Abstract
In this survey, we present the principal results of Krein's spectral theory of a string and describe its development by other authors.
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References
M. S. Birman and V. V. Borzov, “On the asymptotics of discrete spectra of some singular differential operators,”Probl. Mat. Fiz., Issue 5, 24–38 (1971).
V. V. Borzov, “On the quantitative characteristics of singular measures,”Probl. Mat. Fiz., Issue 4, 42–47 (1970).
L. De Branges,Hilbert Spaces of Entire Functions, Prentice-Hall, London (1968).
S. Watanabe, “On time inversion of one-dimensional diffusion processes,”Z. Wahrscheinlichkeitstheor. verw. Geb.,31, 115–124 (1975).
F. R. Gantmakher and M. G. Krein,Oscillation Matrices and Kernels. Small Oscillations of Mechanical Systems [in Russian], Gostekhteorizdat, Moscow-Leningrad (1950).
I. Ts. Gokhberg and M. G. Krein,Theory of Volterra Operators in Hilbert Spaces and Its Applications [in Russian], Nauka, Moscow (1967).
H. Dym and H. P. McKean,Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York (1976).
E. B. Dynkin,Markov Processes [in Russian], Fizmatgiz, Moscow (1963).
K. Itô,Probability Processes [Russian translation], Issue 2, Inostr. Lit., Moscow (1963).
K. Itô and H. P. McKean,Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin-Heidelberg-New York (1965).
Y. Kasahara, “Spectral theory of generalized second order differential operators and its applications to Markov processes,”Jpn. J. Math.,1, No. 1, 67–84 (1975).
Y. Kasahara, “Limit theorems of occupation times for Markov processes,”Publ. RIMS, Kyoto Univ.,12, 801–818 (1977).
Y. Kasahara, S. Kotani, and H. Watanabe, “On the Green functions of 1-dimensional diffusion processes,”Publ. RIMS, Kyoto Univ.,16, 175–188 (1980).
I. S. Kats, “On the existence of spectral functions of singular second-order differential systems,”Dokl. Akad. Nauk SSSR,106, No. 1, 15–18 (1956).
I. S. Kats, “On the behavior of spectral functions of second-order differential systems,”Dokl. Akad. Nauk SSSR,106, No. 2, 183–186 (1956).
I. S. Kats, “General theorems on the behavior of spectral functions of second-order differential systems,”Dokl. Akad. Nauk SSSR,122, No. 6, 974–977 (1958).
I.S. Kats, “On the denseness of the spectrum of a string,”Dokl. Akad. Nauk SSSR,126, No. 6, 1180–1182 (1959).
I. S. Kats, “On the type of the spectrum of a singular string,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 1 (26), 57–64 (1962).
I. S. Kats, “Two general theorems on the asymptotic behavior of spectral functions of second-order differential systems,”Izv. Akad. Nauk SSSR, Ser. Mat.,26, No. 1, 53–78 (1962).
I. S. Kats, “On the multiplicity of the spectrum of a second-order differential operator,”Dokl. Akad. Nauk SSSR,145, No. 3, 510–513 (1962).
I. S. Kats, “Multiplicity of the spectrum of a-second-order differential operator and expansion in eigenfunctions,”Izv. Akad. Nauk SSSR, Ser. Mat.,27, No. 5, 1081–1112 (1963).
I. S. Kats, “On the behavior of spectral functions of second-order differential systems with a boundary condition at a singular end,”Dokl. Akad. Nauk SSSR,157, No. 1, 34–37 (1964).
I. S. Kats, “Existence of spectral functions of generalized second-order differential systems with boundary conditions at a singular end,”Mat. Sb.,68 (110), No. 2, 174–227 (1965).
I. S. Kats, “Some cases of the uniqueness of a solution of the inverse problem for strings with a boundary condition at a singular end,”Dokl. Akad. Nauk SSSR,164, No. 5, 975–978 (1965).
I. S. Kats, “Integral characteristics of the growth of spectral functions of the generalized second-order boundary-value problems with boundary conditions at a regular end,”Izv. Akad. Nauk. SSSR, Ser. Mat.,35, No. 1, 175–205 (1971).
