We study the Schrödinger operator L = −∆ + V on a star-shaped domain Ω in R d with Lipschitz boundary ∂Ω. The operator is equipped with quite general Dirichlet-or Robin-type boundary conditions induced by operators between H 1/2 (∂Ω) and H −1/2 (∂Ω), and the potential takes values in the set of symmetric N × N matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian Grassmannian of the subspace of H 1/2 (∂Ω) × H −1/2 (∂Ω) corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of L (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative eigenvalues of a bilinear form related to the boundary operator.
Assuming a symmetric potential and separated self-adjoint boundary conditions, we relate the Maslov and Morse indices for Schrödinger operators on [0, 1]. We find that the Morse index can be computed in terms of the Maslov index and two associated matrix eigenvalue problems. This provides an efficient way to compute the Morse index for such operators..) However, it is easy to see that R 2n J ∼ = ⊗ R C for any Lagrangian subspace ∈ Λ(n), and we'll take advantage of this correspondence.For a matrix U acting on R 2n J , we denote the adjoint by U J * so thatJ . We denote by U J the space of unitary matrices acting on R 2n J (i.e., the matrices so that U U J * = U J * U = I). In order to clarify the nature of U J , we note that we have the identityEquating real parts, we see that U must be unitary as a matrix on R 2n , while by equating imaginary parts we see that U J = JU . We have, then,
We discuss a definition of the Maslov index for Lagrangian pairs on R 2n based on spectral flow, and develop many of its salient properties. We provide two applications to illustrate how our approach leads to a straightforward analysis of the relationship between the Maslov index and the Morse index for Schödinger operators on [0, 1] and R.Date: September 14, 2018. Key words and phrases. Eigenvalues; Maslov index; Morse index; Schrödinger operators. J ∼ = ℓ ⊗ R C for any Lagrangian subspace ℓ ∈ Λ(n), and we'll take advantage of this correspondence.For a matrix U acting on R 2n J , we denote the adjoint in R 2nJ . We denote by U J the space of unitary matrices acting on R 2n J (i.e., the matrices so that UU J * = U J * U = I). In order to clarify the nature of U J , we note that we have the identityEquating real parts, we see that U must be unitary as a matrix on R 2n , while by equating imaginary parts we see that UJ = JU. We have, then, U J = {U ∈ R 2n×2n | U t U = UU t = I 2n , UJ = JU}.
In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction-diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have nonzero Maslov index, and hence be unstable.
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