Uniqueness for an Inverse Quantum-Dirac Problem with Given Weyl Function

Bohner, Martin; Çetinkaya, Ayça

doi:10.2478/tmmp-2023-0011

Abstract

In this work, we consider a boundary value problem for a q-Dirac equation. We prove orthogonality of the eigenfunctions, realness of the eigenvalues, and we study asymptotic formulas of the eigenfunctions. We show that the eigenfunctions form a complete system, we obtain the expansion formula with respect to the eigenfunctions, and we derive Parseval’s equality. We construct the Weyl solution and the Weyl function. We prove a uniqueness theorem for the solution of the inverse problem with respect to the Weyl function.

References

ABU RISHA, M. H.—ANNABY, M. H.—ISMAIL, M. E. H.—MANSOUR, Z. S.: Linear q-difference equations, Z. Anal. Anwend. 26 (2007), 481–494.
Search in Google Scholar Back to article
AL-TOWAILB, M. A.: A q-fractional approach to the regular Sturm–Liouville problems, Electron. J. Differential Equations (2017), Paper No. 88, 13 pp.
Search in Google Scholar Back to article
ALDWOAH, K. A.—MALINOWSKA, A. B.—TORRES, D. F. M.: The power quantum calculus and variational problems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19 (2012), 93–116.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: One-dimensional q-Dirac equation,Math. Methods Appl. Sci. 40 (2017), 7287–7306.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: Dissipative q-Dirac operator with general boundary conditions, Quaest. Math. 41 (2018), 239–255.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: An expansion theorem for q-Sturm–Liouville operators on the whole line,Turkish J. Math. 42 (2018), 1060–1071.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: Titchmarsh-Weyl theory for Dirac systems with transmission conditions, Mediterr. J. Math. 15 (2018), Paper No. 151, 12 pp.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: On expansion in eigenfunction for q-Dirac systems on the whole line, Math. Rep. (Bucur.) 21(71) (2019), 369–382.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: Dissipative qp-Dirac operator, Palest. J. Math. 9 (2020), 200–211.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: Qualitative spectral analysis of singular q-Sturm--Liouville operators, Bull. Malays. Math. Sci. Soc. 43 (2020), 1391–1402.
Search in Google Scholar Back to article
ALLAHVERDIEV, B. P.—TUNA, H.: Extensions of the matrix-valued q-Sturm–Liouville operators,Turkish J. Math. 45 (2021), 1479–1494.
Search in Google Scholar Back to article
ANNABY, M. H.—BUSTOZ, J.—ISMAIL, M. E. H.: On sampling theory and basic Sturm–Liouville systems, J. Comput. Appl. Math. 206 (2007), 73–85.
Search in Google Scholar Back to article
ANNABY, M. H.—MANSOUR, Z. S.: Basic Sturm–Liouville problems,J. Phys. A 38 (2005), 3775–3797.
Search in Google Scholar Back to article
ANNABY, M. H.—MANSOUR, Z. S.: Asymptotic formulae for eigenvalues and eigenfunctions of q-Sturm–Liouville problems,Math. Nachr. 284 (2011), 443–470.
Search in Google Scholar Back to article
ANNABY, M. H.—MANSOUR, Z. S.: q-Fractional Calculus and Equations,With a fore-word by Mourad Ismail. Lecture Notes in Math. Vol. 2056, Springer, Heidelberg, 2012.
Search in Google Scholar Back to article
BOHNER, M.—ÇETINKAYA, F. A.: A q-Dirac boundary value problem with eigenparameter-dependent boundary conditions, Appl. Anal. Discrete Math. 16 (2022), no. 2, 534–547.
Search in Google Scholar Back to article
ÇETINKAYA, F. A.: A discontinuous q-fractional boundary value problem with eigenparameter dependent boundary conditions, Miskolc Math. Notes 20 (2019), 795–806.
Search in Google Scholar Back to article
CONWAY, J. B.: Functions of One Complex Variable. II, Graduate Texts in Mathematics Vol. 159, Springer-Verlag, New York, 1995.
