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

Authors: Graham Cox, Christopher K. R. T. Jones, Yuri Latushkin and Alim Sukhtayev
Journal: Trans. Amer. Math. Soc. 368 (2016), 8145-8207
MSC (2010): Primary 53D12, 34L40; Secondary 37J25, 70H12
Published electronically: March 11, 2016
MathSciNet review: 3546796
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Abstract: We study the Schrödinger operator $ L=-\Delta +V$ on a star-shaped domain $ \Omega $ in $ \mathbb{R}^d$ with Lipschitz boundary $ \partial \Omega $. The operator is equipped with quite general Dirichlet- or Robin-type boundary conditions induced by operators between $ H^{1/2}(\partial \Omega )$ and $ H^{-1/2}(\partial \Omega )$, and the potential takes values in the set of symmetric $ N\times 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}(\partial \Omega )\times H^{-1/2}(\partial \Omega )$ 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.

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Additional Information

Graham Cox
Affiliation: Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599
Address at time of publication: Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802

Christopher K. R. T. Jones
Affiliation: Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599

Yuri Latushkin
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Alim Sukhtayev
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Schr\"odinger equation, Hamiltonian systems, eigenvalues, stability, differential operators
Received by editor(s): August 22, 2014
Received by editor(s) in revised form: June 3, 2015
Published electronically: March 11, 2016
Additional Notes: This work was partially supported by the grants NSF DMS-0754705, DMS-1067929, DMS-0410267, DMS-1312906 and ONR N00014-05-1-0791, and by the Research Council and the Research Board of the University of Missouri. The third author sincerely thanks Lai-Sang Young for the opportunity to spend his sabbatical at the Courant Institute where this paper was completed
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