Optimal Maximal Spectral Gaps And Optimal Locations Of Nodes Of Sturm-Liouville Operators

The initial boundary value problem of Sturm-Liouville equation is a very classical mathematical physics problem.Its eigenvalues and nodes,namely the interior zeros of the eigenfunctions,are both significant and widely studied physical quantities.The inverse spectral problem and the inverse nodal problem show that the potential can be determined by two sequences of eigenvalues or partial nodes.In quantum mechanics,spectral gaps represent the energy absorbed or released by particles to transition between different energy levels.In this dissertation,we consider the Sturm-Liouville operators-d2/d2x+q(x)on the unit interval and mainly focus on two variational problems around the optimal maximal spectral gaps and the optimal locations of nodes.At first,we consider the gaps λ2n(q)-λ1(q)for the Dirichlet eigenvalues {λm(q)} of Sturm-Liouville operator with potential q.By merely assuming that potentials have the L1 norm r,we will explicitly give the solutions to the maximization problems of λ2n(q)-λ1(q)and also find the optimizers,where n is arbitrary.As a consequence,the solutions can lead to the optimal upper bounds for these spectral gaps.It’s worth mentioning that in the proof we take use of the eigenvalue theory of measure differential equations.Especially,we strengthen an existing result and get an essential lemma about approximating a Radon measure by a sequence of smooth measures.In the rest,nodes are considered as implicitly defined nonlinear functionals of potentials from Lebesgue spaces.For the Sturm-Liouville operators with separate boundary conditions,we will provide two basic results.One is that nodes are continuously Frechet differentiable in potentials when the usual norms of potentials are considered.Moreover,the Frechet derivatives will be given in a concise form using the corresponding eigenfunctions.The other is that nodes are completely continuous in potentials when the weak topologies for potentials are considered.The latter means that nodes are continuously dependent on potentials in a very strong way.At last,we will consider the Sturm-Liouville operators with Dirichlet boundary condition and study the optimization problems to minimize or to maximize nodes subject to the constraint ‖q‖Lp=r.By applying the above results of continuous dependence of nodes on potentials,it will be proved that for the case 1<p<∞,these optimization problems are attained by some potentials.Moreover,a critical equation for optimizers will be derived.It shows that the optimization problem on nodes involves the balance problem between the minimum eigenvalue problem and the maximum eigenvalue problem.When p=1,take the unique node of the second Dirichlet eigenfunction as an example,the optimization problems of partial nodes can be solved by using the results obtained for the case p∈(1,∞)and considering the limit case p→1.To solve the optimization problems of nodes of all orders,this paper will introduce a new variational approach which can turn our infinite dimensional variational problems into finite dimensional ones.It will help us to give answers to our optimization problems when p∈(1,∞)from a new perspective and for the case where p=1,it helps us to solve problems thoroughly and find the optimal values of all nodes as well as the corresponding optimizers.These results are then used to deduce the optimal locations of nodes.