Remarks on the Limit-Circle Classification of Conformable Fractional Sturm-Liouville Operators

2021 ◽  
pp. 10-14
Author(s):  
Bilender P. Allahverdiev ◽  
Hüseyin Tuna ◽  
Yüksel Yalçinkaya
Keyword(s):  
Limit Circle ◽  
Sturm Liouville ◽  
Liouville Operators

m (λ)-functions for complex Sturm-Liouville operators

1980 ◽  
Vol 86 (3-4) ◽  
pp. 275-289 ◽  
Author(s):  
David Race
Keyword(s):  
Limit Point ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Complex Valued ◽  
Well Posed ◽  
Circle Case ◽  
Liouville Operators

SynopsisIn this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.

Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Bilender P. Allahverdiev
Keyword(s):  
Difference Operators ◽  
Boundary Values ◽  
Defect Index ◽  
Order Difference ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Circle Case ◽  
Liouville Operators

AbstractIn this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at ±∞ and limit-point case at ∓∞), acting in the Hilbert space

Livšic’s theorem for q-Sturm—Liouville operators

2016 ◽  
Vol 53 (4) ◽  
pp. 512-524
Author(s):  
Hüseyin Tuna ◽  
Aytekin Eryilmaz
Keyword(s):  
Limit Circle ◽  
Limit Circle Case ◽  
Sturm Liouville ◽  
Circle Case ◽  
Liouville Operators

In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I

1984 ◽  
Vol 99 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
F. V. Atkinson ◽  
C. T. Fulton
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Hypergeometric Function ◽  
Finite Interval ◽  
Confluent Hypergeometric Function ◽  
Limit Circle ◽  
Asymptotic Formulae ◽  
Sturm Liouville

SynopsisAsymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

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