We consider the Schr\"odinger operator $$-\frac{d^2}{d x^2} + V \qquad
\mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where
$V$ is bounded from below and prove a lower bound on the first eigenvalue
$\lambda_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\text{ where }
I_y := \left\{ x \in [a,b]: V(x) \leq y \right\},$ then $$ \lambda_1 \geq
\frac{1}{250} \min_{y > \min V}{\left(\frac{1}{w_V(y)^2} + y\right)}.$$ The
result is sharp up to a universal constant if $\left\{ x \in [a,b]: V(x) \leq y
\right\}$ is an interval for the value of $y$ solving the minimization problem.
An immediate application is as follows: let $\Omega \subset \mathbb{R}^2$ be a
convex domain with inradius $\rho$ and diameter $D$ and let $u:\Omega
\rightarrow \mathbb{R}$ be the first eigenfunction of the Laplacian $-\Delta$
on $\Omega$ with Dirichlet boundary conditions on $\partial \Omega$. We prove
$$ \| u \|_{L^{\infty}} \lesssim \frac{1}{\rho^{}} \left( \frac{\rho}{D}
\right)^{1/6} \|u\|_{L^2},$$ which answers a question of van den Berg in the
special case of two dimensions

Georgiev, Bogdan

Mukherjee, Mayukh

Steinerberger, Stefan

English

arXiv

Georgiev, B.

Mukherjee, M.

Steinerberger, S.

