# zbMATH — the first resource for mathematics

Regular and completely regular differential operators. (English. Russian original) Zbl 1156.34075
Math. Notes 81, No. 4, 566-570 (2007); translation from Mat. Zametki 81, No. 4, 636-640 (2007).
In the space $$L_2(0,1),$$ the authors consider the operator $$L$$ generated by the differential expression
$l(y) = (-i)^n y^{(n)}(x) + p_2(x)y^{(n-2)} + \dots + p_n(x) y$ and $$n$$ linearly independent boundary conditions of the form
$U_j(y) = \sum_{s=0}^{n-1}(a_{j,s}y^{(s)}(0) + b_{j,s}y^{(s)}(1)) = 0,\quad j = 1,\dots ,n.$ Several sufficient and necessary conditions for Birkhoff regularity and complete regularity of the operator $$L$$ are obtained.

##### MSC:
 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
Full Text:
##### References:
 [1] G. Birkhoff, Trans. Amer. Math. Soc. 9(4), 373–395 (1908). · doi:10.1090/S0002-9947-1908-1500818-6 [2] M. A. Naimark, Linear Differential Operators, 2nd ed. (Nauka, Moscow, 1969; Parts I and II, Frederick Ungar Publishing Co., New York, 1967, 1968). [3] Ya. D. Tamarkin, Some General Problems of the Theory of Ordinary Differential Equations and Series Expansion of Arbitrary Functions (Tipografiya M. P. Frolovoi, Petrograd, 1917) [in Russian]. [4] A. A. Shkalikov, Trudy Sem. Petrovsk. 9, 190–229 (1983). [5] L. M. Luzhina, in Candidate’s Dissertation in Mathematics and Physics (Moscow State University, Moscow, 1991) [in Russian]. [6] Ya. B. Lopatinskii, Ukr. Mat. Zh. 5(2), 123–151 (1953). [7] S. Agmon and L. Nirenberg, Comm. Pure Appl. Math. 16, 121–239 (1963). · Zbl 0117.10001 · doi:10.1002/cpa.3160160204 [8] M. S. Agranovich and M. I. Vishik, Uspekhi Mat. Nauk 19(3), 53–161 (1964). [9] A. A. Shkalikov, Trudy Sem. Petrovsk. 14, 140–224 (1989) [J. Soviet Math. 51 (4), 2399–2467 (1990)]. [10] H. E. Benzinger, J. Differential Equations 7(3), 478–496 (1970). · Zbl 0198.12102 · doi:10.1016/0022-0396(70)90096-3 [11] A. M. Minkin, arXiv: math.SP/0409181. [12] A. A. Shkalikov, Uspekhi Mat. Nauk 34(5), 235–236 (1979) [Russian Math. Surveys 34 (5), 249–250 (1979)]. [13] A. A. Vladimirov, Mat. Zametki 75(6), 941–943 (2004) [Math. Notes 75 (5–6), 877–880]. · doi:10.4213/mzm564 [14] E. A. Shiryaev, Mat. Zametki 77(6), 950–954 (2005) [Math. Notes 77 (5–6), 882–886]. · doi:10.4213/mzm2553
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.