Contraction Of The Sturm-liouville Problem On The Domain

In this paper, we discuss the Sturm-Liouville problems on the shrinking intervals. In the first part of this paper, we give a survey of a certain Sturm-Liouville equation on shrinking intervals and its physical background. Some pervious results are introduced briefly. And then we give some basic concept of regular(singular) Sturm-Liouville equation, eigenvalue , eigenfunction and so on.In the second and third part, with the idea of the paper [20], we establish results concerning the detailed asymptotic behavior of the eigenvalues of a class of regular Sturm-Liouville problems with general separated self-adjoim boundary conditions as the length of the interval shrinks to zero. The results are as follows:With aâ†'0⁺, consider equation(1)satisfy BC: Y(0) = 0,Y(a) = 0, we have: if R(0) > 0, thenλ₀(a):-((q₀)/(γ₀))+ O(1/a),λ_γ(a)-((γπ)²p₀)/(γ₀))a^-2;if R(0) = 0,Q(0)≠0, thenλ₀(a)=O(a^-(S+1)),λ_γ(a)-((p₀μ_γ⁰)/(γ_S))a^-(2+S); if R(0) = 0,Q(0) = 0, thenλ_γ(a)-((p₀μ_γ⁰)/(γ_S))a^-(2+S), if N≠S, thenλ₀(a)= O(a^-(S+1)), if N = S, thenλ₀(a)=-((q_N)/(γ_S))+O(a^-(S+1)).(2)satisfy BC: Y(0) = 0, Y(a)H-p(a)Y'(a) = 0, we have: if R(0) > 0, thenλ₀(a)= -((q₀)/(γ₀))-((H²)/p₀γ₀))+O(1/a),λ_γ(a)-((t_γ²P₀)/(γ₀))a^-2; if R(0) = 0,Q(0)≠0, thenλ₀(a)=O(a^-(S+1)),λ_γ(a)-((p₀μ_γ⁰)/(γ_S))a^-(2+S); if R(0) = 0,Q(0) = 0, thenλ_γ(a)-((p₀μ_γ⁰)/(γ_S))a^-(2+S), if N≠S, thenλ₀(a)= O(a^-(S+1)), if N = 5, thenλ₀(a)=-((q_N)/(γ_S))+O(a^-(S+1)).(3)satisfy BC: Y(0)h-P(0)Y'(0) = 0,Y(a) = 0, we have: if R(0) > 0, thenλ₀(a)= -((q₀)/(γ₀))+O(1/(a²)),λ_γ(a)-((t_γ²p₀)/(γ₀))a^-2; if R(0) = 0,Q(0)≠0, thenλ₀(a)=O(a^-(S+2))?λ_γ(a)- ((p₀μ_γ⁰)/(γ_S))a^-(2+S); if R(0) = 0,Q(0) = 0, thenλ_γ(a)-((p₀μ_γ⁰)/(γ_S))a^-(2+S),if N≠S,thenλ₀(a)= O(a^-(S+2)), if N = S, thenλ₀(a)=-((q_N)/(γ_S))+O(a^-(S+2)).(4)satisfy BC: Y(0)h-P(0)Y'(0) = 0,Y(a)H-P(a)Y'(a) = 0, we have: if R(0) > 0, thenλ₀(a)=-((q₀)/(γ₀))+(H-h)O(1/a²),λ_γ(a)-((t_γ²p₀)/(γ₀))a^-2; if R(0) = 0,Q(0)≠0, thenλ₀(a)= O(a^-(S+1)),λ_γ(a)-((p₀μ_γ⁰)/(γ_S))a^-(2+S); if R(0) = 0, Q(0) = 0, thenλ_γ(a)-((p₀μ_γ⁰)/(γ_S))a^-(2+S),if N≠S, thenλ₀(a)=O(a^-(S+1)), if N = S, thenλ₀(a)=-((q_N)/(γ_S))+O(a^-(S+1)?...