References
- Faris WG. Self-adjoint operators. Berlin: Springer; 1975. (Vol. 433, Lecture Notes in Mathematics).
- Kato T. Perturbation theory for linear operators. corr. printing of the 2nd edn. Berlin: Springer; 1980.
- Krein MG. The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I. Mat Sbornik. 1947;20:431–495. (Russian).
- Alonso A, Simon B. The Birman–Krein–Vishik theory of selfadjoint extensions of semibounded operators. J Operator Theory. 1980;4:251–270; ; Addenda: 6, 407 (1981).
- Arlinski Yu. M, Tsekanovski ER. The von Neumann problem for nonnegative symmetric operators. Integr Equ Oper Theory. 2005;51:319–356.
- von Neumann J. Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math Ann. 1929–30;102:49–131.
- Krein MG. The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. II. Mat Sbornik. 1947;21:365–404. (Russian).
- Gesztesy F, Littlejohn LL, Nichols R. On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below. J Differ Equ. 2020;269:6448–6491.
- Gesztesy F, Zinchenko M. Sturm–Liouville Operators, Their Spectral Theory, and Some Applications. in preparation.
- Leighton W, Morse M. Singular quadratic functionals. Trans Am Math Soc. 1936;40:252–286.
- Rellich F. Die zulässigen Randbedingungen bei den singulären Eigenwertproblemen der mathematischen Physik. (Gewöhnliche Differentialgleichungen zweiter Ordnung.). Math Z. 1943/44;49:702–723. (German).
- Rellich F. Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung. Math Ann. 1951;122:343–368. (German).
- Hartman P, Wintner A. On the assignment of asymptotic values for the solutions of linear differential equations of second order. Am J Math. 1955;77:475–483.
- Clark S, Gesztesy F, Nichols R. Principal solutions revisited. in Stochastic and Infinite Dimensional Analysis. C. C. Bernido, M. V. Carpio-Bernido, M. Grothaus, T. Kuna, M. J. Oliveira, and J. L. da Silva (eds.), Trends in Mathematics, Birkhäuser, Springer, pp. 85–117. 2016.
- Dunford N, Schwartz JT. Linear operators. part II: spectral theory. New York: Wiley Interscience; 1988.
- Hartman P. Ordinary differential equations. Philadelphia: SIAM; 2002.
- Niessen H-D, Zettl A. Singular Sturm–Liouville problems: the Friedrichs extension and comparison of eigenvalues. Proc London Math Soc (3). 1992;64:545–578.
- Zettl A. Sturm–Liouville theory. Providence, RI: Amer. Math. Soc.; 2005. (Vol. 121, Mathematical Surveys and Monographs).
- Kalf H. A characterization of the Friedrichs extension of Sturm–Liouville operators. J London Math Soc (2). 1978;17:511–521.
- Rosenberger R. A new characterization of the Friedrichs extension of semibounded Sturm–Liouville operators. J London Math Soc (2). 1985;31:501–510.
- Akhiezer NI, Glazman IM. Theory of linear operators in Hilbert space, Vol. II. Boston: Pitman; 1981.
- Coddington EA, Levinson N. Theory of ordinary differential equations. Malabar, FL: Krieger Publ.; 1985.
- Jörgens K, Rellich F. Eigenwerttheorie Gewöhnlicher Differentialgleichungen. Berlin: Springer-Verlag; 1976.
- Naimark MA. Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space. Transl. by E. R. Dawson, Engl. translation edited by W. N. Everitt, Ungar Publishing, New York. 1968.
- Pearson DB. Quantum scattering and spectral theory. London: Academic Press; 1988.
- Teschl G. Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Math., Vol. 157, Amer. Math. Soc., RI. 2014.
- Weidmann J. Linear operators in Hilbert spaces. New York: Springer; 1980. (Vol. 68, Graduate Texts in Mathematics).
- Weidmann J. Lineare Operatoren in Hilberträumen. Teil II: Anwendungen. Stuttgart: Teubner; 2003.
