2021 UNC Greensboro PDE Conference, Electronic Journal of Differential Equations, Conference 26 (2022), pp. 1–11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LOWER BOUNDS ON THE FUNDAMENTAL SPECTRAL GAP WITH ROBIN BOUNDARY CONDITIONS MOHAMMED AHRAMI, ZAKARIA EL ALLALI Abstract. This article investigates the gap between the first two eigenvalues of Schrödinger operators on an interval subjected to the Robin and Neumann boundary conditions for a class of linear convex potentials. Furthermore, when the potential is constant the gap is minimized. Meanwhile, we establish a link between the first eigenvalues and the real roots of the first derivative of the Airy functions Ai′ and Bi′. 1. Introduction We consider the low-lying eigenvalues λ of a self-adjoint Schrödinger problem on an interval Hu := −d 2u dx2 + q(x)u = λu, x ∈ [0, π]. (1.1) The restriction to an interval of length π is merely a convenient normalization. General bounds on the gap between the first two eigenvalues Γ := λ2 − λ1 of Schrödinger operators have gained considerable attention during the last decades. The quantity Γ has various physical and mathematical applications. For exam- ple, in quantum mechanics, the fundamental gap represents the energy needed to achieve the first excited state from the ground state of the particle described by the considered Schrödinger operators, the so called excitation energy. The fundamen- tal gap is also of importance in quantum field theory and statistical mechanics. To give an example, Van den Berg in [8] applied gap results of the Laplace operator to provide sufficient conditions for a free boson gas to fill the ground state alone under the thermodynamic limit macroscopically. In numerical mathematics, the fundamental gap is used to determine the conver- gence rate of numerical computation methods, such as discretization of the finite element method, which involves the approximation of differential operators by ma- trices. In this setting, the difference between the first two eigenvalues represents the ability to determine the first eigenvalue and eigenvector. Probabilistically, the fundamental gap controls the asymptotic exponential rate of convergence to equi- librium for the associated Markovian semigroup of the considered Schrödinger op- erator, and its related to the log-Sobolev constant (see [7, 20]). 2010 Mathematics Subject Classification. 34B05, 34L15, 34L40. Key words and phrases. Fundamental gap spectral; Schrödinger operators; convex potential; Robin and Neumann boundary conditions; Airy functions. ©2022 This work is licensed under a CC BY 4.0 license. Published August 25, 2022. 1 2 MOHAMMED AHRAMI, ZAKARIA EL ALLALI EJDE-2022/CONF/26 or the problem (1.1) with Dirichlet boundary conditions, Ashbaugh and Ben- guria [5], established that the optimal lower bound for Γ for symmetric single-well potentials is reached if and only if q is constant on (0, π). Lavine [17] investigated the class of convex potentials on [0, π] and demonstrated with either Dirichlet or Neumann boundary conditions, that the constant potential function minimizes Γ. Later Horváth [12] returned to the problem of single-well potentials using Lavine’s methods, but without making any symmetry assumptions, and proved that the constant potential was optimal with some restrictions on the transition point, and in 2015 Yu and Yang [23] extended Horváth’s result by allowing