framework of the multispectral Toda lattice just discussed. As a main result
in Theorem 4.2 we get quasi-determinantal expressions for the transformed
Baker function $(\hat{\Psi}_{1})_{[k]}(t)$ and the quasi-tau matrices
$\hat{H}_{[k]}(t)$.

######  Definition 4.7.

Given two coprime polynomials $\mathcal{Q}_{1}(\boldsymbol{x})$ and
$\mathcal{Q}_{2}(\boldsymbol{x})$, $\deg\mathcal{Q}_{i}=m_{i}$, we consider an
initial condition $G$ and a perturbed one $\hat{G}$ such that

(4.21)
$\displaystyle\hat{G}\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})=\mathcal{Q}_{1}(\boldsymbol{\Lambda})G.$

We can achieve the perturbed semi-infinite matrix $\hat{G}$ in two steps,
using an intermediate matrix $\check{G}$. First, we perform a Geronimus type
transformation

(4.22) $\displaystyle\check{G}\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})=G$

and second, a Christoffel type transformation

(4.23) $\displaystyle\hat{G}=\mathcal{Q}_{1}(\boldsymbol{\Lambda})\check{G}.$

######  Proposition 4.11.

Under the evolution prescribed in (4.4) if (4.21), (4.22) and (4.23) we have

$\displaystyle\hat{G}(t)\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})$
$\displaystyle=\mathcal{Q}_{1}(\boldsymbol{\Lambda})G(t),$
$\displaystyle\check{G}(t)\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})$
$\displaystyle=G(t),$ $\displaystyle\hat{G}(t)$
$\displaystyle=\mathcal{Q}_{1}(\boldsymbol{\Lambda})\check{G}(t).$

###### Proof.

We just check the first as the others follow in an analogous manner:

$\displaystyle\hat{G}(t)\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})=$
$\displaystyle
W_{1}^{(0)}(t_{1})\hat{G}\big{(}W_{2}^{(0)}(t_{2})\big{)}^{-\top}\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})$
$\displaystyle=$ $\displaystyle
W_{1}^{(0)}(t_{1})\hat{G}\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})\big{(}W_{2}^{(0)}(t_{2})\big{)}^{-\top}$
$\displaystyle=$ $\displaystyle
W_{1}^{(0)}(t_{1})\mathcal{Q}_{1}(\boldsymbol{\Lambda})G\big{(}W_{2}^{(0)}(t_{2})\big{)}^{-\top}$
$\displaystyle=$ $\displaystyle\mathcal{Q}_{1}(\boldsymbol{\Lambda})G(t).$

∎

In terms of bilinear forms (4.22) reads

$\displaystyle\big{\langle}\chi(\boldsymbol{x}),\big{(}\chi(\boldsymbol{x})\big{)}^{\top}\mathcal{Q}_{2}(\boldsymbol{x})\big{\rangle}\,\check{}=\big{\langle}\chi(\boldsymbol{x}),\big{(}\chi(\boldsymbol{x})\big{)}^{\top}\big{\rangle}$

so that assuming we can divide by polynomials inside these bilinear forms a
solution to (4.22)

(4.24)
$\displaystyle\check{G}=\Big{\langle}\chi(\boldsymbol{x}),\frac{\big{(}\chi(\boldsymbol{x})\big{)}^{\top}}{\mathcal{Q}_{2}(\boldsymbol{x})}\Big{\rangle}+\big{\langle}v,\chi(\boldsymbol{x})\big{(}\chi(\boldsymbol{x})\big{)}^{\top}\big{\rangle}$

where $v\in\big{(}\mathbb{C}[x]\big{)}^{\prime}$ and
$(\mathcal{Q}_{2}(\boldsymbol{x}))\subset\operatorname{Ker}(v)$. In fact, a
more general case will be

$\displaystyle\check{G}=\Big{\langle}\chi(\boldsymbol{x}),\frac{\big{(}\chi(\boldsymbol{x})\big{)}^{\top}}{\mathcal{Q}_{2}(\boldsymbol{x})}\Big{\rangle}+\big{\langle}v,A\chi(\boldsymbol{x})\big{(}\chi(\boldsymbol{x})\big{)}^{\top}\big{\rangle}$

where $A$ is a semi-infinite matrix with rows having only a finite number of
non vanishing coefficients.

######  Definition 4.8.

We introduce the resolvents

$\displaystyle\omega_{1}(t)$
$\displaystyle:=\hat{S}_{1}(t)\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{(}S_{1}(t)\big{)}^{-1},$
$\displaystyle\omega_{2}(t)$
$\displaystyle:=\Big{(}S_{2}(t)\mathcal{Q}_{2}(\boldsymbol{\Lambda})\big{(}\hat{S}_{2}(t)\big{)}^{-1}\Big{)}^{\top}.$

######  Proposition 4.12.

The resolvent matrices satisfy

(4.25) $\displaystyle\hat{H}(t)\omega_{2}(t)=\omega_{1}(t)H(t).$

The resolvents $\omega_{1}(t),\omega_{2}(t)$ are block banded matrices, having
different from zero only the first $m_{1}$ block superdiagonals and the first
$m_{2}$ block subdiagonals.

