Restricted interpolation by meromorphic inner functions

• Autorzy: Alexei Poltoratski , Rishika Rupam
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Inner functions; Model spaces; Schrödinger operators

• Wydawca: De Gruyter Open
• Czasopismo: Concrete Operators
• Rocznik: 2016
• Tom: 3
• Numer: 1
• Strony: 102-111

Daty

otrzymano

2015-10-29

zaakceptowano

2016-06-18

online

2016-07-11

Twórcy

autor

Alexei Poltoratski

• Texas A&M University, Department of Mathematics, College Station, TX 77843, USA

autor

Rishika Rupam

• Laboratoire Paul Painlevé, Université des Sciences et Technologies Lille 1, 59655 Villeneuve d’Ascq Cédex, France

Bibliografia

• [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.
• [3] V. Ambarzumian. Über eine frage der eigenwerttheorie. Zeitschrift für Physik A: Hadrons and Nuclei, 53(9):690–695, (1929).
• [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).
• [6] A. Beurling and P. Malliavin. On the closure of characters and the zeros of entire functions. Acta Mathematica, 118(1):79–93, (1967).
• [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).
• [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).
• [9] D. N. Clark. One dimensional perturbations of restricted shifts. Journal d’Analyse Mathematique, 25(1):169–191, (1972).
• [10] L. de Branges. Hilbert Spaces of Entire Functions. Prentice Hall, Englewood Cliffs, NJ, (1968).
• [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).
• [12] F. Gesztesy and B. Simon. On the determination of a potential from three spectra. Differential Operators and Spectral Theory, 189:85–92, (1999).
• [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).
• [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).
• [16] N. Levinson. The inverse sturm-liouville problem. Matematisk Tidsskrift. B, pages 25–30, (1949).
• [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).
• [19] N. Makarov and A. Poltoratski. Meromorphic inner functions, toeplitz kernels and the uncertainty principle. In Perspectives in Analysis, pages 185–252. Springer, (2005).
• [20] N. Makarov and A. Poltoratski. Beurling-malliavin theory for toeplitz kernels. Inventiones Mathematicae, 180(3):443–480, (2010).
• [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).
• [23] M. Mitkovski and A. Poltoratski. On the determinacy problem for measures. Inventiones Mathematicae, 202(3):1241–1267, (2015).
• [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).
• [26] A. Poltoratski. A problem on completeness of exponentials. Annals of Mathematics, 178(3):983–1016, (2013).
• [27] A. Poltoratski. Toeplitz Approach to Problems of the Uncertainty Principle, volume 121 of CBMS. American Mathematical Society, (2015).
• [28] A. Poltoratski and D. Sarason. Aleksandrov-clark measures. Contemporary Mathematics, 393:1–14, (2006).
• [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].

Identyfikatory

DOI

10.1515/conop-2016-0012