I am interested to know if there are any general techniques for diagonalizing the operator of the form $$\left({\rm i}\partial_t- \frac{1}{2}t^{3/2}\mathbf{A}\partial_p+ t^{-3/2}p\mathbf{B} \right)$$ where $\mathbf{A}$ is an $n\times n$ diagonal matrix of constants, and $\mathbf{B}$ is an antisymmetric matrix of constants representing nearest neighbor interactions i.e. $\mathbf{B}$ consists of $+1$ on the upper diagonal and $-1$ on the lower diagonal.

I feel like techniques from quantum mechanics aught to work here, hence why I'm asking on PSE.

Edit: The target space of this operator is the $n$-dimensional column vectors whose entries are functions of the variables $p\in[-\infty,\infty]$ and $t\in[0,\infty]$. If we denote the states $\psi$, then the following quantity is conserved in $t$, $$\rho(t) = \sum_{n = -\infty}^\infty\int_{-\infty}^\infty{\rm d}p\int_{-\infty}^\infty{\rm d}x e^{{\rm i} n x}\psi_n(t,p)\,.$$

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    $\begingroup$ What is $\phi$? If it is an angle, how does the third term make sense? Where does this operator come from? $\endgroup$
    – G. Smith
    Aug 14 '20 at 17:56
  • $\begingroup$ $\phi$ represents the canonical momentum conjugate and it is not periodic. This operator comes from a Hamiltonian that is periodic in one of the canonical variables. The shape of $\mathbf{B}$ arrises by representing the solution as a Fourier series in the periodic variable. $\endgroup$
    – David
    Aug 14 '20 at 18:01
  • 1
    $\begingroup$ Can you specify the space on which this operator is meant to act? I'm a bit confused as to what all of these symbols mean. $\endgroup$
    – J. Murray
    Aug 14 '20 at 18:39
  • $\begingroup$ To clear things up: vector space this operator works on is column vectors of length $n$ that are continuous functions of the variables $t$ and $\phi$. $\endgroup$
    – David
    Aug 14 '20 at 18:40
  • 1
    $\begingroup$ You might employ the variable $x\equiv t^{-3/2} \phi$ to simplify the operator. $\endgroup$ Aug 14 '20 at 19:19