other transition points and both Dirichlet and Neumann boundary conditions. Recently Harrell and El Allali [9] used direct optimization methods to prove sharp lower bounds for Γ with general single-well potential q(x), without any restriction on the transition point a ∈ [0, π] and found similar results in the case where the potential is convex. Additionally, Harrell and El Allali analyzed the case where q = q0 + q1, where q0 is a fixed background potential energy, and q1 is assumed either single-well or con- vex. In contrast to the previous studies of single-well potentials, which restrict the transition point in some way, the minimizing potentials they found are in general step functions and not necessarily constant unless additional criteria are imposed. In the classic case where p = 1 they retrieved Lavine’s with different arguments the result of Lavine that Γ is uniquely minimized among convex q by the constant and in the case of single-well potentials, with no restrictions on the position of the minimum, they proved the innovative lower bound Γ ≥ 2.04575 . . . In higher dimensions, the first lower bound on the spectral gap was made by Payne and Weinberger [19] who proved the spectral gap of the Neumann Laplacian on a bounded domain Ω is bounded from below by π2 D2 , where D is the diameter of Ω. Later, Andrews and Clutterbuck [3] showed that Dirichlet Schrödinger operators on a convex domain with a convex potential have a fundamental spectral gap that is always greater than 3π2 D2 . Smits [20] investigated the topic of the lower bounds on the fundamental gap under Robin boundary conditions, which lies between the Dirichlet and Neumann cases. There is an extensive literature on extending these gap results for Neumann and Dirichlet boundary conditions to the case of Robin boundary conditions, see for example Laugesen [16], Andrews, Clutterbuck, and Hauer [2], Chapter 4 (by Bucur, Freitas, and Kennedy) of the book [10], Kielty [15]. For more information about the history of the fundamental gap see [4, 14]. Throughout this article, we consider the following problem under Robin bound- ary conditions. −u′′ + q(x)u = λu x ∈ [0, π], u′(0) = −ηu(0), u′(π) = ζu(π). (1.2) The Robin boundary conditions have the physical interpretation of radiation if η < 0 and ζ > 0, absorption when η > 0 and ζ < 0, and insulation when η = 0 and ζ = 0. We briefly recall the recent literature concerning related works of problem (1.2) subject to the Robin boundary conditions. Andrews, Clutterbuck and Hauer [1] proved that the minimizer of Γ[q] is the constant potential with either convex or single-well potentials. Ashbaugh and Kielty [6] also established that the fundamental gap is an increasing function of the Robin parameters for the class of convex symmetric potentials. Motivated by these results, we shall prove in this EJDE-2018/CONF/26 LOWER BOUNDS ON THE FUNDAMENTAL GAP 3 paper that the optimal lower bound of Γ[q] among convex potentials is the constant potential for two different Robin parameters of the problem (1.2), which is regarded as an open problem of the authors Ashbaugh and Kielty , (in [6], see open problem 3.1). The proof of our main result will be based on the recent work by Andrews, Clutterbuck and Hauer [1] and some techniques used in Master thesis of Höltschl [11] which has not appeared in archival journals in the case of linear potential. This work