑k=1σ‍(|an-kk-ωk|γ+1/2σ+|ank-ωk|γ+1/2σ)).
Summing these two inequalities, we arrive at
(56)∑j=1N-‍|Ej-+Cσ2σ|γ+∑j=1N+‍|Ej+-(4σ-Cσ2σ)|γ≤νσγ(∑n∈Z‍|bn|γ+1/2σ+4∑n∈Z‍∑k=1σ‍|ank-ωk|γ+1/2σ),
where
(57)νσγ=2σ(2σ+1)γ-2Γ((4σ+1)/2σ)Γ(γ+1)Γ(γ+(2σ+1)/2σ),
and the proof of Theorem 5 is complete.



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