Abstract
The paper is concerned with the Neumann eigenvalues for second-order Sturm–Liouville difference equations. By analyzing the new discriminant function, we show the interlacing properties between the periodic, antiperiodic, and Neumann eigenvalues. Moreover, when the potential sequence is symmetric and symmetric monotonic, we show the order relation between the first Dirichlet eigenvalue and the second Neumann eigenvalue, and prove that the minimum of the first Neumann eigenvalue gap is attained at the constant potential sequence.
Highlights
In 1946, Borg showed [5] that, for Hill’s equation, the potential q is constant if and only if all instability intervals, except the zero-th, are absent
Consider the second-order Sturm–Liouville difference equation–∇( yn) + qnyn = λyn on [0, N – 1], (1)where the potential sequence qi ≥ 0 for i = 0, 1, 2, . . . , N – 1, is the forward difference operator, ∇ is the backward difference operator (∇yn = yn – yn–1), and the bracket [0, N – 1] means the integers in [0, N – 1]
Note that equation (1) can be rewritten as the recurrence formula yn+1 = (2 + qn – λ)yn – yn–1 on [0, N – 1] or the matrix formula (D + Q)y = λy, (2020) 2020:599 where y is a vector in RN, Q is a diagonal matrix whose diagonal elements are q0, q1, . . . , qN–1, and D is the N × N tridiagonal matrix of the form
Summary
In 1946, Borg showed [5] that, for Hill’s equation, the potential q is constant if and only if all instability intervals, except the zero-th, are absent. In 1990, Ashbaugh and Benguria [2] studied the comparison of the eigenvalues of two discrete Sturm–Liouville equations whose potential sequences satisfy certain relation.
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