A lower bound on the spectral gap of one-dimensional Schrödinger operators
aa r X i v : . [ m a t h . SP ] F e b A lower bound on the spectral gap ofone-dimensional Schr¨odinger operators onlarge intervals
Joachim Kerner Department of Mathematics and Computer ScienceFernUniversit¨at in Hagen58084 HagenGermany
Abstract
In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schr¨odinger operators with Neumann boundary conditions on large in-tervals and with potentials that decay sufficiently fast at infinity. This complementsprevious results obtained in [KT] where the spectral gap has been studied for Dirich-let boundary conditions. E-mail address:
Introduction
The spectral gap (or gap for short) is defined as the difference between the first twoeigenvalues of a certain operator. In this paper, we focus on (deterministic) Schr¨odingeroperators defined on an interval of length
L >
0, subject to Neumann boundary conditions,and investigate the behaviour of the gap in the limit where L tends to infinity. As recognizedin [KT], one gains a certain flexibility when studying the asymptotics of the gap ratherthan the gap on an interval of fixed length. In particular, one is able to allow for moregeneral potentials which do not satisfy, e.g., a convexity assumption. Assumptions of thistype, on the other hand, are often invoked when studying the influence of generic potentialson the spectral gap on fixed intervals [AB89, Abr91, Lav94].The present paper forms a natural continuation of the investigations started recentlyin [KT]. There, it was shown that the spectral gap of Schr¨odinger operators subject toDirichlet boundary conditions closes strictly faster than in the case of the free DirichletLaplacian for a large class of potentials. This holds, in particular, for non-zero boundedpotentials v : R → R + of compact support. More explicitly, for a symmetric step-potential,lower and upper bounds on the spectral gap were derived showing that the gap closes like ∼ L − and it was conjectured that the gap cannot close faster than ∼ L − for boundedpotentials of compact support. However, no lower bound on the spectral gap was derivedin general and this is what motivates the investigations in this paper. More explicitly,using powerful results from [KS87], our aim is to derive an explicit lower bound on thespectral gap for potentials that decay sufficiently fast at infinity. Since we employ a versionof Harnack’s inequality, the lower bound is exponentially small in the interval length andhence we are still quite far away from a proof of the conjecture as formulated in [KT].Nevertheless, it is a first step providing an explicit lower bound for a large class of potentials. On the interval I = ( − L/ , + L/
2) we consider a Schr¨odinger operator of the form h L = − d d x + v with real, non-negative potentials v ∈ L ∞ ( R ) ∩ L ( R ) and Neumann boundary conditionsat the endpoints x = ± L/
2. Standard operator theory tells us that h L is self-adjoint withpurely discrete spectrum. We denote its eigenvalues as λ ( L ) ≤ λ ( L ) ≤ ... ; the normalizedground state eigenfunction shall be denoted as ϕ L ∈ L ( I ).Our main object of interest is the spectral gapΓ v ( L ) := λ ( L ) − λ ( L ) . We remark that, since the ground state is non-degenerate [LL01], the spectral gap is strictlylarger than zero for every value
L >
0. Also, as shown in [KT], Γ v ( L ) converges to zero2s L → ∞ for potentials that decay sufficiently fast (as shown in [Theorem 2.1,[KT]], itsuffices to assume | v ( x ) | ≤ C | x | for some C > x ∈ R ).In a first result we provide an upper bound on the supremum (or maximum) of theground state eigenfunction for potentials of short range. Proposition 2.1.
Let v ∈ L ∞ ( R ) be non-negative and such that | v ( x ) | ≤ C | x | for some constant C > and almost all x ∈ R . Then there exists a constant δ ≥ suchthat sup x ∈ I | ϕ L ( x ) | ≤ δ √ L for all L > large enough.Proof. In a first step we realize that, for every
L >
0, there exists x L ∈ I such that ϕ L ( x L ) = √ L . Furthermore, ϕ L ( x ) = Z xx L ( ϕ L ( t )) ′ d t + 1 √ L , which yields | ϕ L ( x ) | ≤ p λ ( L ) · √ L + 1 √ L .
Now, for the considered potentials, it was shown in [KT] that λ ( L ) ≤ cL for some constant c > L >
Lemma 2.2.