is arranged as follows: in section 2, we derive some simple proper- ties of the fundamental gap Γ. In section 3, we shall study the optimal estimates of the fundamental gap Γ for Schrödinger operators with Robin boundary condi- tions. In section 4, we establish a relation between the first eigenvalues of Neumann Schrödinger operators on an interval and the real roots of the first derivative of the Airy functions Ai′ and Bi′, where Ai(x) and Bi(x) are the Airy functions on the bounded solution of the ordinary differential equation u”(x) = xu(x). 2. Preliminaries and basics Definition 2.1. The fundamental gap which is denoted by Γ[q], is the difference between the first two eigenvalues Γ[q] = λ2(q)− λ1(q). The fundamental gap for the Schrödinger problem with Robin boundary condi- tions depends on the potential and the length of the underlying interval and the parameters η, ζ. By definition we remark that the fundamental gap of a Schrödinger problem with Robin boundary conditions on an interval [0, π] unaffected by adding a constant Γ[q] = Γ[q+c]. As a consequence of this remark, which will be very important later on, if we have to deal with linear potentials, we only need to consider potentials of the form q(x) = tx with t > 0. Theorem 2.2 ([22]). The spectrum of Schrödinger problem with Robin boundary conditions has a discrete spectrum of simple eigenvalues, satisfies λ1 < λ2 < · · · < λn →∞. Moreover, the eigenfunction un corresponding to the eigenvalue λn has exactly n−1 zeros in (0, π). Now, we give the Hellmann-Feynmann result [13] for the variation of eigenvalues with respect to a family of potentials of the problem (1.2) under Robin boundary conditions. Lemma 2.3. Suppose that q(., t) is a one-parameter family of real-valued, locally L1 function with ∂q ∂t (x, t) ∈ L 1(0, π) and inf q(x, t) > −∞. Then dλn(t) dt = ∫ π 0 ∂q ∂t (x, t)u2n(x, t) dx. Proof. Denote u̇ = du dt , the potential q depends integrably on x and differentiably on t. We define Rt : H1(0, π) \ {0} → R, the Rayleigh quotient by Rt(u) = ∫ π 0 (u′)2dx+ ∫ π 0 qu2dx− (ηu2(0) + ζu2(π))∫ π 0 u2dx 4 MOHAMMED AHRAMI, ZAKARIA EL ALLALI EJDE-2022/CONF/26 and we consider the problem (1.2) with Robin boundary conditions then dλn(t) dt = 1∫ π 0 u2ndx (∫ π 0 (2u′nu̇ ′ n + q̇u2n + 2unu̇nq)dx − (2ηun(0)u̇n(0) + 2ζun(π)u̇n(π)) ) − 1( ∫ π 0 u2ndx )2(∫ π 0 (u′n)2dx+ ∫ π 0 qu2ndx− (ηu2n(0) + ζu2n(π) ) × (∫ π 0 2unu̇ndx ) . Integrating by parts twice∫ π 0 2u′nu̇ ′ ndx = 2 ∫ π 0 λnunu̇ndx− 2 ∫ π 0 qynu̇ndx+ 2(ηun(0)u̇n(0) + ζun(π)u̇n(π)). So dλn(t) dt = 1∫ π 0 u2ndx ( 2 ∫ π 0 λnunu̇ndx+ ∫ π 0 q̇u2dx ) − λn∫ π 0 u2ndx ∫ π 0 2unu̇ndx. Then dλn(t) dt = 1∫ π 0 u2ndx ∫ π 0 q̇u2ndx. Noting that ∫ π 0 u2ndx = 1. So dλn(t) dt = ∫ π 0 ∂q ∂t (x, t)u2n(x, t)dx. � 3. Characterization of optimizers In this section we prove that the minimizer of Γ[q] among convex potentials q is the constant potential. The proof is based on refined arguments from Lavine’s proof of the fundamental gap conjecture with Robin boundary condition. Lemma 3.1. Consider problem (1.2) with Robin boundary conditions such that q(x) = tx. Then for every y satisfying (1.2), we have π((u′(π))2 + (λ− tπ)u2(π)) = ∫ π 0 (2λ− 3tx)u2(x)dx+ ηu2(0) + ζu2(π), (3.1) π((u′(π))2 + (λ− tπ)u2(π)) = 1 π ∫ π 0 x(4λ− 5tx)u2(x)dx− 1 π ( u2(π)− u2(0) ) + 2ζu2(π). (3.2) Proof. We have π((u′(π))2 + (λ− tπ)u2(π)) = ∫ π 0 d dx x ( (u′(x))2 + (λ− tx)u2(x) ) dx = ∫ π 