MPG.PuRe

arXiv:1612.08565v2  [math.SP]  2 Feb 2017A SPECTRAL GAP ESTIMATE AND APPLICATIONSBOGDAN GEORGIEV, MAYUKH MUKHERJEE, AND STEFAN STEINERBERGERAbstract. We consider the Schro¨dinger operator−d2dx2+ V on an interval [a, b] with Dirichlet boundary conditions,where V is bounded from below and prove a lower bound on the first eigenvalue λ1 in terms ofsublevel estimates: if wV (y) = |Iy|, where Iy := {x ∈ [a, b] : V (x) ≤ y} , thenλ1 ≥1250miny>minV(1wV (y)2+ y).The result is sharp up to a universal constant if {x ∈ [a, b] : V (x) ≤ y} is an interval for the valueof y solving the minimization problem. An immediate application is as follows: let Ω ⊂ R2 bea convex domain with inradius ρ and diameter D and let u : Ω → R be the first eigenfunctionof the Laplacian −∆ on Ω with Dirichlet boundary conditions on ∂Ω. We prove‖u‖L∞ .1ρ( ρD)1/6‖u‖L2 ,which answers a question of van den Berg in the special case of two dimensions.1. Introduction and statement of results1.1. Introduction and Result. We consider one-dimensional Schro¨dinger operators of the form− d2dx2+ V on an interval [a, b] with Dirichlet boundary conditions,where V is assumed to be bounded from below. It is intuitively clear that the smallest eigenvalueλ1 will mainly depend on the minimal value attained by the potential as value as the growth ratearound that minimal value. We make this intuition precise. In the case where all sublevel sets areintervals (i.e. which, for instance, is the case for convex potentials V ), this lower bound is sharpup to a universal multiplicative constant. We define a function w : {y ∈ R : y ≥ min V } → R+ bymeasuring the length of sublevel sets viawV (y) = |Iy|, where Iy := {x ∈ [a, b] : V (x) ≤ y} .We will prove that a lower bound on the first eigenvalue can be given as the solution of a mini-mization problem involving the function w(y).yVFigure 1. wV (y) is the length of the interval {x : V (x) ≤ y}.2010 Mathematics Subject Classification. 35P15, 47A75 (primary) and 35P99 (secondary).Key words and phrases. Ground state, Laplacian eigenfunction, convex domain, Schro¨dinger operator.12 A SPECTRAL GAP ESTIMATE AND APPLICATIONSTheorem 1. If V is bounded from below, then the smallest eigenvalue λ1 of −d2/dx2 + V withDirichlet conditions at the endpoints of the interval satisfiesλ1 ≥ 1250miny>minV(1wV (y)2+ y).Moreover, if the sublevel set {x : V (x) ≤ y∗} is an interval for the value y∗ ∈ R solving theminimization problem, thenλ1 ≤ pi2 miny>minV(1wV (y)2+ y).The upper bound is quite simple and follows from a testing argument that actually implies theslightly sharper resultλ1 ≤ miny>minV(pi2wV (y)2+ y).The constant 1/250 in the lower bound seems far from optimal and it could be of interest to obtainbetter results. It could also be of interest to study higher-dimensional analogues. While we do notknow of any such result in the literature, a special case of the result has been established in workof Jerison [5] on the ground state of the Laplacian in convex domains in R2: the eigenvalue (andthe scale of localization) is determined in terms of a geometric characterization that is equivalentto Theorem 1, details are given at the end of the paper.1.2. An application. Let Ω ⊂ Rn be a convex domain with inradius ρ and diameter D and letu ∈ L2(Ω) be the ground state of the Laplacian −∆ in Ω with Dirichlet boundary conditions on∂Ω. A classical inequality of Chiti [1, 2] states that‖u‖L∞ . 1ρn/2‖u‖L2,where the sharp constant is known and assumed for the ball. A natural question, asked by van denBerg [6], is whether this inequality could be improved when the diameter D gets large – motivatedby explicit computations on cones, he conjectured that‖u‖L∞ . 1ρn/2( ρD)1/6‖u‖L2.The second main result of this paper is to prove the conjecture in two dimensions.Theorem 2. Let Ω ⊂ R2 be convex with diameter D and inradius ρ and let u : Ω → R be theground state of the Laplacian with Dirichlet boundary conditions. Then‖u‖L∞ . 