###### Proof.

From the $LU$ factorization we get

$\displaystyle\big{(}\hat{S}_{1}(t)\big{)}^{-1}\hat{H}(t)\big{(}\hat{S}_{2}(t)\big{)}^{-\top}\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})=\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{(}S_{1}(t)\big{)}^{-1}H(t)\big{(}S_{2}(t)\big{)}^{-\top},$

so that

$\displaystyle\hat{H}(t)\Big{(}S_{2}(t)\mathcal{Q}_{2}(\boldsymbol{\Lambda})\big{(}\hat{S}_{2}(t)\big{)}^{-1}\Big{)}^{\top}=\hat{S}_{1}(t)\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{(}S_{1}(t)\big{)}^{-1}H(t).$

∎

In this more general scenario Proposition 3.4 still holds for these new
resolvents, not connected in principle with any linear functional. We have

######  Proposition 4.13 (Connection formulas).

We have

$\displaystyle\omega_{1}(t)P_{1}(t,\boldsymbol{x})$
$\displaystyle=\mathcal{Q}_{1}(\boldsymbol{x})\hat{P}_{1}(t,\boldsymbol{x}),$
$\displaystyle\big{(}\omega_{2}(t)\big{)}^{\top}\hat{P}_{2}(t,\boldsymbol{x})$
$\displaystyle=\mathcal{Q}_{2}(\boldsymbol{x})P_{2}(t,\boldsymbol{x}).$

######  Definition 4.9.

We introduce the semi-infinite matrix

(4.26) $\displaystyle R(t)\coloneq S_{1}(t)\check{G}(t)$

######  Proposition 4.14.

The matrix $R(t)$ can be expressed as follows

(4.27) $\displaystyle
R(t)=\Big{\langle}P_{1}(t,\boldsymbol{x}),\frac{\big{(}\chi(\boldsymbol{x})\big{)}^{\top}}{\mathcal{Q}_{2}(\boldsymbol{x})}\Big{\rangle}+\big{\langle}v,P_{1}(t,\boldsymbol{x})\big{(}\chi(\boldsymbol{x})\big{)}^{\top}\big{\rangle}.$

###### Proof.

Recall (4.24) and (4.26). ∎

######  Proposition 4.15.

We have the following relations

$\displaystyle(\omega_{1}(t)R(t))_{[k],[l]}$ $\displaystyle=0,$ $\displaystyle
l=$ $\displaystyle 0,1,\dots,k-1$ $\displaystyle(\omega_{1}(t)R(t))_{[k],[k]}$
$\displaystyle=\hat{H}_{[k]}(t)$

###### Proof.

Just follow the next chain of equalities

$\displaystyle\omega_{1}(t)R(t)$
$\displaystyle=\hat{S}_{1}(t)\mathcal{Q}_{1}(\boldsymbol{\Lambda})(S_{1}(t))^{-1}S_{1}(t)\check{G}(t)$
$\displaystyle=\hat{S}_{1}(t)\mathcal{Q}_{1}(\boldsymbol{\Lambda})\check{G}(t)$
$\displaystyle=\hat{S}_{1}(t)\hat{G}(t)$ from (4.11) (4.28)
$\displaystyle=\hat{H}(t)(\hat{S}_{2}(t))^{-\top}$

and the matrix $\omega_{1}R$ is an upper triangular block matrix with
$\hat{H}$ as its block diagonal. ∎

Proceeding as we did for (3.8) and (3.9) we can deduce analogous equations in
this new context. For $k<m_{2}$ we can write

$\big{(}(\omega_{1})_{[k],[0]}(t),\dots,(\omega_{1})_{[k],[k+m_{1}-1]}(t)\big{)}=\\\
\begin{multlined}-\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}\big{(}R_{[k+m_{1}],[0]}(t),\dots,R_{[k+m_{1}],[k-1]}(t),\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1}),\dots,\Psi_{[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\big{)}\\\
\times\left(\begin{matrix}R_{[k],[0]}(t)&\dots&R_{[0],[k-1]}(t)&\Psi_{1,[k]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\\\
\vdots&&\vdots&\vdots&&\vdots\\\
R_{[k+m_{1}-1],[0]}(t)&\dots&R_{[k+m_{1}-1],[k-1]}(t)&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\end{matrix}\right)^{-1},\end{multlined}-\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}\big{(}R_{[k+m_{1}],[0]}(t),\dots,R_{[k+m_{1}],[k-1]}(t),\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1}),\dots,\Psi_{[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\big{)}\\\
\times\left(\begin{matrix}R_{[k],[0]}(t)&\dots&R_{[0],[k-1]}(t)&\Psi_{1,[k]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\\\
\vdots&&\vdots&\vdots&&\vdots\\\
R_{[k+m_{1}-1],[0]}(t)&\dots&R_{[k+m_{1}-1],[k-1]}(t)&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\end{matrix}\right)^{-1},$

