Morse–Smale index theorems for elliptic boundary deformation problems

The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

“…(Note that the major conclusions of [DJ11] are all correct, as confirmed by the current paper.) We note that [DP12] also depends on the arguments in [DJ11] (and consequently [DN06,DN08] [DN06]) is given in Proposition 4.10. This requires several preliminary steps, which appear in Lemmas 2.12, 3.13 and 3.14.…”

“…In [JLM13] a similar approach was used to relate the Morse and Maslov indices of a one-dimensional Schrödinger operator with periodic potential and θ-periodic boundary conditions. In [DP12] the results of [DJ11] were extended to general second-order, scalar-valued elliptic operators on star-shaped domains. The paper [CJM15] dealt with the case when Ω is not necessarily star-shaped, and the domains Ω t are obtained from Ω using a general family of diffeomorphisms instead of the linear scaling x → tx.…”

A Morse–Smale index theorem for indefinite elliptic systems and bifurcation

“…Deng and Jones studied in [9] zeroth-order perturbations of the scalar Laplacian for a rather general class of boundary value problems. Subsequently, the first author extended their results for the Dirichlet and Neumann problem in collaboration with Dalbono to general scalar second order elliptic partial differential equations in [8]. The novelty in these investigations is that now, except for the case of the classical Dirichlet condition as treated by Smale in [34], conjugate points can accumulate as in the case of semi-Riemannian geodesics.…”

“…Deng and Jones tried to overcome this problem in [9] by using a Maslov index for curves of Lagrangian subspaces in a symplectic Hilbert space consisting of functions on the boundary of Ω. Note that compared to (1), the equations considered in [9] and [8] correspond for Dirichlet boundary conditions to the case of geodesics in one-dimensional Riemannian manifolds. Finally, the authors studied bifurcation phenomena for scalar semilinear elliptic differential equations on star-shaped domains under shrinking of the domain by variational methods in [29] and [30], and obtained incidentally a new proof of Smale's theorem [34] for scalar elliptic equations (cf.…”

Counting spectrum via the Maslov index for one dimensional $\theta -$periodic Schrödinger operators

“…The signed count of the conjugate points is called the Maslov index [Ar, BF, CLM, F]. Recently the relation between the spectral count and the Maslov index attracted much attention [BM,DJ,DP,CJLS,CJM1,CJM2,CDB1,CDB2,PW]. In particular, formulas relating the number of the negative eigenvalues of the Schrödinger operators with periodic potentials and the Maslov index were given in [JLM, LSS].…”