Abstract
This paper aims to share a method to handle left-definite Hamiltonian systems and to construct nested-ellipsoids related with the corresponding hermitian forms. We share a lower bound for the number of linearly independent Dirichlet-integrable solutions of the Hamiltonian systems with respect to some nonnegative matrices. Moreover, we share the corresponding Titchmarsh-Weyl functions. At the end of the paper we introduce a limit-point criterion.
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References
Weyl, H.: Über gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklungen willkürlicher funktionen. Math. Ann. 68, 220–269 (1910)
Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second-Order Differential Equations. Oxford (1946)
Kodaira, K.: On ordinary differential equations of any even order and the corresponding eigenfunction expansions. Am. J. Math. 72, 502–544 (1950)
Kimura, T., Takahasi, M.: Sur les opérataurs differentiels ordinaires linéaires formellement autoadjoints I. Funkcialaj Ekvacioj. Serio Internacia 7, 35–90 (1965)
Everitt, W.N.: Singular differential equations I, The even order case. Math. Ann. 149, 170–180 (1963)
Everitt, W.N.: Integrable-square, analytic solutions of odd-order, formally symmetric, ordinary differential equations. Proc. Lond. Math. Soc. (3) 25, 156–182 (1972)
Pleijel, A.: Some remarks about the limit point and limit circle theory. Ark. för Mat. 7, 543–550 (1968)
Pleijel, A.: Generalized Weyl circles, Conference on the Theory of Ordinary and Partial Differential Equations, Dundee, Scotland, Springer Lecture Notes, (1974)
Weyl, H.: Über gewöhnliche lineare differentialgleichwgen mit singulären stellen und ihne eigenfunktionen (2. note). Gött. Nachr. 29, 442–467 (1910)
Brauer, F.: Spectral theory for the differential equation \(Ly=\lambda My\). Can. J. Math. 10, 431–446 (1958)
Bennewitz, C.: Generalization of some statements by Kodaira, Proc. Roy. Soc. Edinb. (A), 71, 113-119 (1972/73)
Karlsson, B.: Generalization of a theorem of Everitt. J. Lond. Math. Soc. (2) 9, 131–141 (1974)
Walker, P.: A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square. J. Lond. Math. Soc (2) 9, 151–159 (1974)
Atkinson, F.V.: Discrete and Continuous Boundary Problems. Acad. Press, New York (1964)
Jakubovič, V.A.: Arguments on the group of symplectic matrices. Mat. Sbornik 55, 255–280 (1961)
Jakubovič, V.A.: Oscillation properties of the solutions of canonical equations. Mat. Sbornik 56, 3–42 (1962)
Hinton, D.B., Shaw, J.K.: Titchmarsh-Weyl theory for Hamiltonian systems. In: Knowles, I.W., Lewis, R.E. (eds.) Spectral Theory Diff. Op., pp. 219–230. North Holland (1981)
Hinton, D.B., Shaw, J.K.: Hamiltonian systems of limit point or limit circle type with both endpoints singular. J. Diff. Eq. 50, 444–464 (1983)
Krall, A.M.: \(M(\lambda )\) theory for singular Hamiltonian systems with one singular point. SIAM J. Math. Anal. 20, 664–700 (1989)
Hölder, E.: Zur theorie der randwertaufgaben für lineare kamnische systeme, Jahresberichr der Deutschen Mathematiker Vereinigung, 126-128 (1935)
Schöfke, F.W., Schneider, A.: S-hermitesche Rand-Eigenwertprobleme I. Math. Ann. 162, 9–26 (1965)
Schöfke, F.W., Schneider, A.: S-hermitesche Rand-Eigenwertprobleme II. Math. Ann. 165, 236–260 (1966)
Schöfke, F.W., Schneider, A.: S-hermitesche Rand-Eigenwertprobleme III. Math. Ann. 177, 67–94 (1968)
Bennewitz, C.: A generalization of Niessen’s limit-circle criterion. Proc. R. Soc. Edinb. 78A, 81–90 (1977)
Bennewitz, C.: Spectral theory for hermitean differential systems. In: Knowles, I.W., Lewis, R.T. (eds.) Spectral Theory of Dif. Operators, pp. 61–67. North Holland Publ. Comp. (1981)
Krall, A.M.: Left-definite regular Hamiltonian systems. Math. Nachr. 174, 203–217 (1995)
Vanhoff, R.: Some remarks on left-definite Hamiltonian systems in the regular case. Math. Nachr. 193, 199–210 (1998)
Uğurlu, E., Taş, K., Baleanu, D.: Singular left-definite Hamiltonian systems in the Sobolev space. J. Nonlinear Sci. Appl. 10, 4451–4458 (2017)
Bonanno, G., D’Aguì, G.: A Neumann boundary value problem for the Sturm-Liouville equation. Appl. Math. Comput. 208, 318–327 (2009)
Kravchenko, V.V.: A representation for solutions of the Sturm–Liouville equation. Complex Var. Elliptic Equ. 53, 775–789 (2008)
Aydemir, K., Olğar, H., Muhtaroğlu, O., Muhtarov, F.: Differential operator equations with interface conditions in modified direct sum spaces. Filomat 32, 921–931 (2018)
Krall, A.M.: A limit-point criterion for linear hamiltonian systems. Appl. Anal. 61, 115–119 (1996)
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Uğurlu, E., Bairamov, E. On Dirichlet-integrable solutions of left-definite Hamiltonian systems. Bol. Soc. Mat. Mex. 29, 34 (2023). https://doi.org/10.1007/s40590-023-00507-1
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DOI: https://doi.org/10.1007/s40590-023-00507-1