while for $k\geq m_{2}$

$\big{(}(\omega_{1})_{[k],[k-m_{2}]}(t),\dots,(\omega_{1})_{[k],[k+m_{1}-1]}(t)\big{)}=\\\
\begin{multlined}-\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}\big{(}R_{[k+m_{1}],\boldsymbol{\beta}_{1}}(t),\dots,R_{[k+m_{1}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t),\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1}),\dots,\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\big{)}\\\
\times\left(\begin{matrix}R_{[k-m_{2}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k-m_{2}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\\\
\vdots&&\vdots&\vdots&&\vdots\\\
R_{[k+m_{1}-1],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k+m_{1}-1],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\end{matrix}\right)^{-1},\end{multlined}-\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}\big{(}R_{[k+m_{1}],\boldsymbol{\beta}_{1}}(t),\dots,R_{[k+m_{1}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t),\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1}),\dots,\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\big{)}\\\
\times\left(\begin{matrix}R_{[k-m_{2}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k-m_{2}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\\\
\vdots&&\vdots&\vdots&&\vdots\\\
R_{[k+m_{1}-1],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k+m_{1}-1],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}-1]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})\end{matrix}\right)^{-1},$

We also have

$\displaystyle(\omega_{1}(t))_{[k],[k+m_{1}]}$
$\displaystyle=\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}.$

Then, we extend Definitions 3.4 and 3.5 to this new scenario, and find a
version of Theorem 3.1 in terms of the Baker functions

######  Theorem 4.2 (Christoffel–Geronimus–Uvarov formula for multispectral
Toda hierarchy).

A linear spectral transformation, as in (4.21), for the multispectral Toda
hierarchy has the following effects on the Baker function $\Psi_{1,[k]}(t)$
and the quasi-tau matrices $H_{[k]}(t)$. Given a poised set $\mathcal{S}_{k}$,
of multi-indices and nodes, we have a perturbed Baker function

$\hat{\Psi}_{1,[k]}(t,\boldsymbol{x})=\frac{\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}}{\mathcal{Q}_{1}(\boldsymbol{x})}\\\
\times\Theta_{*}\left(\begin{matrix}R_{[0],[0]}(t)&\dots&R_{[0],[k-1]}(t)&\Psi_{1,[0]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[0]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&\Psi_{1,[0]}(t,\boldsymbol{x})\\\
\vdots&&\vdots&\vdots&&\vdots&\vdots\\\
R_{[k+m_{1}],[k-1]}(t)&\dots&R_{[k+m_{1}],[k-1]}(t)&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{x})\end{matrix}\right),$

and a perturbed quasi-tau matrix

$\hat{H}_{[k]}(t)=\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}\\\
\times\Theta_{*}\left(\begin{matrix}R_{[0],[0]}(t)&\dots&R_{[0],[k-1]}(t)&\Psi_{1,[0]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[0]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&R_{[0],[k]}(t)\\\
\vdots&&\vdots&\vdots&&\vdots&\vdots\\\
R_{[k+m_{1}],[k-1]}(t)&\dots&R_{[k+m_{1}],[k-1]}(t)&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&R_{[k+m_{1}],[k]}(t)\end{matrix}\right).$

When $k\geq m_{2}$ we have the shorter alternative expressions

$\hat{\Psi}_{1,[k]}(t,\boldsymbol{x})=\frac{\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}}{\mathcal{Q}_{1}(\boldsymbol{x})}\\\
\times\Theta_{*}\left(\begin{matrix}R_{[k-m_{2}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k-m_{2}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{x})\\\
\vdots&&\vdots&\vdots&&\vdots&\vdots\\\
R_{[k+m_{1}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k+m_{1}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{x})\end{matrix}\right),$
$\hat{H}_{[k]}(t)=\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}\\\
\times\Theta_{*}\left(\begin{matrix}R_{[k-m_{2}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k-m_{2}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&R_{[k-m_{2}],[k]}(t)\\\
\vdots&&\vdots&\vdots&&\vdots&\vdots\\\
R_{[k+m_{1}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k+m_{1}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&R_{[k+m_{1}],[k]}(t)\end{matrix}\right).$

and

$\hat{H}_{[k]}(t)\Big{(}\big{(}\mathcal{Q}_{2}(\boldsymbol{\Lambda})\big{)}_{[k-m_{2}],[k]}\Big{)}^{\top}=\big{(}\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{)}_{[k],[k+m_{1}]}\\\
\times\Theta_{*}\left(\begin{matrix}R_{[k-m_{2}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k-m_{2}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&H_{[k-m_{2}]}(t)\\\
R_{[k-m_{2}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k-m_{2}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k-m_{2}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&0\\\
\vdots&&\vdots&\vdots&&\vdots&\vdots\\\
R_{[k+m_{1}],\boldsymbol{\beta}_{1}}(t)&\dots&R_{[k+m_{1}],\boldsymbol{\beta}_{r_{2|k,m_{2}}}}(t)&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{1})&\dots&\Psi_{1,[k+m_{1}]}(t,\boldsymbol{p}_{r_{1|k,m_{1}}})&0\end{matrix}\right).$

Regarding the Baker function $\Psi_{2}$ and its behavior under a general
linear spectral transformation, using (4.13), we have for each component

$\displaystyle\hat{\Psi}_{2,[k]}(t,\boldsymbol{z})$
$\displaystyle=\langle\hat{\Psi}_{1,[k]}(t,\boldsymbol{x}),\mathcal{C}(\boldsymbol{z},\boldsymbol{x})\rangle,$

and consequently Theorem 4.2 provides quasi-determinantal expression for
$\hat{\Psi}_{2,[k]}$ performing the following replacements