I. S. Kats, “Power asymptotics of the spectral functions of generalized second-order boundary-value problems,”Dokl. Akad. Nauk SSSR,203, No. 4, 752–755 (1972).
I. S. Kats, “Generalization of the Marchenko asymptotic formula for spectral functions of a second-order boundary-value problem,”Izv. Akad. Nauk SSSR, Ser. Mat.,37, No. 2, 422–436 (1973).
I. S. Kats, “Denseness of the spectrum of a string,”Dokl. Akad. Nauk SSSR,221, No. 3, 16–19 (1973).
I. S. Kats, “Description of the set of spectral functions of a regular string bearing a concentrated mass at its end which is free of boundary conditions,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 7 (146), 27–33 (1974).
I. S. Kats, “Some general theorems on the denseness of the spectrum of a string,”Dokl. Akad. Nauk SSSR, 238, No. 4, 785–788 (1978).
I. S. Kats, “Integral estimates for the growth of spectral functions of a string,”Ukr. Mat. Th.,34, No. 3, 296–302 (1982).
I. S. Kats, “Integral estimates for the distribution of the spectrum of a string,”Sib. Mat. Zh.,27, No. 2, 62–74 (1986).
I. S. Kats, “Denseness of the spectrum of a singular string,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 23–30 (1990).
I. S. Kats and M. G. Krein, “A criterion of the discreteness of the spectrum of a singular string,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 2 (3), 136–153 (1958).
I. S. Kats and M. G. Krein, “R-functions are analytic functions that map the upper half plane onto itself,” Appendix 1 in: F. Atkinson,Discrete and Continuous Boundary-Value Problems [Russian translation], Mir, Moscow (1968).
I. S. Kats and M. G. Krein, “On spectral functions of a string,” Appendix 2 in: F. Atkinson,Discrete and Continuous Boundary-Value Problems [Russian translation], Mir, Moscow (1968).
L. P. Klotz and H. Langer, “Generalized resolvents and spectral functions of a matrix generalization of the Krein — Feller second order derivative,”Math. Nachr.,100, 163–186 (1981).
S. Kotani, “On a generalized Sturm — Liouville operator with a singular boundary,”J. Math. Kyoto Univ.,15, No. 2, 423–454 (1975).
S. Kotani and S. Watanabe, “Krein's spectral theory of strings and generalized diffusion processes,” Springer Lecture Notes Math.,923, 235–259 (1982).
M. G. Krein, “On the representations of functions by Fourier — Stieltjes integrals,”Uchen. Zapiski Kuibyshev. Ped. Inst., Issue 7, 123–148 (1943).
M. G. Krein, “On the problem of extension of spiral arcs in Hilbert spaces,”Dokl. Akad. Nauk SSSR,45, No. 4, 147–150 (1944).
M. G. Krein, “General method for the decomposition of positive definite kernels into elementary products,”Dokl. Akad. Nauk SSSR,53, No. 1, 3–6 (1946).
M. G. Krein, “To the theory of entire functions of exponential type,”Izv. Akad. Nauk SSSR, Ser. Mat.,11, 309–326 (1947).
M. G. Krein, “On Hermitian operators with guiding functionals,”Sb. Tr. Inst. Mat. Akad. Nauk Ukr. SSR,10, 83–106 (1948).
M. G. Krein, “Solution of the inverse Sturm — Liouville problem,”Dokl. Akad. Nauk SSSR,76, No. 1, 21–24 (1951).
M. G. Krein, “Determination of the density of an inhomogeneous symmetric string from the spectrum of its frequencies,”Dokl. Akad. Nauk SSSR,76, No. 3, 345–348 (1951).
M. G. Krein, “On certain problems of maximum and minimum for characteristic numbers and on the Lyapunov regions of stability,”Prikl. Mat. Mekh.,15, Issue 3, 323–348 (1951).
M. G. Krein, “On the indefinite case of the Sturm — Liouville boundary-value problem on the interval (0, •),”Izv. Akad. Nauk. SSSR, Ser. Mat.,16, No. 2, 293–324 (1952).
