Spectral analysis of singular Sturm-Liouville operators on time scales

In this paper, we consider properties of the spectrum of a Sturm-Liouville operator on time scales. We will prove that the regular symmetric Sturm-Liouville operator is semi-bounded from below. We will also give some conditions for the self-adjoint operator associated with the singular Sturm-Liouville expression to have a discrete spectrum. Finally, we will investigate the continuous spectrum of this operator.

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On Schrödinger's factorization method for Sturm-Liouville operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500010143 ◽

1978 ◽

Vol 80 (1-2) ◽

pp. 67-84 ◽

Cited By ~ 23

Author(s):

U.-W. Schmincke

Keyword(s):

Discrete Spectrum ◽

Liouville Operator ◽

Differential Operators ◽

Factorization Method ◽

Friedrichs Extension ◽

First Order ◽

Sturm Liouville ◽

Image Position ◽

Special Case ◽

Liouville Operators

SynopsisWe consider the Friedrichs extension A of a minimal Sturm-Liouville operator L0 and show that A admits a Schrödinger factorization, i.e. that one can find first order differential operators Bk with where the μk are suitable numbers which optimally chosen are just the lower eigenvalues of A (if any exist). With the help of this theorem we derive for the special case L0u = −u″ + q(x)u with q(x) → 0 (|x| → ∞) the inequalityσd(A) being the discrete spectrum of A. This inequality is seen to be sharp to some extent.

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Finite spectrum of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on time scales

Filomat ◽

10.2298/fil1906747a ◽

2019 ◽

Vol 33 (6) ◽

pp. 1747-1757

Author(s):

Ji-Jun Ao ◽

Juan Wang

Keyword(s):

Spectral Analysis ◽

Boundary Conditions ◽

Time Scales ◽

Liouville Equation ◽

Time Scale ◽

The Self ◽

Finite Spectrum ◽

Sturm Liouville

The spectral analysis of a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on bounded time scales is investigated. By partitioning the bounded time scale such that the coefficients of Sturm-Liouville equation satisfy certain conditions on the adjacent subintervals, the finite eigenvalue results are obtained. The results show that the number of eigenvalues not only depend on the partition of the bounded time scale, but also depend on the eigenparameter-dependent boundary conditions. Both of the self-adjoint and non-self-adjoint cases are considered in this paper.

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Trace Formulas for Powers of a Sturm-Liouville Operator

Canadian Journal of Mathematics ◽

10.4153/cjm-1964-043-4 ◽

1964 ◽

Vol 16 ◽

pp. 412-422 ◽

Cited By ~ 4

Author(s):

Richard C. Gilbert ◽

Vernon A. Kramer

Keyword(s):

Hilbert Space ◽

Differential Operator ◽

Positive Integer ◽

Boundary Conditions ◽

Liouville Operator ◽

Adjoint Operator ◽

The Self ◽

Trace Formulas ◽

Sturm Liouville

Let H0 be the mth power (m a positive integer) of the self-adjoint operator defined in the Hilbert space L2(0, π) by the differential operator — (d2/dx2) and the boundary conditions u(0) = u(π) = 0. The eigenvalues of H0 are μn = n2m and the corresponding eigenfunctions are ϕn = (2/π)1/2 sin nx, n = 1 , 2 , . . ..Let p be a (2m — 2)-times continuously differentiate real valued function defined over the interval [0, π] satisfying the conditions p(j)(0) = p(j)(π) = 0 for j odd and less than 2m — 4.

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