Beginning with the Schrodinger equation for N particles in one dimension interacting via a δ-function potential(adsbygoogle = window.adsbygoogle || []).push({});

##(-\sum_{1}^{N}\frac{\partial^2}{\partial x_i^2}+2c\sum_{<i,j>}\delta(x_i-x_j))\psi=E\psi##

The boundary condition equivalent to the ##\delta## function potential is

##\left(\frac{\partial}{\partial x_j}-\frac{\partial}{\partial x_k}\right)\psi |_{x_j=x_{k+}}-\left(\frac{\partial}{\partial x_j}-\frac{\partial}{\partial x_k}\right)\psi |_{x_j=x_{k-}}=2c\psi |_{x_j=x_k}.##

Integrate ##\int_{x_k-\varepsilon}^{x_k+\varepsilon}##, here, ##x_k## is a integrate limit. Why ##x_k## is considered as a derivative ##\frac{\partial}{\partial x_k}##? It says that we can integrate the ordinate of j's particle with the boundary of k's particle?

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# The boundary condition for ##\delta## function

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