Relevant to the NLS is the differential equation,(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \left( -\sum^N_{i=1} \frac{\partial^2}{\partial x^2_i} +c \sum_{i\neq j} \delta(x_i-x_j)\right)f_N = E_Nf_N[/tex](2.87)

How does one show that

[tex] \left(\prod_{i<j}(\theta(x_i - x_j) + e^{i\Delta(k_j-k_i)}\theta(x_j-x_i))\right)\exp\left(i\sum^N_{j=1}k_jx_j\right) [/tex](2.90)

where [tex]\theta(x)=\frac{|x|+x}{2x}[/tex] and

[tex]e^{i\Delta(q)}=\frac{q-ic}{q+ic}}[/tex](2.91)

is a solution? The textbook just asserts but does not calculate.

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