scholarly journals A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs

2018 ◽  
Vol 32 (2) ◽  
pp. 311-349 ◽  
Author(s):  
Christophe Sabot ◽  
Xiaolin Zeng
Keyword(s):  
Schrödinger Operator ◽  
Jump Process ◽  
Infinite Graphs ◽  
Schrodinger Operator ◽  
Random Schrödinger Operator ◽  
Random Schrodinger Operator

scholarly journals Green kernel for a random Schrödinger operator

2016 ◽  
Vol 18 (05) ◽  
pp. 1550082
Author(s):  
Carlos G. Pacheco
Keyword(s):  
Schrödinger Operator ◽  
Homogeneous Problem ◽  
Random Operator ◽  
Liouville Theory ◽  
Inner Product ◽  
Schrodinger Operator ◽  
Green Kernel ◽  
Random Schrödinger Operator ◽  
Discrete Spectra ◽  
Random Schrodinger Operator

We find explicitly the Green kernel of a random Schrödinger operator with Brownian white noise. To do this, we first handle the random operator by defining it weakly using the inner product of a Hilbert space. Then, using classic Sturm–Liouville theory, we can build the Green kernel with linearly independent solutions of a homogeneous problem. As a corollary, we have that the random operator has a discrete spectra.

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