1ρ( ρD)1/6‖u‖L2.Roughly, our proof consists of two ingredients: we use a result of Grieser & Jerison [3] to reduce theproblem to the problem of analyzing the ground state of a Schro¨dinger operator −d2/dx2+V on acompact interval; further, we use Theorem 1 to estimate the first eigenvalue and first eigenfunctionof such an operator.2. Proofs2.1. Proof of Theorem 1.Proof. Assuming that{x : V (x) ≤ y} is an interval,the upper bound is fairly easy and follows immediately from choosing a using suitable test functionin the Rayleigh-Ritz quotient: for every y > minV , definef(x) ={sin(pix−inf Iy|Iy|)if x ∈ Iy = {x : V (x) ≤ y}0 otherwise.A SPECTRAL GAP ESTIMATE AND APPLICATIONS 3A direct computation using the Rayleigh-Ritz quotient showsλ1∫ baf(x)2dx ≤∫ baf ′(x)2 + V (x)f(x)2dx≤ pi2w(y)2∫ baf(x)2dx+ y∫ baf(x)2dx ≤ pi2(1w(y)2+ y)∫ baf(x)2dx.Since this is true for all y, we can take the minimum of the arising expression as an upper boundon the first eigenvalue, which implies the upper bound on the eigenvalue.Let now f∗ denote the symmetrically decreasing rearrangement of f around (a+ b)/2 and let V∗denote the symmetrically increasing rearrangement of V around (a+ b)/2. The Hardy-Littlewoodrearrangement inequality implies∫ baV (x)f(x)2dx ≥∫ baV∗(x)f∗(x)2dxand the Polya-Szego˝ inequality implies∫ baf(x)2dx =∫ baf∗(x)2dx and∫ ba(ddxf∗(x))2dx ≤∫ ba(ddxf(x))2dx.From this, we can infer thatλ1(− d2dx2+ V ∗)≤ λ1(− d2dx2+ V)and will now proceed to bound the smallest eigenvalue of the rearranged potential from below.The argument will keep track of three a priori unspecified real constants α, β, γ (an optimizationover which will then yield an explicit lower bound). Let f denote the L2−normalized ground stateof −∆ + V with Dirichlet conditions at a and b. It is classical (and follows from the ordinarydifferential equation and the unimodal structure of V ) that the ground state is first monotonicallyincreasing and then monotonically decreasing. Let J ⊂ [a, b] be the smallest interval such that∫Jf(x)2dx ≥ α‖f‖2L2,where α ∈ (0, 1) is a constant that will be determined later. We start by observing that thefunction f has the same value on both boundary points of J : if it did not, then continuity off suffices to ensure that we could slightly slide the interval to increase the contained L2−norm,which would then allow shrinking the interval, thus contradicting the minimality of J .We consider the intervalJ∗ =[a+ b2− |J |2,a+ b2+|J |2]and observe that∫Jf(x)2dx =∫J∗f∗(x)2dx = α‖f‖2L2.Clearly, the L2−normalization of f implies that∫[a,b]\J∗f∗(x)2dx = (1− α)‖f‖2L2and therefore ∫ baV∗(x)f∗(x)2dx ≥∫[a,b]\J∗V∗(x)f∗(x)2dx≥ w−1∗ (|J∗|)(1 − α)‖f‖2L2 = (1− α)w−1(|J |)‖f‖2L2.The remainder of the proof deals with the gradient term and proceeds via case distinction: eitherthere is a lot of oscillation on the interval J or there is not. More precisely, for a constant β > 0to be determined later, we are either dealing witheither∫Jf ′(x)2dx ≥ β|J |2∫Jf(x)2dx or∫Jf ′(x)2dx ≤ β|J |2∫Jf(x)2dx.4 A SPECTRAL GAP ESTIMATE AND APPLICATIONSCase 1 (Lots of Oscillation). As it turns out, this case is easy to deal with since we can estimate∫ baf ′(x)2dx ≥∫Jf ′(x)2dx ≥ β|J |2∫Jf(x)2dx =αβ|J |2 ‖f‖2L2.Altogether, this means that we have∫ baf ′(x)2 + V (x)f(x)2dx ≥(αβ|J |2 + (1 − α)w−1(|J |))‖f‖2L2≥ min(αβ, 1 − α)(1|J |2 + w−1(|J |))‖f‖2L2.However, a change of variables immediately shows that(1|J |2 + w−1(|J |))≥ miny>minV(1w(y)2+ y),which is the desired statement with a constant min(αβ, 1 − α).Case 2 (Almost flat). The remaining case is where f has relatively little oscillation on J∫Jf(x)2dx = α‖f‖2L2 and∫Jf ′(x)2dx ≤ β|J |2∫Jf(x)2dx.The remainder of the argument is as follows: we will show that the function has to decay outsideof J with at least a certain speed (otherwise, if the function were to remain close to constant,it would eventually violate the L2−normalization of f). As proved above, f has the same valueat the two endpoints of J , which we will denote by ε := f(∂J). We start by showing that thisnumber cannot be too small and consider g = f − ε on J . Using the Cauchy-Schwarz inequality,we see that ∫Jg(x)2dx =∫J(f(x)− ε)2dx ≥∫Jf(x)2dx− 2ε∫Jf(x)dx≥ α− 2ε|J |1/2(∫Jf(x)2dx)1/2≥ α− 2ε|J |1/2α1/2.At the same time, we have that∫Jg′(x)2dx =∫Jf ′(x)2dx ≤ β|J |2∫Jf(x)2dx =αβ|J |2 .The classical Poincare´ inequality for the Dirichlet-Laplacian on an interval implies that∫Jg′(x)2dx ≥ pi2|J |2∫Jg(x)2dxand