I am recently interested in the concept of bispectrality (or self-duality) in quantum integrable systems, but some concepts are not clear to me. I may have a (big) lack of precision and rigor in my post, but I'll try to explain what I have in mind.

The spectral problem

Suppose that we have two sets of $n$ commuting Hamiltonians $H_i$ and $\tilde{H}_i$ acting on a certain Hilbert space $\mathcal{H}$. For me, the spectral problem consists in constructing a basis $\left |x\right>$ of the Hilbert space which solves the spectral problem for $H_i$,

$$H_i \left |x \right>=e_i(x) \left|x \right>, \\ \tilde{H}_i \left |x \right>= \tilde{H}_i(x,\partial_x) \left|x \right>, $$

while $\tilde{H}_i(x,\partial_x)$ acts as a differential/difference operator,

and of course constructing the second basis $\left |y\right>$ of $\mathcal{H}$ which solves the spectral problem for $\tilde{H}_i$:

$$H_i \left |y \right>=H_i(y,\partial_y) \left|y \right>, \\ \tilde{H}_i \left |y \right>= e_i(y) \left|y \right>, $$

The bispectral problem

If we consider the wave functions of $H_i$ and $\tilde{H}_i$, denoted by $\psi(x) = \left< x | \psi \right>$ and $\psi(y) = \left< y | \psi \right>$, we can map one wave-function to another using a kind of generalized Fourier transform:

$$\left< x | \psi \right> = \int dy \left<x|y\right> \left<y|\psi\right>. $$

The kernel of the integral is the solution of the bispectral problem, because of the equations $$\left<x|H_i|y\right>, $$ $$\left<x|\tilde{H}_i|y\right>.$$

The two quantum integrable systems from a self-dual system when the kernel $\left<x|y\right>$ is an involution. Examples of self-dual quantum integrable systems are a class of Ruijsenaars-Schneider models (see https://www1.maths.leeds.ac.uk/~siru/papers/p68.pdf for instance).

My question is

In the few self-dual systems I studied, I found the equivalent of $\left<x|y\right>$ (Macdonald polynomials for the trigonometric Ruijsenaars-Schneider models, the $\epsilon$-function described in the paper for the hyperbolic one), but I never found the equivalent of $\psi(x)$ and $\psi(y)$. The best I could find is in the Ruijsenaars paper, where the so called $\epsilon$-function is an isometry $L^2(\mathbb{R}_+,dx) \rightarrow L^2(\mathbb{R}_+,dy)$. But I would have expected explicit "wave-functions" $\psi(x)$ and $\psi(y)$, where we can actually explicitly compute the integral written before.

For me it is a problem, but I could not find something related to this. What am I missing here?



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