1D nonnegative Schrodinger operators with point interactions

Author Yu. G. Kovalev
yury_kovalev@ukr.net, yury.kovalev.lugansk@gmail.com
Department of Mathematical Analysis, East Ukrainian National University

Abstract Let $Y$ be an infinite discrete set of points in $\mathbb R $, satisfying the condition $\inf\{|y-y'|,\; y,y'\in Y, y'\ne y\}>0.$ In the paper we prove that the systems $\{\delta(x-y)\}_{y\in Y}, \;\{\delta'(x-y)\}_{y\in Y}, \{\delta(x-y),\;\delta'(x-y)\}_{y\in Y}$ form Riesz bases in the corresponding closed linear spans in the Sobolev spaces $W_2^{-1}(\mathbb R )$ and $W_2^{-2}(\mathbb R )$. As an application, we prove the transversalness of the Friedrichs and Kre\u\i n nonnegative selfadjoint extensions of the nonnegative symmetric operators $A_0$, $A'$, and $H_0$ defined as restrictions of the operator $A =-\frac{ d^2}{ dx^2},$ $\mathop{\rm dom} (A)=W^2_2(\mathbb R )$ to the linear manifolds $\mathop{\rm dom} (A_0)=\left\{ f\in W_2^2(\mathbb{R})\colon f(y)=0,\; y\in Y \right\}$, $\mathop{\rm dom} (A')=\{ g\in W_2^2(\mathbb{R})\colon g'(y)=0,\; y\in Y \},$ and $\mathop{\rm dom} (H_0)=\left\{f\in W_2^2(\mathbb{R})\colon f(y)=0,\;f'(y)=0,\; y\in Y \right\}$, respectively. Using the divergence forms, the basic nonnegative boundary triplets for $A^*_0$, $A'^*$, and $H^*_0$ are constructed.
Keywords point interaction; Riesz basis; boundary triplet; the Friedrichs extension; the Krein extension
Reference 1. Albeverio S., Gesztesy F., Hegh-Krohn R., Holden H., Solvable models in quantum mechanics. - Texts and Monographs in Physics. Springer-Verlag, New York.

2. Arlinskii Yu.M. Positive spaces of boundary values and sectorial extensions of a nonnegative symmetric operator// Ukrain. Math. Journ. - 1988. - V.40, ¹1. - P. 8-14. (in Russian)

3. Arlinskių Yu., Hassi S., Sebestyen Z., de Snoo H. On the class of extremal extensions of a nonnegative operators// Operator Theory: Advan. and Appl. - 2001. - V.127. - P. 41-81.

4. Arlinskii Yu.M., Tsekanovskii E.R. The von Neumann problem for nonnegative symmetric operators// Integral equation and operator theory. - 2005. - P. 315-356.

5. Arlinskii Yu.M., Kovalev Yu.G. Operators in divergence form and their Friedrichs and Krein extensions// Opuscula mathematica. - 2011. - V.31, ¹4. - P. 501-517.

6. Berezansky Yu.M., Expansions in eigenfunction of selfadjoint operators. - "Naukova Dumka", Kiev, 1965 (in Russian). English translation in Translations of Mathematical Monographs. Amer. Math. Soc. Providence, 1968, V.17.

7. Bruk V.M. On one class of boundary value problems with a spectral parameter in the boundary condition// Mat. Sbornik. - 1976. - V.100, ¹2. - P. 210-216. (in Russian)

8. Gohberg I.C., Krein M.G., Introduction to the theory of linear nonselfadjoint operators. - M: Nauka, 1965 (in Russian). English translation in Translations of Mathematical Monographs, V.18, Amer. Math. Soc., Providence, R.I., 1969.

9. Gorbachuk M.L., Gorbachuk V.I., Boundary value problems for differential-operator equations. - K.: Naukova Dumka, 1984. (in Russian)

10. Kato T., Perturbation theory for linear operators. - Springer-Verlag, 1966.

11. Kochubei A.N. On extensions of symmetric operators and symmetric binary relations// Math. Zametki. - 1975. - V.17, ¹1. - P. 41-48. (in Russian)

12. Kochubei A.N. One dimensional point interactions// Ukr. Math. J. - 1989. - V.41, ¹10. - P. 90-95.

13. Kostenko A.S., Malamud M.M. 1-D Schrodinger operators with local point interactions on a discrete set// J. Differ. Equations. - 2010. - V.249, ¹2. - P. 253-304.

14. Krein M.G. The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, I// Mat. Sbornik. - 1947. - V.20, ¹3. - P. 431-495. (in Russian)

15. Lyantse V.E., Storozh O.G., Methods of the theory of unbounded operators. - K.: Naukova Dumka, 1983. (in Russian)

16. Malamud M.M. On some classes of Hermitian operators with gaps// Ukrainian Mat. J. - 1992. - V.44, ¹2. - P. 215-234. (in Russian)

17. Pipa H.M., Storozh O.G. Accretive perturbations of proper extensions for positively definite operator// Mat. Stud. - 2006. - V.25, ¹2. - P. 181-190. (in Ukrainian)

Pages 150-163
Volume 39
Issue 2
Year 2013
Journal Matematychni Studii
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