Article
Kyungpook Mathematical Journal 2020; 60(2): 297-305
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Comparative Analysis of Spectral Theory of Second Order Difference and Differential Operators with Unbounded Odd Coefficient
Fredrick Oluoch Nyamwala∗, David Otieno Ambogo, Joyce Mukhwana Ngala
Department of Mathematics, Moi University, 3900-30100, Eldoret, Kenya
e-mail : foluoch2000@yahoo.com or foluoch2000@mu.ac.ke
Department of Pure and Applied Mathematics, Maseno University, 333-40105, Maseno, Kenya
e-mail : otivoe@yahoo.com
Department of Mathematics, Moi University, 3900-30100, Eldoret, Kenya
e-mail : mukhwana.ngala@gmail.com
Received: February 20, 2019; Revised: April 21, 2020; Accepted: April 22, 2020
We show that selfadjoint operator extensions of minimal second order difference operators have only discrete spectrum when the odd order coefficient is unbounded but grows or decays according to specific conditions. Selfadjoint operator extensions of minimal differential operator under similar growth and decay conditions on the coefficients have a absolutely continuous spectrum of multiplicity one.
Keywords: differential operators, selfadjoint realisations, deficiency indices, spectrum
1. Introduction
We consider the second order symmetric differential operators generated by
defined on ℒ2([0, ∞)) and their discrete counterparts
defined on ℓ2(ℕ). In (
Further, we assume that the coefficients of (
with their discrete counterparts, coefficients of (
In order to obtain deficiency indices and spectral results, we have solved the equations
In this particular case . One, therefore, defines a symmetric operator
So
This condition will hold for any constant
Here, we require that
where
A similar regularity condition is achieved for difference operators generated by (
where
On the other hand the maximal difference operator generated by (
Assume that for some natural number
On the other hand, asymptotic summation is based on a theorem of Levinson-Benzaid-Lutz which states that if
Our main results show that when
2. Results
Theorem 2.1
-
Eigenvalues of ( .1.1 ) satisfy the uniform dichotomy condition -
If |q 1(x )|−1is integrable, then defT = (2, 2)and σ (H )is discrete . -
If |q 1(x )|−1is not integrable, then defT = (1, 1). Suppose q 1(x ) > 0then σ ac (H ) ⊂ [p̄ 0, ∞)and if q 1(x ) < 0then σ ac (H ) = ℝwith spectral multiplicity 1. Here p̄ 0 =lim sup p 0(x ).
(i) Here, we apply asymptotic integration. This requires that (
This leads to a first order system of the form
The coefficients are functions of
There exists finitely many values of
Here,
(ii) The first order system can now be diagonalised twice using eigenvectors. Approximately, the diagonalising matrix of
Here,
Thus assume |
(iii) If |
for some positive constant
Theorem 2.2
In this particular case, we apply asymptotics. This requires that (
The coefficients are functions of
For the eigenvalues of (
By application of binomial expansion and approximating to
Explicitly, this implies that
These two eigenvalues satisfy the
The system can now be converted into Levinson’s-Benzaid-Lutz form [4, 5] through diagonalisations. In this case, the diagonalising matrix, if the first component of the eigenvectors are normalised, is of the form
After diagonalisation we have a first order of the form
The
This leads to lim
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