## COMMUTATORS, EIGENVALUE GAPS, AND MEAN CURVATURE IN THE THEORY OF SCHRÖDINGER OPERATORS (2005)

Citations: | 5 - 1 self |

### BibTeX

@MISC{Harrell05commutators,eigenvalue,

author = {Evans M. Harrell and II},

title = {COMMUTATORS, EIGENVALUE GAPS, AND MEAN CURVATURE IN THE THEORY OF SCHRÖDINGER OPERATORS},

year = {2005}

}

### OpenURL

### Abstract

Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, H = − ∆ + V (x) , acting on functions of a smooth, compact d-dimensional manifold M immersed in R d+1. Here ∆ denotes the Laplace-Beltrami operator, and the real-valued potential–energy function V (x) acts by multiplication. The manifold M may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed. It is found that the mean curvature of a manifold poses tight constraints on the spectrum of H. Further, a special algebraic rôle is found to be played by a Schrödinger operator with potential proportional to the square of the mean curvature: Hg: = − ∆ + gh 2, where g is a real parameter and h:= d� κj, j=1 with {κj}, j = 1,..., d denoting the principal curvatures of M. For instance, by Theorem 2.1 and Corollary 3.4, each eigenvalue gap of an arbitrary Schrödinger operator is bounded above by an expression using H 1/4. The “isoperimetric ” parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.

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Citation Context ... are achieved when the potential energy is of this form. Low-lying eigenvalues, λ1 < λ2 ≤ · · · , of (1.1) have been studied, and sharp estimates have been discovered relating to the geometry of M in =-=[10, 6, 8, 13, 14, 11]-=-. For certain operators of the type (1.1) with quadratic functions q, the estimates of the gap Γ in this article will likewise be sharp. Section 2 will be devoted to the fundamental eigenvalue gap in ... |

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Citation Context ...ify formulae (1.3)–(1.6). With k = j = 1, the inequality is thus equivalent to 〈Gu1 | (H − λ1)(λ2 − λ1)Gu1〉 ≤ 〈 Gu1 | (H − λ1) 2 〉 Gu1 , which follows from the spectral functional calculus (e.g., see =-=[7, 9, 21]-=-), since for µ ∈ σ(H), (µ − λ1)) (λ2 − λ1)) ≤ (µ − λ1)) 2 . q.e.d. Whereas identities (1.3)–(1.6) and Lemma 1.1 are general facts about operator algebra, if H is a differential operator, then its comm... |

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Citation Context ...nections between eigenvalue gaps and operators are algebraic, through calculations of commutators. Commutators and their relation to eigenvalue gaps and inverse spectral problems have been studied in =-=[1, 2, 12, 15, 16, 17, 19]-=-, which partly inspired this work. Here H will be defined on a hypersurface, and commutators will be used to connect eigenvalue gaps and certain other functions defined on the spectrum σ(H) to the mea... |

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Citation Context ...ify formulae (1.3)–(1.6). With k = j = 1, the inequality is thus equivalent to 〈Gu1 | (H − λ1)(λ2 − λ1)Gu1〉 ≤ 〈 Gu1 | (H − λ1) 2 〉 Gu1 , which follows from the spectral functional calculus (e.g., see =-=[7, 9, 21]-=-), since for µ ∈ σ(H), (µ − λ1)) (λ2 − λ1)) ≤ (µ − λ1)) 2 . q.e.d. Whereas identities (1.3)–(1.6) and Lemma 1.1 are general facts about operator algebra, if H is a differential operator, then its comm... |

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