Search in Google Scholar Back to article
ERYILMAZ, A.: Spectral analysis of q-Sturm–Liouville problem with the spectral parameter in the boundary condition, J. Funct. Spaces Appl. (2012), Art. ID 736437, 17 pp.
Search in Google Scholar Back to article
EXTON, H.: q-hypergeometric Functions and Applications. With a foreword by L. J. Slater. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1983.
Search in Google Scholar Back to article
FREILING, G.—YURKO, V.: Inverse Sturm–Liouville Problems and their Applications. Nova Science Publishers, Inc., Huntington, NY, 2001.
Search in Google Scholar Back to article
GASPER, G.—RAHMAN, M.: Basic Hypergeometric Series. (With a foreword by Richard Askey). Second edition. Encyclopedia of Mathematics and its Applications Vol. 96. Cambridge University Press, Cambridge, 2004.
Search in Google Scholar Back to article
HAHN, W.: Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der Hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation,Math. Nachr. 2 (1949), 340–379.
Search in Google Scholar Back to article
ISMAIL, M. E. H.: The basic Bessel functions and polynomials, SIAM J. Math. Anal. 12 (1981), 454–468.
Search in Google Scholar Back to article
ISMAIL,M.E.H.: On Jackson’s third q-Bessel function, Preprint (1996).
Search in Google Scholar Back to article
JACKSON, F. H.: On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
Search in Google Scholar Back to article
JACKSON, F. H.: q-difference equations,Amer. J. Math. 32 (1910), 305–314.
Search in Google Scholar Back to article
KAC, V.—CHEUNG, P.: Quantum Calculus. Universitext. Springer-Verlag, New York, 2002.
Search in Google Scholar Back to article
KARAHAN, D.—MAMEDOV, K. R.: Sampling theory associated with q-Sturm-Liouville operator with discontinuity conditions, J. Contemp. Appl. Math. 10 (2020), 40–48.
Search in Google Scholar Back to article
KARAHAN, D.—MAMEDOV, K. R.: On a q-Boundary Value Problem with Discontinuity Conditions, Bulletin of the South Ural State University, Ser. Mathematics, Mechanics, Physics 13 (2021), 5–12.
Search in Google Scholar Back to article
KOELINK, H. T.—SWARTTOUW, R. F.: On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials, J. Math. Anal. Appl. 186 (1994), 690–710.
Search in Google Scholar Back to article
LAVAGNO, A.: Basic-deformed quantum mechanics,Rep. Math. Phys. 64 (2009), 79–91.
Search in Google Scholar Back to article
LEVITAN, B. M.—SARGSJAN, I. S.: Sturm–Liouville and Dirac operators.Translated from the Russian. Mathematics and its Applications (Soviet Series) Vol. 59, Kluwer Academic Publishers Group, Dordrecht, 1991.
Search in Google Scholar Back to article
MANSOUR, Z. S. I.: On fractional q-Sturm–Liouville problems, J. Fixed Point Theory Appl. 19 (2017), 1591–1612.
Search in Google Scholar Back to article
MATSUO, A.: Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations, Comm. Math. Phys. 151 (1993), 263–273.
Search in Google Scholar Back to article
MOSAZADEH, S.: Spectral properties and a Parseval’s equality in the singular case for q--Dirac problem, Adv. Difference Equ. (2019), Paper No. 522, 14 pp.
Search in Google Scholar Back to article
PALAMUT KOŞAR, N.: The Parseval identity for q-Sturm–Liouville problems with transmission conditions, Adv. Difference Equ. (2021), Paper No. 251, 12 pp.
Search in Google Scholar Back to article
PÓLYA, G.—ALEXANDERSON, G. L.: Gaussian binomial coefficients,Elem. Math. 26 (1971), 102–109.
Search in Google Scholar Back to article
TARIBOON, J.—NTOUYAS, S. K.: Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Difference Equ. (2013), 2013:282, 19 pp.
Search in Google Scholar Back to article

Uniqueness for an Inverse Quantum-Dirac Problem with Given Weyl Function

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