0 (u′(π))2 + (λ− tx)u2(x))dx+ ∫ π 0 x(2u′(x)u′′(x) + 2u(x)u′(x)(λ− tx)− tu2(x))dx = ∫ π 0 ((u′(π))2 + (λ− tx)u2(x))dx− ∫ π 0 txu2(x)dx. EJDE-2018/CONF/26 LOWER BOUNDS ON THE FUNDAMENTAL GAP 5 Since u satisfies problem (1.2), it follows that∫ π 0 (u′(x))2dx− ∫ π 0 (λ− tx)u2(x)dx = ∫ π 0 (u(x)u′(x))′dx = ηu2(0) + ζu2(π). So − ∫ π 0 (u′(x))2dx+ ∫ π 0 (λ− tx)u2(x)dx+ ηu2(0) + ζu2(π) = 0. Thus, π((u′(π))2 + (λ− tπ)u2(π)) = ∫ π 0 (2λ− 3tx)u2(x)dx+ ηu2(0) + ζu2(π). In the same way as formula (3.1), we can prove (3.2). � Theorem 3.2. Consider problem (1.2) with Robin boundary conditions. Then for every convex and non-affine potential q, there exists a linear potential qt = tx such that Γ[q] ≥ Γ[qt]. Proof. Let q be a convex non affine potential and Lq(x) = tx+b the linear potential such that q(x±) = Lq(x±). We know by [6] that there exist 0 ≤ x− < x+ ≤ π and u22(x) ≥ u21(x) on (0, x−) ∪ (x+, π), u21(x) > u22(x) on (x−, x+). By convexity of q, q − Lq ≥ 0 on (0, x−) ∪ (x+, π), q − Lq ≤ 0 on (x−, x+). So ∫ π 0 (q − Lq)(u22 − u21)dx > 0. Let Lq(θ) = θq + (1− θ)Lq for θ ∈ (0, 1). Therefore L̇q(θ) = q − Lq. Then dΓ(Lq(θ)) dθ = ∫ π 0 (q − Lq)(u22 − u21)dx > 0. Integrating this inequality with respect to θ over (0, 1), we find that∫ 1 0 dΓ(Lq(θ)) dθ = Γ[q]− Γ[Lq] ≥ 0. Then Γ[q] ≥ Γ[Lq] = Γ[qt]. � Theorem 3.3. Consider problem (1.2) with Robin boundary conditions. Then for every convex and non-affine potential q we have Γ[q] ≥ Γ[0] if and only if η > 0 and ζ < 0. 6 MOHAMMED AHRAMI, ZAKARIA EL ALLALI EJDE-2022/CONF/26 Proof. We consider a family of potentials qθ = θtx. From Lemma 2.3 we have d(λ2(qθ)− λ1(qθ)) dθ = t ∫ π 0 x(u22(x)− u21(x))dx. Suppose that the critical point of the gap is achieved at some t 6= 0, then∫ π 0 x(u22(x)− u21(x))dx = 0. By Lemma 3.1 we find that∫ π 0 (2λ− 3tx)u2(x)dx+ ηu2(0) + ζu2(π) = 1 π ∫ π 0 x(4λ− 5tx)u2(x)dx− 1 π (u2(π)− u2(0)) + 2ζu2(π). Then 2λn + ζu2n(π) + ηu2n(0)− 2ζu2n(π) + 1 π (u2n(π)− u2n(0)) = −5t π ∫ π 0 x2u2n(x)dx+ ( 3t+ 4λn π )∫ π 0 xu2n(x)dx. Since ∫ π 0 x(u22(x)− u21(x))dx = 0, it follows that 2(λ2 − λ1)− ζ(u22(π)− u21(π)) + η(u22(0)− u21(0)) + 1 π (u22(π)− u21(π)− u22(0) + u21(0)) = −5t π ∫ π 0 x2(u22(x)− u21(x))dx = −5t π ∫ π 0 (x2 −Ax−B)(u22(x)− u21(x))dx < 0 if t > 0, choosing A and B. Which yields a contradiction with η > 0 and ζ < 0 and (u22(π)−u21(π))−(u22(0)−u21(0)) > 0. Then the fundamental gap for linear potentials qt(x) = tx achieves its minimum at t = 0. � Example and numerical simulation. By Theorem 3.3, we have Γ[q] ≥ Γ[0]. The eigenfunctions of problem (1.2) for t = 0 are given by u(x) = c1 sin( √ λx) + c2 cos( √ λx). The Robin boundary conditions u′(0) = −ηu(0), u′(π) = ζu(π), (3.3) give tan(kπ) (ηζ k − k ) = η + ζ with k = √ λ. Using Mathematica, we can calculate the approximates of non- negative real roots of the transcendental equation tan(kπ)(ηζk − k) = η + ζ. For simplicity we fix ζ = −1, obtaining the results in Table 1. Then for η → 0, we have Γ[q] ≥ 1.335526 . . . . EJDE-2018/CONF/26 LOWER BOUNDS ON THE FUNDAMENTAL GAP 7 Table 1. η λ1 λ2 Γ 0.01 0.142009 1.479635 1.337626 0.1 0.092975 1.428501 1.335526 10 0.654731 2.964465 2.309734 100 0.623687 2.810751 2.187064 1000 0.620702 2.795907 2.175205 4. Estimates on the fundamental gap for linear potentials In this section, we establish a relation between the first two eigenvalues of the Neumann Schrödinger operators and the real roots of the first derivative