$\displaystyle\Psi_{1,[l]}(t,\boldsymbol{x})$
$\displaystyle\rightarrow\Big{\langle}\frac{\Psi_{1,[l]}(t,\boldsymbol{x})}{\mathcal{Q}_{1}(\boldsymbol{x})},\mathcal{C}(\boldsymbol{z},\boldsymbol{x})\Big{\rangle},$
$\displaystyle l$ $\displaystyle\in\\{k-m_{2},\dots,k+m_{1}\\}.$

Alternative expressions are achieved if the relation (4.28) is recalled.
Indeed, it implies

(4.29)
$\displaystyle\hat{\Psi}_{2}(t,\boldsymbol{z})=\omega_{1}R\big{(}W^{(0)}_{2}(t_{2})\big{)}^{\top}\chi^{*}(\boldsymbol{z}).$

Then, using (4.27) we conclude that the replacements to perform in Theorem 4.2
to find a quasi-determinantal expression for $\hat{\Psi}_{2,[k]}$ are

$\displaystyle\Psi_{1,[l]}(t,\boldsymbol{x})$
$\displaystyle\rightarrow\Big{\langle}P_{1,[l]}(t,\boldsymbol{x}),\operatorname{e}^{t_{2}(\boldsymbol{x})}\frac{\mathcal{C}(\boldsymbol{z},\boldsymbol{x})}{\mathcal{Q}_{2}(\boldsymbol{x})}\Big{\rangle}+\big{\langle}v,\operatorname{e}^{t_{2}(\boldsymbol{x})}P_{1,[l]}(t,\boldsymbol{x})\mathcal{C}(\boldsymbol{z},\boldsymbol{x})\big{\rangle},$
$\displaystyle l$ $\displaystyle\in\\{k-m_{2},\dots,k+m_{1}\\}.$

In this general setting $G$ is not restricted by a Hankel type constraint,
thus given a polynomial
$\mathcal{Q}(\boldsymbol{x})\in\mathbb{R}[\boldsymbol{x}]$ we have

$\displaystyle
G\mathcal{Q}(\boldsymbol{\Lambda}^{\top})\not=\mathcal{Q}(\boldsymbol{\Lambda})G.$

For example, instead of (4.21) we may have considered

$\displaystyle\mathcal{Q}_{2}(\boldsymbol{\Lambda})\hat{G}=G\mathcal{Q}_{1}(\boldsymbol{\Lambda}^{\top}).$

In this case a transposition formally gives

$\displaystyle\hat{G}^{\top}\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})=\mathcal{Q}_{1}(\boldsymbol{\Lambda})G^{\top},$

which can be gotten from (4.21) by the replacement $G\mapsto G^{\top}$ and
$\hat{G}\mapsto\hat{G}^{\top}$; i.e., at the level of the Gauss–Borel
factorization (4.5)

$\displaystyle S_{1}$ $\displaystyle\mapsto S_{2},$ $\displaystyle H$
$\displaystyle\mapsto H^{\top},$ $\displaystyle S_{2}$ $\displaystyle\mapsto
S_{1},$ $\displaystyle\hat{S}_{1}$ $\displaystyle\mapsto\hat{S}_{2},$
$\displaystyle\hat{H}$ $\displaystyle\mapsto\hat{H}^{\top},$
$\displaystyle\hat{S}_{2}$ $\displaystyle\mapsto\hat{S}_{1}.$

Thus, previous formulæ hold by replacing $P_{1}$ by $P_{2}$ and transposing
the matrices $H_{[k]}$ and $\hat{H}_{[k]}$.

A quite general transformation, which we will not explore in this paper,
corresponds to

$\displaystyle\mathcal{Q}_{2}^{L}(\boldsymbol{\Lambda})\hat{G}\mathcal{Q}_{2}^{R}(\boldsymbol{\Lambda}^{\top})=\mathcal{Q}_{1}^{R}(\boldsymbol{\Lambda})G\mathcal{Q}_{1}^{L}(\boldsymbol{\Lambda}^{\top}),$

for polynomials
$\mathcal{Q}_{1}^{L}(\boldsymbol{x}),\mathcal{Q}^{R}_{1}(\boldsymbol{x}),\mathcal{Q}_{2}^{L}(\boldsymbol{x}),\mathcal{Q}_{2}^{R}(\boldsymbol{x})\in\mathbb{R}[\boldsymbol{x}]$.
This transformation is preserved by the integrable flows introduced above;
i.e.,

$\displaystyle\mathcal{Q}_{2}^{L}(\boldsymbol{\Lambda})\hat{G}(t)\mathcal{Q}_{2}^{R}(\boldsymbol{\Lambda}^{\top})=\mathcal{Q}_{1}^{R}(\boldsymbol{\Lambda})G(t)\mathcal{Q}_{1}^{L}(\boldsymbol{\Lambda}^{\top}).$

Notice that this transformation for a multi-Hankel reduction
$\Lambda_{a}G=G(\Lambda_{a})^{\top}$, $a\in\\{1,\dots,D\\}$, is just the one
considered in previous sections.