thereforeαβ|J |2 ≥pi2|J |2 (α− 2ε|J |1/2α1/2)and after rearrangementε ≥ α1/22|J |1/2(1− βpi2).We now show that f has to decay at least with a certain speed outside of J , otherwise there wouldbe too much L2−mass outside of J . We fix another positive constant γ and look at the values off at points, whose distance to J is γ|J |. Assuming that J = [j1, j2] and using the monotonicityof f , we have1 = ‖f‖2L2([a,b]) ≥∫ j2+γ|J|j1f(x)2dx ≥ (1 + γ)|J |f(j2 + γ|J |)2A SPECTRAL GAP ESTIMATE AND APPLICATIONS 5and thusf(x) ≤ 1(1 + γ)1/2|J |1/2 for all x at distance at least γ|J | from the interval J.Using Euler-Lagrange equations it is a basic exercise to showinf{∫ dch′(x)2dx∣∣ h(c) = C, h(d) = D}=(D − C)2d− cbecause the minimizer is simply the linear function with appropriate values at the endpoints.For our problem, this implies that∫ baf ′(x)2dx ≥∫ j2+γ|J|j2f ′(x)2dx≥∫ j2+γ|J|j2h′(x)2dx ≥ 1γ|J | (ε− f(j2 + γ|J |))2≥ 1γ|J |2[α1/22(1− βpi2)− 1(1 + γ)1/2]2=:1|J |2F (α, β, γ),where we have to assume thatα1/22(1− βpi2)≥ 1(1 + γ)1/2.We can argue as in Case 1 and obtain that∫ baf ′(x)2 + V (x)f(x)2dx ≥(1|J |2F (α, β, γ) + (1 − α)w−1(|J |))‖f‖2L2≥ min (F (α, β, γ) , (1− α))(1|J |2 + w−1(|J |))‖f‖2L2,which is the desired result.The numerical constant. We conclude by untangling the relationship between the implicitconstants. Altogether, we have the following system of constraintsα ∈ (0, 1), β ∈ (0, pi2), γ ≥ 4α(1− βpi2)−2− 1.and are trying to estimatemaxα,β,γmin{1γ[α1/22(1− βpi2)− 1(1 + γ)1/2]2, 1− α, αβ}.The lower bound 1/250 follows from settingα =99100, β =71000and γ = 14.1327.L∞−bounds. We finally establish bounds on the L∞−norm of the eigenfunction in the casewhere the potential is nonnegative V ≥ 0, which is an easy combination of the Cauchy-Schwarzinequality and the fact that the ground state vanishes on the endpoints. More precisely, for everya ≤ x ≤ b we distinguish between the cases∫ xaf(y)2dy ≤ 12or∫ bxf(y)2dy ≤ 12.6 A SPECTRAL GAP ESTIMATE AND APPLICATIONSIn the first case, we estimatef(x)2 = f(x)2 − f(a)2 =∫ xa(ddyf(y)2)dx =∫ xa2f(y)f ′(y)dy≤ 2(∫ xaf(y)2dy)1/2 (∫ xaf ′(y)2dy)1/2≤√2(∫ baf ′(x)2 + V (x)f(x)2dx)1/2≤√2λ1.In the second case, we change signs and arguef(x)2 = f(x)2 − f(b)2 = −∫ bx(ddyf(y)2)dx = −∫ bx2f(y)f ′(y)dy≤ 2(∫ bxf(y)2dy)1/2(∫ bxf ′(y)2dy)1/2≤√2(∫ baf ′(x)2 + V (x)f(x)2dx)1/2≤√2λ1and therefore‖f‖L∞ ≤ (2λ1)1/4.2.2. Proof of Theorem 2.Proof. We begin by giving a short overview of the overall idea. We can use scaling to restrictourselves to convex domains with inradius ρ = 1. We first recall Theorem 1.6 of [3]: assume werotate the convex domain Ω so that the projection onto the y-axis has least length and dilate (ifnecessary) so that this length is 1. The boundary of Ω can then be written as the union of thegraphs of two functions f1(x) ≤ f2(x) on [a, b], which satisfy0 ≤ f1(x) ≤ f2(x) ≤ 1 for a ≤ x ≤ b,mina≤x≤bf1(x) = 0, maxa≤x≤bf2(x) = 1.Their result then states that the profile of the eigenfunction u on Ω is essentially given by theground state φ of the Schro¨dinger operator− d2dx2+pi2h(x)2, where h(x) := f2(x) − f1(x).Formally, let L be the length of the longest interval I ⊂ [a, b] such thath(x) ≥ 1− 1L2,on I and letα(x, y) = piy − f1(x)h(x).Theorem 3 ([3], Theorem 1.6). Normalize u and φ, so that max u = maxφ = 1.Then there is an absolute constant C such that|u(x, y)− φ(x) sinα(x, y)| ≤ CL,for all x ∈ I ′ where I ′ is the interval concentric with I of half the length.A SPECTRAL GAP ESTIMATE AND APPLICATIONS 7Our proof will now proceed as follows: we first study exclusively the ground states of Schro¨dingeroperators that can possibly arise from convex domains and establish some basic properties. Af-terwards, we use the above result to transfer the results to the profile of the eigenfunction on theconvex domain.Step 1 (Bound on the ground state).We make use of Theorem 2 and replace our study of the first eigenvalue of the Schro¨dinger operatorassociated to the domain with the study ofminy>minV(1wV (y)2+ y).It is easy to see that a larger potential in the Schro¨dinger operator V1 ≥ V2 leads to shorter sublevelswV1(y) ≤ wV2 (y), which can only increase the minimal value in the minimization