Open Access

# Scale-invariant self-adjoint extensions of scale-invariant symmetric operators: continuous versus discrete

### Abstract

We continue our study of a $q$-difference version of a second-order differential operator which depends on a real parameter. This version was introduced in our previous three articles on the subject. First we study general symmetric and scale-invariant operators on a Hilbert space. We show that if the index of defect of the operator under consideration is $(1,1)$, then the operator either does not admit any scale-invariant self-adjoint extension, or it admits exactly one scale-invariant self-adjoint extension, or it admits exactly two scale-invariant self-adjoint extensions, or all self-adjoint extensions are scale invariant. We then apply these results to the differential operator and the corresponding difference operator under consideration. For the continuous case, we show that the interval of the parameter, for which the differential operator is not semi-bounded, contains an infinite sequence of values for which all self-adjoint extensions are scale-invariant, while for the remaining values of the parameter from that interval, there are no scale-invariant self-adjoint extensions. For the corresponding difference operator, we show that if it is not semi-bounded, then it does not admit any scale-invariant self-adjoint extension. We also show that both differential and difference operators, at value(s) of the parameter that cor espond to the endpoint(s) of the interval(s) of semi-boundedness, have exactly one scale-invariant self-adjoint extension.

Key words: q-difference operator, self-adjoint, scale-invariant, discrete spectrum, simple spectrum.

### Article Information

 Title Scale-invariant self-adjoint extensions of scale-invariant symmetric operators: continuous versus discrete Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 41-55 MathSciNet MR3407919 zbMATH 06533466 Milestones Received 16/09/2014; Revised 24/10/2014 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

Miron B. Bekker
Department of Mathematics, University of Pittsburgh at Johnstown, Johnstown, PA, USA

Martin J. Bohner
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

Mark A. Nudel'man
Integrated Banking Information Systems, Odessa, 65125, Ukraine

Hristo Voulov
Department of Mathematics and Statistics, University of Missouri-Kansas City, Kansas City, MO, USA

### Citation Example

Miron B. Bekker, Martin J. Bohner, Mark A. Nudel'man, and Hristo Voulov, Scale-invariant self-adjoint extensions of scale-invariant symmetric operators: continuous versus discrete, Methods Funct. Anal. Topology 21 (2015), no. 1, 41-55.

### BibTex

@article {MFAT750,
AUTHOR = {Bekker, Miron B. and Bohner, Martin J. and Nudel'man, Mark A. and Voulov, Hristo},
TITLE = {Scale-invariant self-adjoint extensions of scale-invariant symmetric operators: continuous versus discrete},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {1},
PAGES = {41-55},
ISSN = {1029-3531},
MRNUMBER = {MR3407919},
ZBLNUMBER = {06533466},
URL = {http://mfat.imath.kiev.ua/article/?id=750},
}

### References

1. Milton Abramowitz, Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.  MathSciNet
2. N. I. Akhiezer, I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications, Inc., New York, 1993.  MathSciNet
3. G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Tom II. Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny., Spravochnaya Matematicheskaya Biblioteka, Nauka, Moscow [Higher transcendental functions. Vol. II. Bessel functions, parabolic cylinder functions, orthogonal polynomials. Translated from the English by N. Ja. Vilenkin, Second edition, unrevised. Mathematical Reference Library], 1974.  MathSciNet
4. Borislava Bekker, Miron B. Bekker, On selfadjoint homogeneous operators, Complex Anal. Oper. Theory 7 (2013), no. 1, 9-31.  MathSciNet CrossRef
5. Miron B. Bekker, On a class of nondensely defined Hermitian contractions, Adv. Dyn. Syst. Appl. 2 (2007), no. 2, 141-165.  MathSciNet
6. Miron B. Bekker, Martin J. Bohner, Alexander N. Herega, Hristo Voulov, Spectral analysis of a $q$-difference operator, J. Phys. A 43 (2010), no. 14, 145207, 15.  MathSciNet CrossRef
7. Miron B. Bekker, Martin J. Bohner, Hristo Voulov, A $q$-difference operator with discrete and simple spectrum, Methods Funct. Anal. Topology 17 (2011), no. 4, 281-294.  MathSciNet
8. Miron B. Bekker, Martin J. Bohner, Hristo Voulov, Extreme self-adjoint extensions of a semibounded $q$-difference operator, Math. Nachr. 287 (2014), no. 8-9, 869-884.  MathSciNet CrossRef
9. Nelson Dunford, Jacob T. Schwartz, Linear operators. Part II, John Wiley & Sons, Inc., New York, 1988.  MathSciNet
10. Seppo Hassi, Sergii Kuzhel, On symmetries in the theory of finite rank singular perturbations, J. Funct. Anal. 256 (2009), no. 3, 777-809.  MathSciNet CrossRef
11. A. N. Kocubei, Symmetric operators commuting with a family of unitary operators, Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 77-78.  MathSciNet
12. M. G. Krein, Ju. L. Smul′jan, Fractional linear transformations with operator coefficients, Mat. Issled 2 (1967), no. vyp. 3, 64-96.  MathSciNet
13. A. V. Kuzhel, S. A. Kuzhel, Regular extensions of Hermitian operators, VSP, Utrecht, 1998.  MathSciNet
14. K. A. Makarov, E. Tsekanovskii, On $\mu$-scale invariant operators, Methods Funct. Anal. Topology 13 (2007), no. 2, 181-186.  MathSciNet
15. M. A. Naimark, Lineinye differentsialnye operatory, Izdat. Nauka'', Moscow, 1969.  MathSciNet
16. Ju. L. Smul′jan, Operator balls, Integral Equations Operator Theory 13 (1990), no. 6, 864-882.  MathSciNet CrossRef
17. Joachim Weidmann, Spectral theory of ordinary differential operators, Springer-Verlag, Berlin, 1987.  MathSciNet