###  4.6. Generalized bilinear equations and linear spectral transformations

We are ready to show that the Baker functions at different times and their
linear spectral transforms satisfy a bilinear equation as in the KP theory,
see [24, 26, 25]. In the standard formulation [24, 26, 25] discrete times
appeared in the bilinear equation, which in this case are identified, see for
example [30], with the linear spectral transformations. To deduce the bilinear
equations we use a similar method as in [4, 56, 59].

We begin with the following observation

######  Proposition 4.16.

Wave matrices $W_{i}(t)$, $i\in\\{1,2\\}$ and their linear spectral
transformed wave matrices $\hat{W}_{i}(t^{\prime})$, $i\in\\{1,2\\}$,
according to the coprime polynomials
$\mathcal{Q}_{1}(\boldsymbol{x}),\mathcal{Q}_{2}(\boldsymbol{x})\in\mathbb{C}[\boldsymbol{x}]$,
fulfill

$\displaystyle\hat{W}_{1}(t^{\prime})\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{(}W_{1}(t)\big{)}^{-1}=\hat{W}_{2}(t^{\prime})\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})\big{(}W_{2}(t)\big{)}^{-1}.$

###### Proof.

We have

$\displaystyle G=$ $\displaystyle\big{(}W_{1}(t)\big{)}^{-1}W_{2}(t),$
$\displaystyle\hat{G}=$
$\displaystyle\big{(}\hat{W}_{1}(t^{\prime})\big{)}^{-1}\hat{W}_{2}(t^{\prime}).$

Hence, using (4.21) we deduce

$\displaystyle\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{(}W_{1}(t)\big{)}^{-1}W_{2}(t)=\big{(}\hat{W}_{1}(t^{\prime})\big{)}^{-1}\hat{W}_{2}(t^{\prime})\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top}).$

∎

Now, we need

######  Lemma 4.1.

Given two semi-infinite matrices $U$ and $V$ we have

$\displaystyle UV=$
$\displaystyle\frac{1}{(2\pi\operatorname{i})^{D}}\oint_{\mathbb{T}^{D}(\boldsymbol{r})}U\chi(\boldsymbol{z})\big{(}V^{T}\chi^{*}(\boldsymbol{z})\big{)}^{\top}\operatorname{d}z_{1}\cdots\operatorname{d}z_{D}=\frac{1}{(2\pi\operatorname{i})^{D}}\oint_{\mathbb{T}^{D}(\boldsymbol{r})}U\chi^{*}(\boldsymbol{z})\big{(}V^{T}\chi(\boldsymbol{z})\big{)}^{\top}\operatorname{d}z_{1}\cdots\operatorname{d}z_{D}.$

###### Proof.

Observe that

$\displaystyle\chi(\chi^{*})^{\top}$
$\displaystyle=\left(\begin{matrix}Z_{[0],[0]}&Z_{[0],[1]}&\dots\\\
Z_{[1],[0]}&Z_{[1],[1]}&\dots\\\ \vdots&\vdots&\end{matrix}\right),$
$\displaystyle Z_{[k],[\ell]}$ $\displaystyle\coloneq\frac{1}{z_{1}\cdots
z_{D}}\left(\begin{matrix}\boldsymbol{z}^{\boldsymbol{k}_{1}-\boldsymbol{\ell}_{1}}&\boldsymbol{z}^{\boldsymbol{k}_{1}-\boldsymbol{\ell}_{2}}&\dots&\boldsymbol{z}^{\boldsymbol{k}_{1}-\boldsymbol{\ell}_{|[\ell]|}}\\\
\boldsymbol{z}^{\boldsymbol{k}_{2}-\boldsymbol{\ell}_{1}}&\boldsymbol{z}^{\boldsymbol{k}_{2}-\boldsymbol{\ell}_{2}}&\dots&\boldsymbol{z}^{\boldsymbol{k}_{2}-\boldsymbol{\ell}_{|[\ell]|}}\\\
\vdots&\vdots&&\vdots\\\
\boldsymbol{z}^{\boldsymbol{k}_{|[k]|}-\boldsymbol{\ell}_{1}}&\boldsymbol{z}^{\boldsymbol{k}_{|[k]|}-\boldsymbol{\ell}_{2}}&\dots&\boldsymbol{z}^{\boldsymbol{k}_{|[k]|}-\boldsymbol{\ell}_{|[\ell]|}}\end{matrix}\right).$

If we now integrate in the polydisk distinguished border
$\mathbb{T}^{D}(\boldsymbol{r})$ using the Fubini theorem we factor each
integral in a product of $D$ factors, where the $i$-th factor is an integral
over $z_{i}$ on the circle centered at origin of radius $r_{i}$. This is zero
unless the integrand is $z_{i}^{-1}$ which occurs only in the principal
diagonal. Consequently, we have