of the Airy functions Ai and Bi. We take the Robin boundary conditions and sending η → 0, ζ → 0 so, we recover the Neumann boundary conditions. We denote by Λ = ⋃ n∈N α ′ n the set of zeros of Ai′ and by ∨ = ⋃ n∈N β ′ n the set of zeros of Bi′. For x ∈ R \ Λ, we introduce the function f(x) = Bi′(x) Ai′(x) . The zeros β′n of Bi′ coincide with the zeros of f , while the zeros α′n of Ai′ give the singularities of f . We call f/(β′ n+1,β ′ n) the n+ 1-th branch of f and f/(β′ 1,0) the first branch of f . Lemma 4.1. The function f is strictly increasing on every branch. Proof. Differentiate f we obtain f ′(x) = Bi′′(x)Ai′(x)− Bi′(x)Ai′′(x) Ai′(x)2 , f ′(x) = −xW (Ai(x),Bi(x)) Ai′(x)2 . The Wronskian of Ai and Bi is known to be π−1. Thus we obtain f ′(x) = −xπ−1 Ai′(x)2 > 0 for each x < 0. Let D : (−∞, α′1) → (0,∞) be the function describing the distance of two consec- utive branches of f , defined by D(α′i) = α′i−1 − α′i such that D is continuous on (−∞, α′1). � Theorem 4.2. Consider the Neumann Schrödinger problem with linear potential q(x) = tx with t > 0. Let α′n denote the n-th zero of Ai′. Then Γ[q] ≥ (α′1 − α′2)t2/3. Proof. Consider the Schrödinger equation −u′′(x) + q(x)u(x) = λu(x). Using the change of variable ε = λ/t2/3 and x = z/t1/3, we obtain the new equation u′′(z) = (z − ε)u(z). 8 MOHAMMED AHRAMI, ZAKARIA EL ALLALI EJDE-2022/CONF/26 The eigenfunctions of Schrödinger equation with linear potentials have the form u(z) = c1Ai(z − ε) + c2Bi(z − ε) with c1, c2 ∈ R. u(x) = c1Ai(t1/3x− λt−2/3) + c2Bi(t1/3x− λt−2/3). Applying Neumann boundary conditions at the endpoints u′(0) = u′(π) = 0, we find that Bi′(t1/3π − λt−2/3)Ai′(−λt−2/3)− Bi′(−λt−2/3)Ai′(t1/3π − λt−2/3) = 0. It follows that Ai′(−λt−2/3)Bi′ ( t1/3 ( π − λ t )) − Bi′(−λt−2/3)Ai′ ( t1/3 ( π − λ t )) = 0. (4.1) According to the asymptotic behavior of Airy functions, the equation (4.1) takes the following form as t goes to +∞, sin θ(−λt−2/3) exp (2 3 [ t1/3(π − λ t ) ]3/2) + 1 2 cos θ(−λt−2/3) exp ( − 2 3 [ t1/3(π − λ t ) ]3/2) = 0. Then we obtain tan θ(−λt−2/3) = −1 2 exp ( − 4 3 [ t1/3(π − λ t ) ]3/2) . As exp ( − 4 3 [t1/3(π − λ t )]3/2 ) has a limit 0 when t→∞. Then lim t→∞ tan θ(−λt−2/3) = lim t→∞ Ai′(−λt−2/3) = 0, which implies that limt→∞−λnt−2/3 = α′n. Equivalently λn = −α′nt2/3 + o(t2/3), as t→∞. Overall, we obtain a lower bound on the fundamental gap Γ[q] = λ2 − λ1 = −α′2t2/3 + o(t2/3) + α′1t 2/3 + o(t2/3) ≥ (α′1 − α′2)t2/3. � Remark 4.3. By Theorem 4.2, we conclude that the fundamental gap is un- bounded when t goes to infinity. Lemma 4.4. The eigenvalues of Neumann Schrödinger operator with q(x) = tx on [0, 1] are given by λ1 = −τ1t2/3 with τ1 ∈ (β′2, β ′ 1). Proof. Let τ1 = max{τ ∈ (−∞, β′1)/D(τ) = t1/3}. Assume that λn = −τ1t2/3 with n ≥ 2, so there exists a solutions τ̃ of f(τ+πt1/3) = f(τ) with τ̃ > τ such that λ1 = −τ̃ t2/3. Note that by definition of τ1, D(τ̃) 6= t1/3 implies that is τ̃ and τ̃ + πt1/3 do not lie in the domains of consecutive branches of f , i.e. if τ̃ ∈ (β′n+1, β ′ n) then τ̃ + πt1/3 > β′n−1. By the definition of D, τ̃ and τ̃+D(τ̃) lie in the domains of consecutive branches of f that is τ̃ ∈ (β′n+1, β ′ n) implies τ̃ +D(τ̃) ∈ (β′n, β ′ n−1). Then D(τ̃) < t1/3. Noting that D is continuous, then there exists ν > τ̃ such that D(ν) = t1/3 so ν < τ1. This contradicts τ̃ > τ1. � EJDE-2018/CONF/26 LOWER BOUNDS ON THE FUNDAMENTAL GAP 9 We can generalize the