# Eigenvalues of a differential operator [closed]

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 $$t$$, $$\rho(t) = \sum_{n = -\infty}^\infty\int_{-\infty}^\infty{\rm d}p\int_{-\infty}^\infty{\rm d}x e^{{\rm i} n x}\psi_n(t,p)\,.$$

• What is $\phi$? If it is an angle, how does the third term make sense? Where does this operator come from? Aug 14 '20 at 17:56
• $\phi$ 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. Aug 14 '20 at 18:01
• 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. Aug 14 '20 at 18:39
• 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 $\phi$. Aug 14 '20 at 18:40
• You might employ the variable $x\equiv t^{-3/2} \phi$ to simplify the operator. Aug 14 '20 at 19:19