$\displaystyle\oint_{\mathbb{T}^{D}(\boldsymbol{r})}\chi(\boldsymbol{z})\chi^{*}(\boldsymbol{z})^{\top}\operatorname{d}z_{1}\cdots\operatorname{d}z_{D}=\oint_{\mathbb{T}^{D}(\boldsymbol{r})}\chi^{*}(\boldsymbol{z})\chi(\boldsymbol{z})^{\top}\operatorname{d}z_{1}\cdots\operatorname{d}z_{D}=(2\pi\operatorname{i})^{D}\mathbb{I},$

and the result follows. ∎

We notice that $\Psi_{1}$ and $\Psi_{2}^{*}$ lead to the computation of finite
sums, i.e., polynomials, but $\Psi_{1}^{*}$ and $\Psi_{2}$ involve Laurent
series. We will denote by $\mathscr{D}_{2,{\boldsymbol{\alpha}}}(t)$ and
$\mathscr{D}_{1,{\boldsymbol{\alpha}}}^{*}(t)$ the domains of convergence of
$\Psi_{2,{\boldsymbol{\alpha}}}(t,\boldsymbol{z})$ and
$\Psi_{1,{\boldsymbol{\alpha}}}^{*}(t,\boldsymbol{z})$, respectively. Recall
that these domains are Reinhardt domains; i.e., if
$\mathscr{D}\subset\mathbb{C}^{D}$ is the domain of convergence then for any
$\boldsymbol{c}=(c_{1},\dots,c_{D})^{\top}\in\mathscr{D}$ we have that
$\mathbb{T}^{D}(|c_{1}|,\dots,|c_{D}|)\subset\mathscr{D}$.

######  Theorem 4.3 (Generalized bilinear equations).

For any pair of times $t$ and $t^{\prime}$, points
$\boldsymbol{r}_{1}\in{\mathscr{D}}^{*}_{1,{\boldsymbol{\alpha}}}(t)$ and
$\boldsymbol{r}_{2}\in\hat{\mathscr{D}}_{2,{\boldsymbol{\alpha}}}(t^{\prime})$
in the respective Reinhardt domains and $D$-dimensional tori
$\mathbb{T}^{D}(\boldsymbol{r}_{1})$ and $\mathbb{T}^{D}(\boldsymbol{r}_{2})$,
and multi-indices
${\boldsymbol{\alpha}},{\boldsymbol{\alpha}}^{\prime}\in\mathbb{Z}_{+}^{D}$,
the Baker and adjoint Baker functions and their linear spectral
transformations satisfy the following bilinear identity

$\displaystyle\oint_{\mathbb{T}^{D}(\boldsymbol{r}_{1})}\hat{\Psi}_{1,{\boldsymbol{\alpha}}^{\prime}}(t^{\prime},\boldsymbol{z})\Psi_{1,{\boldsymbol{\alpha}}}^{*}(t,\boldsymbol{z})\mathcal{Q}_{1}(\boldsymbol{z})\operatorname{d}z_{1}\cdots\operatorname{d}z_{D}=\oint_{\mathbb{T}^{D}(\boldsymbol{r}_{2})}\hat{\Psi}_{2,{\boldsymbol{\alpha}}^{\prime}}(t^{\prime},\boldsymbol{z})\Psi_{2,{\boldsymbol{\alpha}}}^{*}(t,\boldsymbol{z})\mathcal{Q}_{2}(\boldsymbol{z})\operatorname{d}z_{1}\cdots\operatorname{d}z_{D}.$

###### Proof.

From Definition 4.3 and Lemma 4.1, choosing
$U=\hat{W}_{1}(t^{\prime})\mathcal{Q}_{1}\boldsymbol{\Lambda})$ and
$V=\big{(}W_{1}(t)\big{)}^{-1}$ we get

$\displaystyle\hat{W}_{1}(t^{\prime})\mathcal{Q}_{1}(\boldsymbol{\Lambda})\big{(}W_{1}(t)\big{)}^{-1}$
$\displaystyle=\frac{1}{(2\pi\operatorname{i})^{D}}\oint_{\mathbb{T}^{D}(\boldsymbol{r}_{1})}\hat{\Psi}_{1}(t^{\prime},\boldsymbol{z})\Psi_{1}^{*}(t,\boldsymbol{z})\mathcal{Q}_{1}(\boldsymbol{z})\operatorname{d}z_{1}\cdots\operatorname{d}z_{D},$

and choosing $U=\hat{W}_{2}(t^{\prime})$ and
$V=\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})\big{(}W_{2}(t)\big{)}^{-1}$ we
get

$\displaystyle\hat{W}_{2}(t^{\prime})\mathcal{Q}_{2}(\boldsymbol{\Lambda}^{\top})\big{(}W_{2}(t)\big{)}^{-1}$
$\displaystyle=\frac{1}{(2\pi\operatorname{i})^{D}}\oint_{\mathbb{T}^{D}(\boldsymbol{r}_{2})}\hat{\Psi}_{2}(t^{\prime},\boldsymbol{z})\Psi_{2}^{*}(t,\boldsymbol{z})\mathcal{Q}_{2}(\boldsymbol{z})\operatorname{d}z_{1}\cdots\operatorname{d}z_{D}.$