previous lemma by the following result. Lemma 4.5. The eigenvalues of Neumann Schrödinger operator with q(x) = tx on [0, 1] are given by λn = −τnt2/3 with τn ∈ (β′n+1, β ′ n). Proposition 4.6. Consider the Neumann Schrödinger problem with linear poten- tial q(x) = tx with t > 0 on [0, 1]. Let α′n and β′n denote the n-th zero of Ai′ and Bi′ respectively. If t ≥ (α′1 − α′2)3 then Γ[q] ≥ (α′2 − β′2)t2/3. Proof. We have D(α′2) = α′1 − α′2, the fact that D is increasing we obtain λ1 < −α′2t2/3. Observing that λ2 > −β′2t2/3. This yields that Γ[q] = λ2(q)− λ1(q) ≥ α′2t2/3 − β′2t2/3. Then we conclude that Γ[q] ≥ (α′2 − β′2)t2/3. � Proposition 4.7. Consider the Neumann Schrödinger problem with linear poten- tial q(x) = tx with t > 0 on [0, π]. Let α′n and β′n denote the n-th zero of Ai′ and Bi′ respectively. If t ≥ (α′ 1−α ′ 2) 3 π3 then Γ[q, π] ≥ (α′2 − β′2)t2/3. Proof. It is easy to show that Γ[q, π] = 1 π2 Γ[π2q(πx)]. From Proposition 4.6, for π3t ≥ (α′1 − α′2)3 we deduce that Γ[π2q(πx)] ≥ (α′2 − β′2)π2t2/3. Consequently, for t ≥ (α′ 1−α ′ 2) 3 π3 we obtain Γ[q, π] ≥ (α′2 − β′2)t2/3. � Example and numerical simulation. Using Mathematica, for β′1 = −2.2944, β′2 = −4.0731, α′2 = −3.2481, t > 0.1814, we obtain q(x) = 1 2x x 0, 1814x Γ ≥ 0.5197 0.825 0.2643 Thus Γ[q0] ≥ 0.2643 for q0(x) = 0.1814x. Remark 4.8. Our main results include improvements of the lower bound on the fundamental gap of Robin Schrödinger operators with a convex potential. Mean- while, when we establish the link between the first eigenvalues and the real roots of the first derivative of the Airy functions Ai′ and Bi′, how small can the fundamental eigenvalue gap be? 10 MOHAMMED AHRAMI, ZAKARIA EL ALLALI EJDE-2022/CONF/26 5. Appendix: Airy functions [18, 21] The Airy functions can be defined as the linearly independent solutions to the differential equation du2 dx2 (x) = xu(x). The first Airy function Ai and the second Airy function Bi have representations as improper Riemann integrals Ai(x) = 1 π ∫ ∞ 0 cos ( t3 3 + xt ) dt, Bi(x) = 1 π ∫ ∞ 0 exp ( − t3 3 + xt ) sin ( t3 3 + xt ) dt. The Airy functions Ai and Bi have infinitely many zeros on the negative real axis. We denote those zeros by αn and βn, n ∈ N, respectively, in decreasing order, so that 0 > β1 > α1 > β2 > α2 > β3 . . . . and let α′n and β′n denote the n-th zero of the first derivative of the Airy functions Ai′ and Bi′ respectively, so that 0 > α′1 > β′1 > α′2 > β′2 > α′3 . . . . The asymptotic behavior of the Airy functions Ai and Bi as t→∞ is given by Ai(t) ∼ exp(− 2 3 t 3/2) 2 √ πt1/4 , Bi(t) ∼ exp( 2 3 t 3/2) √ πt1/4 . And the asymptotic behavior of the Airy functions Ai′ and Bi′ for t→∞ is given by Ai′(t) ∼ − exp(− 2 3 t 3/2) 2 √ πt−1/4 , Bi′(t) ∼ exp( 2 3 t 3/2) √ πt−1/4 . For a negative t, Ai′(t) = N(t) sin θ(t), Bi′(t) = N(t) cos θ(t) where N(t) = √ (Ai′(t))2 + (Bi′(t))2, θ(t) = arctan ( Ai′(t) Bi′(t) ) . Acknowledgements. We would like to thank Evans M. Harrell II for useful con- versations and helpful comments. References [1] B. Andrews, J. Clutterbuck, D. Hauer; The fundamental gap for a one-dimensional Schrödinger operator with Robin boundary conditions, Proc. Amer. Math. Soc., Februray 1, (2021), https://doi.org/10.1090/proc/15140. [2] B. Andrews, J. Clutterbuck, D. Hauer; Non-concavity of Robin eigenfunctions, Cambridge J. Math. 8, 243–310 (2020). [3] B. Andrews, J. Clutterbuck; Proof of the fundamental gap