$\mathbf {\operatorname {SL} _2({\mathbb {R}})}$, exponential Herglotz representations, and spectral averaging
HTML articles powered by AMS MathViewer
- by Fritz Gesztesy and Konstantin A. Makarov
- St. Petersburg Math. J. 15 (2004), 393-418
- DOI: https://doi.org/10.1090/S1061-0022-04-00814-3
- Published electronically: March 30, 2004
- PDF | Request permission
Abstract:
We revisit the concept of spectral averaging and point out its relationship with one-parameter subgroups of $\operatorname {SL}_2({\mathbb {R}})$ and the corresponding Möbius transformations. In particular, we identify exponential Herglotz representations as the basic ingredient for the absolute continuity of averaged spectral measures with respect to Lebesgue measure; the associated spectral shift function turns out to be the corresponding density for the averaged measure. As a by-product of our investigations we unify the treatment of rank-one perturbations of selfadjoint operators and that of selfadjoint extensions of symmetric operators with deficiency indices $(1,1)$. Moreover, we derive separate averaging results for absolutely continuous, singular continuous, and pure point measures and conclude with an averaging result for the $\kappa$-continuous part (with respect to the $\kappa$-dimensional Hausdorff measure) of singular continuous measures.References
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495â503. MR 0000089
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771â782. MR 19, DOI 10.2307/2371335
- N. Aronszajn and W. F. Donoghue, On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Anal. Math. 5 (1956â57), 321â388.
- Peter Scherk, Bemerkungen zu einer Note von Besicovitch, J. London Math. Soc. 14 (1939), 185â192 (German). MR 29, DOI 10.1112/jlms/s1-14.3.185
- Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635â646. MR 98
- T. RadĂł and P. Reichelderfer, On a stretching process for surfaces, Amer. J. Math. 61 (1939), 645â650. MR 47, DOI 10.2307/2371318
- S. Losinsky, Sur le procĂ©dĂ© dâinterpolation de FejĂ©r, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318â321 (French). MR 0002001
- Alfred Rosenblatt, Sur les points singuliers des Ă©quations diffĂ©rentielles, C. R. Acad. Sci. Paris 209 (1939), 10â11 (French). MR 85
- R. H. J. Germay, GĂ©nĂ©ralisation de lâĂ©quation de Hesse, Ann. Soc. Sci. Bruxelles SĂ©r. I 59 (1939), 139â144 (French). MR 86
- W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202â279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63â97. MR 95, DOI 10.1007/BF02547750
- J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560â566. MR 97, DOI 10.2307/1968940
- I. M. Kamenetzky, Sur lâinterpolation au moyen des dĂ©rivĂ©es et sur les procĂ©dĂ©s dâinterpolation correspondants. I, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25 (1939), 356â358 (French). MR 0002003
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712â730. MR 12, DOI 10.2307/1968951
- I. M. Kamenetzky, Sur lâinterpolation au moyen des dĂ©rivĂ©es et sur les procĂ©dĂ©s dâinterpolation correspondants. I, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25 (1939), 356â358 (French). MR 0002003
- J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560â566. MR 97, DOI 10.2307/1968940
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63â97. MR 95, DOI 10.1007/BF02547750
- R. H. J. Germay, GĂ©nĂ©ralisation de lâĂ©quation de Hesse, Ann. Soc. Sci. Bruxelles SĂ©r. I 59 (1939), 139â144 (French). MR 86
- Radu BÇdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571â577. MR 0000087
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479â494. MR 0000088
- Louis Weisner, Condition that a finite group be multiply isomorphic with each of its irreducible representations, Amer. J. Math. 61 (1939), 709â712. MR 32, DOI 10.2307/2371325
- P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294â297. MR 2000, DOI 10.1073/pnas.26.4.294
- S. Losinsky, Sur le procĂ©dĂ© dâinterpolation de FejĂ©r, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318â321 (French). MR 0002001
- M. Gontcharoff, Sur quelques sĂ©ries dâinterpolation gĂ©nĂ©ralisant celles de Newton et de Stirling, Uchenye Zapiski Moskov. Gos. Univ. Matematika 30 (1939), 17â48 (Russian, with French summary). MR 0002002
- S. Losinsky, Sur le procĂ©dĂ© dâinterpolation de FejĂ©r, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318â321 (French). MR 0002001
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63â97. MR 95, DOI 10.1007/BF02547750
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63â97. MR 95, DOI 10.1007/BF02547750
- J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560â566. MR 97, DOI 10.2307/1968940
- Reinhold Baer, Groups with Abelian norm quotient group, Amer. J. Math. 61 (1939), 700â708. MR 34, DOI 10.2307/2371324
- Stefan Kempisty, Sur les fonctions Ă variation bornĂ©e au sens de Tonelli, Bull. SĂ©m. Math. Univ. Wilno 2 (1939), 13â21 (French). MR 46
- S. Losinsky, Sur le procĂ©dĂ© dâinterpolation de FejĂ©r, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318â321 (French). MR 0002001
- Katharine E. OâBrien, Some problems in interpolation by characteristic functions of linear differential systems of the fourth order, Bull. Amer. Math. Soc. 46 (1940), 281â290. MR 2004, DOI 10.1090/S0002-9904-1940-07197-6
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479â494. MR 0000088
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495â503. MR 0000089
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425â428. MR 6, DOI 10.2307/2303037
- R. H. J. Germay, GĂ©nĂ©ralisation de lâĂ©quation de Hesse, Ann. Soc. Sci. Bruxelles SĂ©r. I 59 (1939), 139â144 (French). MR 86
- I. M. Kamenetzky, Sur lâinterpolation au moyen des dĂ©rivĂ©es et sur les procĂ©dĂ©s dâinterpolation correspondants. I, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25 (1939), 356â358 (French). MR 0002003
- J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560â566. MR 97, DOI 10.2307/1968940
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425â428. MR 6, DOI 10.2307/2303037
- HidegorĂŽ Nakano, Ăber Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357â375 (German). MR 94
- H. S. Vandiver, Certain congruences involving the Bernoulli numbers, Duke Math. J. 5 (1939), 548â551. MR 21
- L. E. Dickson, All integers except $23$ and $239$ are sums of eight cubes, Bull. Amer. Math. Soc. 45 (1939), 588â591. MR 28, DOI 10.1090/S0002-9904-1939-07041-9
- Radu BÇdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571â577. MR 0000087
- J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560â566. MR 97, DOI 10.2307/1968940
- Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635â646. MR 98
- Radu BÇdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571â577. MR 0000087
- M. Gontcharoff, Sur quelques sĂ©ries dâinterpolation gĂ©nĂ©ralisant celles de Newton et de Stirling, Uchenye Zapiski Moskov. Gos. Univ. Matematika 30 (1939), 17â48 (Russian, with French summary). MR 0002002
- Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549â559 (German). MR 96, DOI 10.2307/1968939
- HidegorĂŽ Nakano, Ăber Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357â375 (German). MR 94
Bibliographic Information
- Fritz Gesztesy
- Affiliation: Department of Mathematics, University of Missouri, Columbia, MO 65211
- MR Author ID: 72880
- Email: fritz@math.missouri.edu
- Konstantin A. Makarov
- Affiliation: Department of Mathematics, University of Missouri, Columbia, MO 65211
- Email: makarov@math.missouri.edu
- Published electronically: March 30, 2004
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 15 (2004), 393-418
- MSC (2000): Primary 34B20, 47A11; Secondary 34L05, 47A10
- DOI: https://doi.org/10.1090/S1061-0022-04-00814-3
- MathSciNet review: 2052165