Eigenvalue inequalities for Klein-Gordon Operators
Eigenvalue inequalities for Klein-Gordon Operators
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DOI：
10.1016/j.jfa.2008.12.008
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发表时间：
2008-10
期刊：
影响因子：
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通讯作者：
E. Harrell;Selma Yildirim Yolcu
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文献类型：
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作者：
E. Harrell;Selma Yildirim Yolcu
我们考虑由量子力学的规定与Klein-Gordon哈密顿量|P|2+m2相联系的拟微分算符Hm，Ω，当它被限制在有界的开域Ω∈Rd时。当质量m为0时，算子H0，Ω与柯西随机过程的生成元重合，且满足∂Ω上的KILIT条件。(算符H0，Ω有时称为幂为12的分数拉普拉斯算子。[R.巴纽埃洛斯，T.Kulczycki，柯西过程的特征值间隙和一个Poincaré不等式，J.Funct.肛门。211(2)(2004)355-423；R.巴涅洛斯，T.Kulczycki，柯西过程和Steklov问题，J.Funct。肛门。E.Giere，《应用中的分数拉普拉斯》，http://www.eckhard-giere.de/math/publications/review.pdf].)证明了HM，β的特征值0<；β1<；⩽⋯2Ω及其均值βk‘：=1k∑ℓ=1kβℓ的几个普遍的不等式.证明了：对于只依赖于维d的显式最优“半经典”常数，对于任意维d⩾2和任意k，当d⩾2和k⩾2j时，我们给出了某些函数类中V(X)的包含外部势能场的算子的类似估计，即Hm，Ω+V(X).
We consider the pseudodifferential operators Hm,Ωassociated by the prescriptions of quantum mechanics to the Klein–Gordon Hamiltonian |P|2+m2when restricted to a bounded, open domain Ω∈Rd. When the mass m is 0 the operator H0,Ωcoincides with the generator of the Cauchy stochastic process with a killing condition on ∂Ω. (The operator H0,Ωis sometimes called the fractional Laplacian with power 12, cf. [R. Bañuelos, T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincaré inequality, J. Funct. Anal. 211 (2) (2004) 355–423; R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 234 (2006) 199–225; E. Giere, The fractional Laplacian in applications, http://www.eckhard-giere.de/math/publications/review.pdf].) We prove several universal inequalities for the eigenvalues 0<β1<β2⩽⋯ of Hm,Ωand their means βk¯:=1k∑ℓ=1kβℓ. Among the inequalities proved are: for an explicit, optimal “semiclassical” constant depending only on the dimension d. For any dimension d⩾2 and any k, Furthermore, when d⩾2 and k⩾2j, Finally, we present some analogous estimates allowing for an operator including an external potential energy field, i.e., Hm,Ω+V(x), for V(x) in certain function classes.