- Ando T, Nishio K. Positive selfadjoint extensions of positive symmetric operators. Tohoku Math J (2). 1970;22:65–75.
- Yu.M., Arlinskii, S., Hassi, Z., Sebestyén, H.S.V., de Snoo: ‘On the class of extremal extensions of a nonnegative operator, in Recent Advances in Operator Theory and Related Topics. L. Kérchy, C. Foias, I. Gohberg, and H. Langer (eds.), Operator Theory: Advances and Applications, Vol. 127, Birkhäuser, Basel, 2001, pp. 41–81.
- Yu., Arlinski, E., Tsekanovski: ‘M.Kren's research on semibounded operators, its contemporary developments, and applications’, in V., Adamyan, Y.M., Berezansky, I., Gohberg, M.L., Gorbachuk, V., Gorbachuk, A.N., Kochubei, H., Langer, G., Popov (Eds.): ‘Modern analysis and applications. the Mark Krein centenary conference. Operator theory: advances and applications’, vol. 1 (Birkhäuser, Basel, 2009), pp. 65–112.
- Ashbaugh MS, Gesztesy F, Mitrea M, Teschl G. Spectral theory for perturbed Krein Laplacians in nonsmooth domains. Adv Math. 2010;223:1372–1467.
- Ashbaugh MS, Gesztesy F, Mitrea M, Shterenberg R, Teschl G. The Krein–von Neumann extension and its connection to an abstract buckling problem. Math Nachr. 2010;283:165–179.
- Ashbaugh MS, Gesztesy F, Mitrea M, Shterenberg R, Teschl G. A survey on the Krein–von Neumann extension, the corresponding abstract buckling problem, and Weyl-type spectral asymptotics for perturbed Krein Laplacians in non smooth domains. in Mathematical Physics, Spectral Theory and Stochastic Analysis, M. Demuth and W. Kirsch (eds.), Operator Theory: Advances and Applications, Vol. 232, Birkhäuser, Springer, Basel, pp. 1–106. 2013.
- Ashbaugh MS, Gesztesy F, Laptev A, Mitrea M, Sukhtaiev S. A bound for the eigenvalue counting function for Krein–von Neumann and Friedrichs extensions. Adv Math. 2017;304:1108–1155.
- Behrndt J, Hassi S, de Snoo H. Boundary value problems, Weyl functions, and differential operators. Springer: Birkhäuser; 2020. (vol. 105, Monographs in Math.).
- Sh. Birman M. On the theory of self-adjoint extensions of positive definite operators. Mat Sbornik. 1956;38:431–450. (Russian.).
- Derkach VA, Malamud MM. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J Funct Anal. 1991;95:1–95.
- Derkach VA, Malamud MM. The extension theory of Hermitian operators and the moment problem. J Math Sci. 1995;73:141–242.
- Grubb G. Spectral asymptotics for the “soft” selfadjoint extension of a symmetric elliptic differential operator. J Oper Theory. 1983;10:9–20.
- Grubb G. Distributions and operators. New York: Springer; 2009. (Vol. 252, Graduate Texts in Mathematics).
- Hassi S, Malamud M, de Snoo H. On Kren's extension theory of nonnegative operators. Math Nachr. 2004;274–275:40–73.
- Hassi S, Sandovici A, de Snoo H, Winkler H. A general factorization approach to the extension theory of nonnegative operators and relations. J Operator Theory. 2007;58:351–386.
- Nenciu G. Applications of the Kren resolvent formula to the theory of self-adjoint extensions of positive symmetric operators. J Operator Theory. 1983;10:209–218.
- Prokaj V, Sebestyén Z. On extremal positive operator extensions. Acta Sci Math (Szeged). 1996;62:485–491.
- Sebestyén Z, Sikolya E. On Krein–von Neumann and Friedrichs extensions. Acta Sci Math (Szeged). 2003;69:323–336.