problem. Everyconvex body of inradius 1 and diameter D contains a disk of radius 1 and a point at distance∼ D/2, which gives rise to a cone by convexity – this cone is universal in the sense that it isstrictly contained in every other convex domain with inradius 1 and diameter D and its potentialtherefore dominates all other potentials.Figure 2. A description of the object contained in every convex set with inradius1 and diameter D: a ball with inradius 1 and a cone with height (D − 1)/2.A simple computation shows that in this case the relevant potential is, up to universal constants,V ∼ D2(D − x)2 on [0, D].We now use the fact that adding or subtracting constants to the potential does not change theground state and replace the potential under consideration byV ∼ D2(D − x)2 − 1and apply Theorem 2 to this potential instead. We then obtainwV (y) = D(1− 1√1 + y)and thus (1wV (y)2+ y)∼ D−2(1− 1√1 + y)−2+ y ∼ 1D2y2+ ywhich is minimized for y ∼ D−2/3 and gives the upper bound λ1 . D−2/3. A classical L∞−bound,proven for convenience of the reader at the end of the paper, gives‖φ‖L∞ . λ1/41 ‖φ‖L2.This gives the desired result ‖φ‖L∞ . D−1/6‖φ‖L2 for the ‘profile’-eigenfunction of the Schro¨dingeroperator. The second part of Theorem 2 implies the following: if we take J to be the shortestinterval containing half of the L2−mass of φ, then|J | & 1√λ1.8 A SPECTRAL GAP ESTIMATE AND APPLICATIONSThese two facts imply that φ is essentially constant on an interval of length L ∼ 1/√λ1 and thata constant proportion of L2−mass is contained on that interval.Step 2 (Using Grieser-Jerison). A simple computation (mentioned in [3], see also [5, Lemma 2.4])shows that the comparison is accurate at least on length scale ∼ λ−1/21 around the maximum.Suppose now that the functions u1, φ1 are rescaled from the eigenfunctions in such a way thatmaxu1 = maxφ1 = 1and that|u1(x, y)− φ1(x) sinα(x, y)| ≤ CL.We know from the first part that φ is essentially constant on scale & λ−1/21 , that ‖φ‖L∞ . λ1/41and that a constant proportion of the L2−norm is there. This implies for the rescaled function φ1that ∫Jφ1(x)2dx &∫Jλ−1/21 φ(x)2dx ∼ λ−1/21 ,where J is the interval of length λ−1/21 on which φ is essentially constant. This, however, implies∫Ωu1(x, y)2dxdy & λ−1/21and therefore‖u1‖2L∞ = 1 . λ1/21 ‖u1‖2L2 . D−1/3‖u1‖2L2which is the desired result. We emphasize a useful connection between the estimateλ1 ∼ miny>minV(1wV (y)2+ y)and the length L of the longest interval I ⊂ [a, b] such thath(x) ≥ 1− 1L2in the Grieser-Jerison Schro¨dinger operator [3, 5]. Note thatL ≤∣∣∣∣{x :1h(x)2− 1 ≤ 1L2}∣∣∣∣ and thus wV (L−2) ∼ L.As a consequence, for y = L−2, we observe that1wV (y)2=1wV (L−2)2∼ L−2 = ywhich corresponds to a balancing of terms in the functional and immediately yields that λ1 ∼ L−2.Acknowledgements. The first and second authors gratefully acknowledge the Max Planck Insti-tute for Mathematics, Bonn for providing ideal working conditions. The third author was partiallysupported by an AMS Simons Travel grant and INET Grant #INO15-00038References[1] G. Chiti, A reverse Ho¨lder inequality for the eigenfunctions of linear second order elliptic operators, J. AppliedMathematics and Physics, 33 (1982), 143 - 148.[2] G, Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un.Mat. Ital. A, 1 (1982), 145 - 151.[3] D. Grieser and D. Jerison, The size of the first eigenfunction of a convex planar domain, J. Amer. Math. Soc.,11 (1998), no. 1, 41- 72.[4] D. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013),no. 4, 601 - 667.[5] D. Jerison, The diameter of the first nodal line of a convex domain, Ann. of Math. (2), 141 (1995), no. 1, 1 -33.A SPECTRAL GAP ESTIMATE AND APPLICATIONS 9[6] M. van den Berg, On the L∞−Norm of the First Eigenfunction of the Dirichlet Laplacian, Potential Analysis13 (2000), 361- 366.Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, GermanyE-mail address: bogeor@mpim-bonn.mpg.deMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, GermanyE-mail address: mukherjee@mpim-bonn.mpg.deDepartment of Mathematics, Yale University, 06511 New Haven, CT, USAE-mail address: stefan.steinerberger@yale.edu

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