Path integral representation for schrödinger operators with bernstein functions of the laplacian
Path integral representation for schrödinger operators with bernstein functions of the laplacian
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DOI：
10.1142/s0129055x12500134
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发表时间：
2009-05
影响因子：
1.8
通讯作者：
F. Hiroshima;T. Ichinose;J. Lőrinczi
中科院分区：
文献类型：
--
作者：
F. Hiroshima;T. Ichinose;J. Lőrinczi
建立了一类拉普拉斯Bernstein函数下广义薛定谔算子的路径积分表示。利用Bernstein函数与Levy从属函数的一一对应，使得布朗运动进入标准费曼-卡茨公式的作用在这里由从属的布朗运动代替。作为具体的例子，包括分数和相对论薛定谔算子与磁场和自旋。在允许奇异磁场和奇异外势以及任意整数和半整数自旋值的条件下，得到了这些算子的自伴随性。该方法还允许提出广义Kato类的概念，并给出了相关广义薛定谔半群的Lp-Lq界。因此，还导出了抗磁性和能量比较不等式。
Path integral representations for generalized Schrodinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Levy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrodinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrodinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.