On Weidmann's Conjecture And Differential Operators With Transmission Conditions

In this paper, we study three important problems in the field of differential operators: self-adjoint domains, spectral analysis and the differential operators with transmission conditions. Since the spectrum of self-adjoint operators is real, the characterization of self-adjoint domains by the real-parameter solutions is very important for investigating the null space and the range of operator associated with spectral analysis. At the same time, we notice that the discreteness of spectrum, deficiency indices and the dimension of the real-parameter L²-solution space of the differential equation Ï„u = λu (λ∈R) are only determined by the coefficients of differential expression. There should be close connection among them. In 1987, Weidmann in his book "Spectral theory of ordinary differential operators" [93] raised a famous conjecture: "If for every λ ∈ (μ₁, μ₂) there exist 'sufficient many' L²-solutions of (Ï„-λ)u = 0, then (μ₁, μ₂) contains no points of the essential spectrum" . Here, we investigate these important problems of differential operators with middle deficiency indices. We give the complete characterization of self-adjoint domains by the real-parameter L²-solutions, including the construction of separate self-adjoint boundary conditions. The key point is that when A is not an eigenvalue we determine the "standard" pattern of initial value of real-parameter L²-solutions of the equation Ï„u = λu, and construct the self-adjoint operator A_t with separate boundary conditions in terms of these properties. Using the estimate of inequality and the approaching of operator in the sense of strong resolvent convergence, we prove that if for every λ∈ (μ₁,μ₂), the dimension of real-parameter L²-solution space of the differential equation Ï„u = λu is equal to the deficiency index, then for every self-adjoint...