M. G. Krein, “On inverse problems for inhomogeneous strings,”Dokl. Akad. Nauk SSSR,82, No. 5, 669–672 (1952).
M. G. Krein, “On a generalization of Stieltjes results,”Dokl. Akad. Nauk SSSR,87, No. 6, 881–884 (1952).
M. G. Krein, “New problems in the theory of oscillations of Sturm systems,”Prikl. Mat. Mekh.,16, Issue 5, 555–568 (1952).
M. G. Krein, “On the transition function of a second-order one-dimensional boundary-value problem,”Dokl. Akad. Nauk SSSR,88, No. 3, 405–408 (1953).
M. G. Krein, “An analog of the Chebyshev — Markov inequality in a one-dimensional boundary-value problem,”Dokl. Akad. Nauk SSSR,89, No. 1, 5–8 (1953).
M. G. Krein, “Some cases of effective determination of the density of an inhomogeneous string from its spectral function,”Dokl. Akad. Nauk SSSR,93, No. 4, 617–620 (1953).
M. G. Krein, “On the inverse problems in filter theory and λ-domains of stability,”Dokl. Akad. Nauk SSSR,93, No. 5, 767–770 (1953).
M. G. Krein, “Principal approximation problem in the theory of extrapolation and filtration of stationary random processes,”Dokl. Akad. Nauk SSSR,94, No. 1, 13–16 (1954).
M. G. Krein, “An effective method for the solution of inverse boundary-value problems,”Dokl. Akad. Nauk SSSR,97, No. 6, 987–990 (1954).
M. G. Krein, “Integral equations that generate second-order differential equations,”Dokl. Akad. Nauk SSSR,97, No. 4, 21–24 (1954).
M. G. Krein, “Determination of the potential of a particle according to itsS-function,”Dokl. Akad. Nauk SSSR,105, No. 3, 433–436 (1955).
M. G. Krein, “Continual analogs of the assertions on polynomials orthogonal on the unit circle,”Dokl. Akad. Nauk SSSR,105, No. 4, 637–640 (1955).
M. G. Krein, “On the theory of accelerants andS-matrices of canonical differential systems,”Dokl. Akad. Nauk SSSR,111, No. 6, 1167–1170 (1956).
M. G. Krein, “Chebyshev-Markov inequalities in the theory of spectral functions of a string,”Mat. Issled.,5, No. 1, 77–101 (1970).
U. Küchler, “Some asymptotic properties of the transition densities of one-dimensional quasidiffusions,”Publ. RIMS, Kyoto Univ.,16, No. 1, 245–268 (1980).
H. Langer,Zur Spectraltheorie Verallgemeinerter Cewönnlicher Differenzialoperatoren Zweiter Ordung mit Einer Nichtmonotonen Gewichtfunktion, University of Jyväskylä, Department of Mathematics, Report 14 (1972).
H. Langer, L. Partzsch, and D. Schütze, “Über verallgemeinerte gewönliche Differentialoperatoren mit nichtlokalen Randbedingungen und die von ihnen erzeugten Markov-Processe,”Publ. RIMS, Kyoto Univ.,7, No. 3, 660–702 (1972).
B. M. Levitan, “On the asymptotic behavior of the spectral function of a self-adjoint second-order differential equation,”Izv. Akad. Nauk SSSR, Ser. Mat.,16, 325–352 (1952).
V. A. Marchenko, “Some problems in the theory of one-dimensional second-order linear differential operators. I,”Tr. Mosk. Mat. Obshch.,1, 381–422 (1952).
H. P. McKean and D. B. Ray, “Spectral distribution of a differential operator,”Duke Math. J.,29, 281–292 (1962).
M. A. Naimark,Linear Differential Operators [in Russian], Nauka, Moscow (1969).
M. Tomisaki, “On the asymptotic behavior of transition probability densities of one-dimensional diffusion processes,”Publ. RIMS Kyoto Univ.,12, 819–834 (1977).
T. Uno and I. Hong, “Some consideration of eigenvalues for the equationd 2 u/dx 2+ λρ(x)u=0”Jpn. J. Math.,29, 152–164 (1959).
W. Feller, “On second order differential operators,”Ann. Math.,61, 90–105 (1955).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 155–176, March, 1994.
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Kats, I.S. Spectral theory of a string. Ukr Math J 46, 159–182 (1994). https://doi.org/10.1007/BF01062233
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DOI: https://doi.org/10.1007/BF01062233