Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.
@article{bwmeta1.element.doi-10_1515_conop-2016-0012,
author = {Alexei Poltoratski and Rishika Rupam},
title = {Restricted interpolation by meromorphic inner functions},
journal = {Concrete Operators},
volume = {3},
year = {2016},
pages = {102-111},
zbl = {1345.30083},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_conop-2016-0012}
}
Alexei Poltoratski; Rishika Rupam. Restricted interpolation by meromorphic inner functions. Concrete Operators, Tome 3 (2016) pp. 102-111. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_conop-2016-0012/
[1] A. B. Aleksandrov. Multiplicity of boundary values of inner functions. Izv. Akad. Nauk Arm. SSR, 22:490–503, (1987).
[2] A. B. Aleksandrov. Isometric embeddings of coinvariant subspaces of the shift operator. Zapiski Nauchnykh Seminarov, S.- Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 232:5–15, 1996. no. Issled. po Linein. Oper. i Teor. Funktsii. 24, 5–15, 213; English transl., Journal of Mathematical Sciences (New York) 92 (1998), no. 1, 3543–3549. | Zbl 0888.30023
[3] V. Ambarzumian. Über eine frage der eigenwerttheorie. Zeitschrift für Physik A: Hadrons and Nuclei, 53(9):690–695, (1929). | Zbl 55.0868.01
[4] A. D. Baranov. De branges’ mistake. Preprint (Private Communications), (2011).
[5] A. Beurling and P. Malliavin. On fourier transforms of measures with compact support. Acta Mathematica, 107(3):291–309, (1962). | Zbl 0127.32601
[6] A. Beurling and P. Malliavin. On the closure of characters and the zeros of entire functions. Acta Mathematica, 118(1):79–93, (1967). | Zbl 0171.11901
[7] G. Borg. Uniqueness theorems in the spectral theory of y" +(λ - q(x))y =0. In Proc. of the 11th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo, pages 276–287, (1952). | Zbl 0048.06802
[8] J. A. Cima, A. L. Matheson, and W. T. Ross. The Cauchy transform. Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, (2006). | Zbl 1096.30046
[9] D. N. Clark. One dimensional perturbations of restricted shifts. Journal d’Analyse Mathematique, 25(1):169–191, (1972). | Zbl 0252.47010
[10] L. de Branges. Hilbert Spaces of Entire Functions. Prentice Hall, Englewood Cliffs, NJ, (1968). | Zbl 0157.43301
[11] R. del Rio, F. Gesztesy, and B. Simon. Inverse spectral analysis with partial information on the potential. III. updating boundary conditions. International Mathematics Research Notices, 1997(15):751–758, (1997). | Zbl 0898.34075
[12] F. Gesztesy and B. Simon. On the determination of a potential from three spectra. Differential Operators and Spectral Theory, 189:85–92, (1999). | Zbl 0922.34008
[13] F. Gesztesy and B. Simon. Inverse spectral analysis with partial information on the potential, II. the case of discrete spectrum. Transactions of the American Mathematical Society, 352(6):2765–2787, (2000). | Zbl 0948.34060
[14] M. Horváth. Inverse spectral problems and closed exponential systems. Annals of Mathematics, 162(2):885–918, (2005).
[15] M. G. Krein. On the trace formula in perturbation theory. Matematicheskii Sbornik, 75(3):597–626, (1953). | Zbl 0052.12303
[16] N. Levinson. The inverse sturm-liouville problem. Matematisk Tidsskrift. B, pages 25–30, (1949). | Zbl 0045.36402
[17] B. M. Levitan and I. S. Sargsjan. Sturm-Liouville and Dirac operators. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, (1991).
[18] I. M. Lifshits. On a problem of the theory of perturbations connected with quantum statistics. Uspekhi Matematicheskikh Nauk, 7(1):171–180, (1952). | Zbl 0046.21203
[19] N. Makarov and A. Poltoratski. Meromorphic inner functions, toeplitz kernels and the uncertainty principle. In Perspectives in Analysis, pages 185–252. Springer, (2005). | Zbl 1118.47020
[20] N. Makarov and A. Poltoratski. Beurling-malliavin theory for toeplitz kernels. Inventiones Mathematicae, 180(3):443–480, (2010). | Zbl 1186.47025
[21] V. A. Marchenko. Some questions in the theory of one-dimensional linear differential operators of the second order. Trudy Mosk. Mat. Obsch. 1 327–420, American Mathematical Society Translations, 2(101):1–104, (1952).
[22] M. Mitkovski and A. Poltoratski. Pólya sequences, toeplitz kernels and gap theorems. Advances in Mathematics, 224(3):1057– 1070, (2010). | Zbl 1204.30018
[23] M. Mitkovski and A. Poltoratski. On the determinacy problem for measures. Inventiones Mathematicae, 202(3):1241–1267, (2015). | Zbl 1346.30015
[24] M. Mitkovski and R. Rupam. Defining sets for mifs and uniqueness of potentials. In Preparation.
[25] A. Poltoratski. Spectral gaps for sets and measures. Acta Mathematica, 208(1):151–209, (2012). | Zbl 1245.42008
[26] A. Poltoratski. A problem on completeness of exponentials. Annals of Mathematics, 178(3):983–1016, (2013). | Zbl 1285.42009
[27] A. Poltoratski. Toeplitz Approach to Problems of the Uncertainty Principle, volume 121 of CBMS. American Mathematical Society, (2015). | Zbl 1317.47003
[28] A. Poltoratski and D. Sarason. Aleksandrov-clark measures. Contemporary Mathematics, 393:1–14, (2006). | Zbl 1102.30032
[29] R. Rupam. Existence of meromorphic inner functions based on spectral data. In Preparation.
[30] R. Rupam. Uniform boundedness of derivatives of meromorphic inner functions on the real line. Journal d’Analyse Mathematique, to appear. arXiv preprint :1309.6728
[5].