Littlejohn and Wellman developed a general abstract left-definite theory for a self-adjoint operator A that is bounded below in a Hilbert space (H,(‧,‧)). More specifically, they construct a continuum of Hilbert spaces {(H_r,(‧,‧)_r)}_r>0 and, for each r>0, a self-adjoint restriction A_r of A in H_r. The Hilbert space H_r is called the rth left-definite Hilbert space associated with the pair (H,A) and the operator A_r is called the rth left-definite operator associated with (H,A). We apply this left-definite theory to the self-adjoint Legendre type differential operator generated by the fourth-order formally symmetric Legendre type differential expression ℓ[y](x):=((1-x²)²y″(x))″-((8+4A(1-x²))y′(x))′ +λy(x), where the numbers A and λ are, respectively, fixed positive and non-negative parameters and where x ∈ (-1,1).

Mathematics.

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