On the first two eigenvalues of Sturm-Liouville operators (original) (raw)
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On the first two eigenvalues of the Sturm-Liouville operators
2008
Among the Schrödinger operators with single-well potential defined on (0, π) with transition point at π 2 , the gap between the first two eigenvalues of the Dirichlet problem is being investigated. We also show how this extends former results with symmetric potential. Finally we will consider an analogous Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.
Non-localization of eigenfunctions for Sturm–Liouville operators and applications
Journal of Differential Equations, 2018
In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouville operators Aa = −∂xx + a(•) Id with Dirichlet boundary conditions, where a(•) runs over the bounded nonnegative potential functions on the interval (0, L) with L > 0. More precisely, we address the extremal spectral problem of minimizing the L 2-norm of a function e(•) on a measurable subset ω of (0, L), where e(•) runs over all eigenfunctions of Aa, at the same time with respect to all subsets ω having a prescribed measure and all L ∞ potential functions a(•) having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.
Eigenvalues of Schrödinger operators on finite and infinite intervals
Mathematische Nachrichten
We consider a Sturm-Liouville operator a with integrable potential q on the unit interval I = [0, 1]. We consider a Schrödinger operator with a real compactly supported potential on the half line and on the line, where this potential coincides with q on the unit interval and vanishes outside I. We determine the relationships between eigenvalues of such operators and obtain estimates of eigenvalues in terms of potentials.
Communications in Mathematical Physics, 1989
Consider the Schrodinger equationu" + V(x)u = λu on the interval la [R, where K(x)^0 for xel and where Dirichlet boundary conditions are imposed at the endpoints of /. We prove the optimal bound-f-^n 2 for n = 2,3,4,... λ 1 on the ratio of the n ih eigenvalue to the first eigenvalue for this problem. This leads to a complete treatment of bounds on ratios of eigenvalues for such problems. Extensions of these results to singular problems are also presented. A modified Priifer transformation and comparison techniques are the key elements of the proof.
Eigenvalue ratios for Schrödinger operators with indefinite potentials
Applied Mathematics Letters, 2018
We consider the Schrödinger equation −y′′ + q(x)y = λy, on a finite interval with Dirichlet boundary conditions, where q(x) is of indefinite sign. In the case of symmetric potentials, we prove the optimal lower bound λn λ1 ≥ n2 (resp. upper bound λn λ1 ≤ n2) for single-well q with λ1 > 0 and μ1 ≤ 0 (resp. single-barrier q and μ1 ≥ 0), where μ1 is the first eigenvalue of the Neumann boundary problem. In the case of nonsymmetric potentials, we prove the optimal lower bound λ2 λ1 ≥ 4 for single-well q with transition point at x = 1 2 , λ1 > 0 and μ = max(ˆμ1, ˜μ1) ≤ 0, where ˆμ1 and ˜μ1 are the first eigenvalues of the Neumann boundary problems defined on [0, 1 2 ] and [12 , 1], respectively.
A bound for ratios of eigenvalues of Schrodinger operators with single-barrier potentials
M. Horvath and M. Kiss [Proc. Amer. Math. Soc, (2005)] , proved that the upper estimate λn λm ≤ n 2 m 2 (n > m ≥ 1) of Dirichlet Shrodinger operators with nonnegative and single-well potentials. In this paper we discuss the case of nonpositive potentials q(x) defined and continuous on the interval [0, 1]. Namely, we prove that if q(x) ≤ 0 and single-barrier then λn λm ≥ n 2 m 2 for λn > λm ≥ −2q , where q = min {q(x), x ∈ [0, 1]}. Moreover we show that there exixst l 0 ∈ (0, 1] such that if q(x) ≤ 0 and single-barrier, then the associated eigenvalues (λn (ℓ 0)) n≥1 satisfy λ 1 (ℓ 0) > 0 and λn(ℓ 0) λm(ℓ 0) ≥ n 2 m 2 for n > m ≥ 1.
Optimal bounds on the fundamental spectral gap with single-well potentials
Proceedings of the American Mathematical Society
We characterize the potential-energy functions V (x) that minimize the gap Γ between the two lowest Sturm-Liouville eigenvalues for H(p, V)u := − d dx p(x) du dx + V (x)u = λu, x ∈ [0, π], where separated self-adjoint boundary conditions are imposed at end points, and V is subject to various assumptions, especially convexity or having a "single-well" form. In the classic case where p = 1 we recover with different arguments the result of Lavine that Γ is uniquely minimized among convex V by the constant, and in the case of single-well potentials, with no restrictions on the position of the minimum, we obtain a new, sharp bound, that Γ > 2.04575. .. .
Hacettepe Journal of Mathematics and Statistics, 2019
The spectral problem\[-y''+q(x)y=\lambda y,\ \ \ \ 0<x<1\]\[y(0)=0, \quad y'(0)=\lambda(ay(1)+by'(1)),\]is considered, where lambda\lambdalambda is a spectral parameter, q(x)inL1(0,1)q(x)\in{{L}_{1}}(0,1)q(x)inL1(0,1) is a complex-valued function, aaa and bbb are arbitrary complex numbers which satisfy the condition ∣a∣+∣b∣ne0|a|+|b|\ne 0∣a∣+∣b∣ne0. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in Lp(0,1){{L}_{p}}(0,1)Lp(0,1)) of the above-mentioned Sturm-Liouville problem.
A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator
Communications in Mathematical Physics, 2006
Let λ i (Ω, V ) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω ⊂ R n and with the positive potential V . Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V ⋆ , we prove that λ 2 (Ω, V ) ≤ λ 2 (S 1 , V ⋆ ). Here S 1 denotes the ball, centered at the origin, that satisfies the condition λ 1 (Ω, V ) = λ 1 (S 1 , V ⋆ ).
Extremal values of eigenvalues of Sturm–Liouville operators with potentials in balls
Journal of Differential Equations, 2009
This paper is a continuation of Zhang [M. Zhang, Continuity in weak topology: Higher order linear systems of ODE, Sci. China Ser. A 51 (2008) 1036-1058; M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in L 1 balls, J. Differential Equations 246 (2009) 4188-4220]. Given a potential q ∈ L p ([0, 1], R), p ∈ [1, ∞]. We use λ m (q) to denote the mth Dirichlet eigenvalue of the Sturm-Liouville operator with potential q(t), where m ∈ N. The minimal value L m,p (r) and the maximal value M m,p (r) of λ m (q) with potentials q in the L p ball of radius