Then, Proposition 4.16 implies the result. ∎

##  Appendix A Uvarov perturbations

Uvarov considered in §2 of [75] the addition of a finite number of masses,
Dirac deltas, to a given measure in the OPRL situation. In this appendix we
discuss some elements of the multivariate extension of this construction.
There is an immediate extension when one considers masses, see [27]. A bit
more involved case is to consider higher multipoles, i.e., derivatives of the
Dirac distributions. In [27] a Sobolev type modification was considered, see
for example equation (2.16) in that paper, but this can not be modeled by a
perturbation $\hat{u}=u+v$ of a linear functional $u$ (or measure in that
case, $u=\operatorname{d}\mu$). All this can be considered as a 0-dimensional
additive perturbation. However, more interesting and less trivial extensions
are to consider higher dimensional additive perturbations. For example,
1D-Uvarov perturbations, i.e., additive perturbations supported over curves.
For the 0D-Uvarov perturbations, as was found in [75], one needs to solve a
linear system constructed in terms of the non-perturbed Christoffel–Darboux
kernel evaluated at the 0D discrete support of the perturbation. We will se
that for the 1D scenario the linear system of the 0D case is replaced by a
Fredholm integral equation evaluated at the 1D support of the perturbation..

Our approach to the problem is based on a simple relation among perturbed and
non perturbed MVOPR which involves the non perturbed Christoffel–Darboux
kernel. Let us consider a generalized function
$u\in\big{(}\mathbb{C}[\boldsymbol{x}]\big{)}^{\prime}$ such that is
quasidefinite and consider an additive perturbation of it given by another
generalized function $v\in\big{(}\mathbb{C}[\boldsymbol{x}]\big{)}^{\prime}$

$\displaystyle\hat{u}=u+v.$

######  Proposition A.1.

For an additive perturbation we have

(A.1) $\displaystyle\hat{P}_{[n]}(\boldsymbol{x})$
$\displaystyle=P_{[n]}(\boldsymbol{x})-\left\langle
v,\hat{P}_{[n]}(\boldsymbol{y})K_{n-1}(\boldsymbol{y},\boldsymbol{x})\right\rangle,$
(A.2) $\displaystyle\hat{H}_{[n]}$ $\displaystyle=H_{[n]}+\left\langle
v,\hat{P}_{[n]}(\boldsymbol{x})\big{(}P_{[n]}(\boldsymbol{x})\big{)}^{\top}\right\rangle.$

###### Proof.

From (1.3) and (1.4) we deduce

$\displaystyle\left\langle\hat{u},\hat{P}_{[n]}(\boldsymbol{x})\Big{(}P_{[m]}(\boldsymbol{x})\Big{)}^{\top}\right\rangle$
$\displaystyle=0,$ $\displaystyle m\in\\{0,1,\dots,n-1\\},$
$\displaystyle\left\langle\hat{u},\hat{P}_{[n]}(\boldsymbol{x})\Big{(}P_{[n]}(\boldsymbol{x})\Big{)}^{\top}\right\rangle$
$\displaystyle=\hat{H}_{[n]},$

and, consequently,

$\displaystyle\left\langle
u,\hat{P}_{[n]}(\boldsymbol{x})\big{(}P_{[m]}(\boldsymbol{x})\big{)}^{\top}\right\rangle$
$\displaystyle=-\left\langle
v,\hat{P}_{[n]}(\boldsymbol{x})\big{(}P_{[m]}(\boldsymbol{x})\big{)}^{\top}\right\rangle,$
$\displaystyle m\in\\{0,1,\dots,n-1\\}.$

Thus, in terms of the Christoffel–Darboux kernel, see Definition 1.4

$\displaystyle\left\langle
u,\hat{P}_{[n]}(\boldsymbol{x})K_{n-1}(\boldsymbol{x},\boldsymbol{y})\right\rangle=-\left\langle
v,\hat{P}_{[n]}(\boldsymbol{x})K_{n-1}(\boldsymbol{x},\boldsymbol{y})\right\rangle.$

Observe that $\hat{P}_{[n]}(\boldsymbol{x})-P_{[n]}(\boldsymbol{x})$ is a
multivariate polynomial of degree $n-1$ and, according to (1.8), we conclude

$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})-P_{[n]}(\boldsymbol{x})$
$\displaystyle=\left\langle
u,\big{(}\hat{P}_{[n]}(\boldsymbol{y})-P_{[n]}(\boldsymbol{y})\big{)}K_{n-1}(\boldsymbol{y},\boldsymbol{x})\right\rangle$
$\displaystyle=\left\langle
u,\hat{P}_{[n]}(\boldsymbol{y})K_{n-1}(\boldsymbol{y},\boldsymbol{x})\right\rangle$
$\displaystyle=-\left\langle
v,\hat{P}_{[n]}(\boldsymbol{y})K_{n-1}(\boldsymbol{y},\boldsymbol{x})\right\rangle.$

Finally, we have

$\displaystyle\hat{H}_{[n]}$ $\displaystyle=\left\langle
u,\hat{P}_{[n]}(\boldsymbol{x})\big{(}P_{[n]}(\boldsymbol{x})\big{)}^{\top}\right\rangle+\left\langle
v,\hat{P}_{[n]}(\boldsymbol{x})\big{(}P_{[n]}(\boldsymbol{x})\big{)}^{\top}\right\rangle$
$\displaystyle=H_{[n]}+\left\langle
v,\hat{P}_{[n]}(\boldsymbol{x})\big{(}P_{[n]}(\boldsymbol{x})\big{)}^{\top}\right\rangle.$