- Simon B. The classical moment problem as a self-adjoint finite difference operator. Adv Math. 1998;137:82–203.
- Skau CF. Positive self-adjoint extensions of operators affiliated with a von Neumann algebra. Math Scand. 1979;44:171-195.
- Storozh OG. On the hard and soft extensions of a nonnegative operator. J Math Sci. 1996;79:1378–1380.
- traus AV. On extensions of a semibounded operator. Sov Math Dokl. 1973;14:1075–1079.
- Tsekanovskii ER. Friedrichs and Krein extensions of positive operators and holomorphic contraction semigroups. Funct Anal Appl. 1981;15:308–309.
- Viik ML. On general boundary problems for elliptic differential equations. Trudy Moskov Mat Obsc. 1952;1:187–246 (Russian); Engl. transl. in Amer. Math. Soc. Transl. (2), 24, 107–172 (1963).
- Gesztesy F, Kalton N, Makarov K, Tsekanovskii E. Some applications of operator-valued Herglotz functions. in Operator Theory, System Theory and Related Topics. The Moshe Livsic Anniversary Volume, D. Alpay and V. Vinnikov (eds.), Operator Theory: Advances and Applications, Vol. 123, Birkhäuser, Basel, pp. 271–321. 2001.
- Clark S, Gesztesy F, Nichols R, Zinchenko M. Boundary data maps and Krein's resolvent formula for Sturm–Liouville operators on a finite interval. Oper Matrices. 2014;8:1–71.
- Gesztesy F, Nichols R, Stanfill J. A Survey of Some Norm Inequalities. Comp Anal Oper Theo. 2021;15(23).
- Kamke E. Differentialgleichungen. Lösungsmethoden und Lösungen. gewöhnliche Differentialgleichungen. 7th edn. Leipzig: Akademische Verlagsgesellschaft; 1961.
- Abramowitz M, Stegun IA. Handbook of mathematical functions. New York: Dover; 1972.
- Belinskiy BP, Hinton DB, Nichols R. Singular Sturm–Liouville operators with extreme properties that generate black holes. Stud Appl Math. 2021:1–29. DOI:https://doi.org/10.1111/sapm.12385.
- Gesztesy F, Littlejohn LL, Piorkowski M, Stanfill J. The Jacobi operator and its Weyl–Titchmarsh–Kodaira m-functions. preprint. 2020.
- Bush M, Frymark D, Liaw C. Singular boundary conditions for Sturm–Liouville operators via perturbation theory. preprint. 2020.
- Everitt WN. A catalogue of Sturm–Liouville differential equations. in Sturm–Liouville Theory: Past and Present. W. O. Amrein, A. M. Hinz, D. B. Pearson (eds.), Birkhäuser, Basel, pp. 271–331. 2005.
- Everitt WN, Kwon KH, Littlejohn LL, Wellman R, Yoon GJ. Jacobi Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression. J Comput Appl Math. 2007;208:29–56.
- Frymark D. Boundary triples and Weyl m-functions for powers of the Jacobi differential operator. J Differ Equ. 2020;269:7931–7974.
- Grünewald U. Jacobische Differentialoperatoren. Math Nachr. 1974;63:239–253.
- Koornwinder T, Kostenko A, Teschl G. Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv Math. 2018;333:796–821.
- Kuijlaars A, Martinez-Finkelshtein A, Orive R. Orthogonality of Jacobi polynomials with general parameters. Electron Trans Numer Anal. 2005;19:1–17.
- Olver FW, Lozier DW, Boisvert RF, Clark CW. NIST Handbook of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.26 of 2020-03-15.
- Szeg G. Orthogonal Polynomials. 4th edn, Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, RI. 1975.
- Olver FWJ, Lozier DW, Boisvert RF, Clark CW (eds.) NIST Handbook of Mathematical Functions. National Institute of Standards and Technology (NIST), U.S. Dept. of Commerce, and Cambridge Univ. Press. 2010.