On the Spectrum of a Class of Differential Operators and Some Imbedding Theorems

Embedding theorems and the spectra of certain differential operators

“…VIII, §4) and suppose that Tis the semi-bounded, self-adjoint operator in L2(R) associated with the differential operator Ly = -y"+qy via the form t. The celebrated criterion due to Molcanov (1953) asserts that the spectrum of T is discrete if, and only if, f*X + (i) lim q(y) oo \X\^C C J X for all 0) > 0 . This result has been developed by many authors, in particular b Birman & Pavlov (1961), Maz'ja (1985), Apyshev & Otelbaev (1979) and, very recently, by Oinarov & Otelbaev (1988), who give necessary and sufficient conditions for the spectrum of the problem -y"+q2y = \r2y in R, ryeL 2(R), to be discrete, q and r being continuous functions on R. These conditions follow directly from certain embedding theorems which they establish.…”

Correctness conditions for high-order differential equations with unbounded coefficients

“…So, it follows that, for eachf ∈ L 2 ,ŷ = P -1 mf ∈ D(P m ) andŷ is a solution of equation (17). By (19), we deduce that…”

Weighted Second-Order Differential Inequality on Set of Compactly Supported Functions and Its Applications

“…Thus, the oscillatory properties of Equation ( 4) and the spectral properties of the operator L can be easily derived. When r ≡ 1 and the function υ −p is strongly singular at infinity, the oscillatory properties of the equation in form (7) are studied in [10], and the spectral properties of the operator in form (8) are investigated in [9,11] (Chapters 29 and 34), [12][13][14]. When the functions υ −p and r −1 are weakly singular at infinity, then there exists one of the limits lim 3) is of second-order, but there exist three boundary conditions.…”