conjecture, J. Am. Math. Soc. 24(3), (2011), 899–916. [4] M. Ashbaugh; The fundamental gap, AIM, Report, 2006. [5] M. Ashbaugh, R. Benguria; Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials, Proc. Amer. Math. Soc., 105(1989), 419-424. EJDE-2018/CONF/26 LOWER BOUNDS ON THE FUNDAMENTAL GAP 11 [6] M. Ashbaugh, D. Kielty; spectral gaps of 1-D Robin Schrödinger operators with single-well potentials, Journal of Mathematical Physics, 61, 091507 (2020). [7] R. Bañuelos, Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger opera- tors, J. Funct. Anal. 100 (1991), no. 1, 181-206. [8] M. van den Berg; On condensation in the free-Boson gas and the spectrum of the Laplacian, Journal of Statistical Physics 31 (1983), 623-637. [9] Z. El Allali, E. M. Harrell II; Optimal bounds on the fundamental spectral gap with single-well potentials, Proc. Amer. Math. Soc., 150 (2022), 575-587. [10] A. Henrot (editor); Shape Optimization and Spectral Theory, De Gruyter, Berlin, Boston, 2017. [11] A. Höltschl; On the fundamental gap of one-dimensional Schrödinger operators, University of Stuttgart Master Thesis, 2015. [12] M. Horváth; On the first two eigenvalues of Sturm-Liouville operators, Proc. Amer. Math. Soc. 131 4 (2002) 1215–1224. [13] T. Kato; Perturbation theory for the linear operator, Springer-Verlag, 1980. [14] J. Kerner; A lower bound on the spectral gap of Schrödinger operators with weak potentials of compact support, Preprint 2021; https://arxiv.org/abs/2103.03813. [15] D. Kielty; Degeneration on the spectral gap with negative Robin parametre, Preprint 2022; https://arxiv.org/abs/2105.02323. [16] R. S. Laugesen; The Robin Laplacian– Spectral conjectures, rectangular theorems, Journal of Mathematical Physics, 60(12), (2019), 121507. [17] R. Lavine; The eigenvalue gap for one-dimensional convex potentials. Proceedings of the American Mathematical Society (1994); 121, 815-821. [18] Frank W. J. Olver; Asymptotics and Special Functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997, Reprint of the 1974 original [Academic Press, New York]. [19] L. E. Payne, H. F. Weinberger; An optimal Poincaré inequality for convex domains, Arch. Ration. Mech. Anal., 5, (1960), 286–292. [20] R. G. Smits; Spectral gaps and rates to equilibrium for diffusions in convex domains, Michigan Mathematical Journal, 43(1), (1996), 141–157. [21] O. Vallée, M. Soares; Airy functions and applications to physics. Imperial College Press, 2010. [22] J. Weidmann; Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathe- matics 1258, Springer-Verlag, Berlin, 1987. [23] X. J. Yu, C. F. Yang; The gap between the first two eigenvalues of Schrödinger operators with single-well potential, Applied Mathematics and Computation, 268 (2015), 275-283. Mohammed Ahrami Team of Modeling and Scientific Computing, Department of Mathematics, Multidisci- plinary Faculty of Nador, University of Mohammed First, Morocco Email address: m.ahrami@ump.ac.ma Zakaria El Allali Team of Modeling and Scientific Computing, Department of Mathematics, Multidisci- plinary Faculty of Nador, University of Mohammed First, Morocco Email address: z.elallali@ump.ma 1. Introduction 2. Preliminaries and basics 3. Characterization of optimizers Example and numerical simulation 4. Estimates on the fundamental gap for linear potentials Example and numerical simulation 5. Appendix: Airy functions Frank,Vallee Acknowledgements References