Assume v ∈ L ∞ ( R ) ∩ L ( R ) and v ≥ . Then inf x ∈ I | ϕ L ( x ) | ≥ e − L k v k L I ) √ L (2.1) holds for all L > .Proof. We follow the strategy outlined in [BM05] and, in particular, the proof of [Theo-rem 1.2,[BM05]] . [Eq. (2.3),[BM05]] implies, for all x ∈ I , Z I ( q ( x ) − | q ( x ) | ) d x ≤ ( ϕ L ) ′ ( x ) ϕ L ( x ) ≤ k q k L ( I ) , q ( x ) := λ ( L ) − v ( x ). This immediately gives (cid:12)(cid:12)(cid:12)(cid:12) ( ϕ L ) ′ ( x ) ϕ L ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k q k L ( I ) . Note that k q k L ( I ) ≤ k v k L ( I ) since λ ( L ) ≤ k v k L ( I ) L by the minmax-principle (using 1 / √ L as a test function in the Rayleigh quotient). Further-more, this shows that the constant C in the last steps of the proof of [Theorem 1.2,[BM05]]can be chosen to be 4 k v k L ( I ) showing that ϕ L ( y ) ϕ L ( x ) ≤ e L k v k L I ) , x, y ∈ I .
Finally, using sup x ∈ I | ϕ L ( x ) | ≥ √ L then yields the statement. Remark 2.3.
Lemma 2.2 is sharp: choosing v ≡ √ L . On theother hand, for the zero-potential, the ground state is given by ϕ L ( x ) = √ L .Using [Theorem 1.4,[KS87]] in combination with Proposition 2.1 and Lemma 2.2 thengives the main result. Theorem 2.4 (Lower bound spectral gap) . Let v ∈ L ∞ ( R ) be non-negative and such that | v ( x ) | ≤ C | x | for some constant C > and almost all x ∈ R . Then, with the constant δ ≥ fromProposition 2.1, Γ v ( L ) ≥ π δ L · e − L k v k L R ) for all L > large enough.Proof. The idea is to compare the spectral gap of the operator h L with the spectral gap ofthe Neumann Laplacian (i.e., setting v ≡
0) which is given by π /L . Such a comparisonresult has been provided in [Theorem 1.4,[KS87]]; more explicitly, one hasΓ v ( L ) ≥ (cid:18) inf x ∈ I | ϕ L ( x ) | sup x ∈ I | ϕ L ( x ) | (cid:19) · π L , taking into account that the normalized ground state eigenfunction for the Neumann Lapla-cian is the constant function 1 / √ L . The result then follows with Proposition 2.1 andLemma 2.2. 4 emark 2.5. As already indicated in the introduction, Theorem 2.4 establishes a lowerbound for bounded potentials of short-range which is exponentially small in the intervallength. Hence, this bound is still far away from a lower bound as established, for example,in [Proposition 2.9,[KT]] for a symmetric step-potential. In this case, the lower bound reads αL − for some constant α > . However, due to the fact that a Harnack-type inequalityhas been used in the proof of Theorem 2.4, an exponential factor seems expectable.Also, for the zero-potential v ≡ , the lower bound in Theorem 2.4 reads π L (taking intoaccount that δ = 1 in this case). Hence, the lower bound is sharp. Acknowledgement
JK would like to thank M. T¨aufer for interesting discussions that led to an improvementof the paper.
References [AB89] M. S. Ashbaugh and R. Benguria,
Optimal lower bound for the gap between the firsttwo eigenvalues of one-dimensional Schr¨odinger operators with symmetric single-wellpotentials , Proc. Amer. Math. Soc. (1989), no. 2, 419–424.[Abr91] S. Abramovich,
The gap between the first two eigenvalues of a one-dimensionalSchr¨odinger operator with symmetric potential , Proc. Amer. Math. Soc. (1991),no. 2, 451–453.[BHE08] J. Blank, M. Havliˇcek, and P. Exner,
Hilbert space operators in quantum physics ,Springer, 2008.[BM05] S. Berhanu and A. Mohammed,
A Harnack inequality for ordinary differential equations ,Amer. Math. Monthly (2005), no. 1, 32–41.[KS87] W. Kirsch and B. Simon,
Comparison theorems for the gap of Schr¨odinger operators ,J. Funct. Anal. (1987), no. 2, 396–410.[KT] J. Kerner and M. T¨aufer, On the spectral gap of one-dimensional Schr¨odinger operatorson large intervals , arXiv:2012.09060.[Lav94] R. Lavine,
The eigenvalue gap for one-dimensional convex potentials , Proc. Amer. Math.Soc. (1994), no. 3, 815–821.[LL01] E. L. Lieb and M. Loss,
Analysis , American Mathematical Society, 2001., American Mathematical Society, 2001.