∎

###  A.1. 0D-Uvarov multipolar perturbations. Masses (or charges) and dipoles

Here we discuss the more general additive perturbation with finite discrete
support. As we have a finite number of points for the support we say that is a
0 dimensional perturbation. Let us proceed and consider a set of couples
$S=\big{\\{}\boldsymbol{x}_{i},\boldsymbol{\beta}_{i}\big{\\}}_{i=1}^{q}\subset\mathbb{R}^{D}\times\mathbb{Z}_{+}^{D}$
and define the associated generalized function

$\displaystyle
v_{S}:=\sum_{i=1}^{q}\sum_{{\boldsymbol{\alpha}}\leq\boldsymbol{\beta}_{i}}\frac{(-1)^{|{\boldsymbol{\alpha}}|}}{{\boldsymbol{\alpha}}!}\xi_{i,{\boldsymbol{\alpha}}}\delta^{({\boldsymbol{\alpha}})}(\boldsymbol{x}-\boldsymbol{x}_{i}).$

Here the sum over multi-indices extend to all those multi-indices below a
given one. The Dirac delta distribution and its derivatives are given by

$\displaystyle\langle\delta^{({\boldsymbol{\alpha}})}(\boldsymbol{x}-\boldsymbol{x}_{j}),P(\boldsymbol{x})\rangle$
$\displaystyle:=(-1)^{|{\boldsymbol{\alpha}}|}\frac{\partial^{|{\boldsymbol{\alpha}}|}P}{\partial\boldsymbol{x}^{\boldsymbol{\alpha}}}\Big{|}_{\boldsymbol{x}=\boldsymbol{x}_{i}},$
$\displaystyle\forall P(\boldsymbol{x})\in\mathbb{C}[\boldsymbol{x}],$

and we have used the lexicographic order for the set of integer multi-indices.
Observe that this is the more general distribution with support on
$\\{\boldsymbol{x}_{j}\\}_{j=1}^{q}=\operatorname{supp}(v_{S})$. From a
physical point of view, the delta functions can be understood as point masses.
For higher order derivatives, we have an electromagnetic interpretation, for
zero order derivatives we have point charges, and first order derivatives
could be understood as dipoles, and in general for $j$-th order derivatives we
are dealing with $2^{j}$-multipoles (for $j=2$ we have quadropoles, for $j=3$
we have octopoles, and so on and so forth).111Given a charge density
$\rho(\boldsymbol{x})$ we get the multipolar expansion by writing
$\rho(\boldsymbol{x})=\int\delta(\boldsymbol{x}-\boldsymbol{x}^{\prime})\rho(\boldsymbol{x}^{\prime})\operatorname{d}\boldsymbol{x}^{\prime}$
and recalling
$\delta(\boldsymbol{x}-\boldsymbol{x}^{\prime})=\sum\limits_{{\boldsymbol{\alpha}}\in\mathbb{Z}_{+}^{D}}\frac{(-1)^{|{\boldsymbol{\alpha}}|}}{{\boldsymbol{\alpha}}!}(\boldsymbol{x}^{\prime})^{\boldsymbol{\alpha}}\delta^{({\boldsymbol{\alpha}})}(\boldsymbol{x})$.
Then, the charge density is expressed as
$\rho(\boldsymbol{x})=\sum\limits_{{\boldsymbol{\alpha}}\in\mathbb{Z}_{+}^{D}}\frac{(-1)^{|{\boldsymbol{\alpha}}|}}{{\boldsymbol{\alpha}}!}\xi_{\boldsymbol{\alpha}}\delta^{({\boldsymbol{\alpha}})}(\boldsymbol{x})$
with multipole moments the _tensors_
$\xi_{\boldsymbol{\alpha}}=\int\rho(\boldsymbol{x})\boldsymbol{x}^{\boldsymbol{\alpha}}\operatorname{d}\boldsymbol{x}$.

######  Definition A.1.

  1. (1)

Given a multi-index $\boldsymbol{\beta}\in\mathbb{Z}_{+}^{D}$ with
$|\boldsymbol{\beta}|=k\in\mathbb{Z}_{+}$ we have a corresponding
lexicographic ordered set of multi-indices

$\displaystyle\big{\\{}{\boldsymbol{\alpha}}^{(0)}_{1},{\boldsymbol{\alpha}}^{(1)}_{1},\dots,{\boldsymbol{\alpha}}^{(1)}_{D},{\boldsymbol{\alpha}}^{(2)}_{1},\dots,{\boldsymbol{\alpha}}^{(k)}_{1},\dots,{\boldsymbol{\alpha}}^{(k)}_{m}=\boldsymbol{\beta}\big{\\}},$

where $m$ denotes the position of the multi-index $\boldsymbol{\beta}$ among
those of length $k$, see §1.3.

  2. (2)

Given a couple
$(\boldsymbol{x},\boldsymbol{\beta})\in\mathbb{R}^{D}\times\mathbb{Z}_{+}^{D}$
and a polynomial $P\in\mathbb{C}[\boldsymbol{x}]$ we define the jet

$\displaystyle\mathcal{J}_{P}^{\boldsymbol{\beta}}(\boldsymbol{x})=\begin{bmatrix}\dfrac{1}{({\boldsymbol{\alpha}}^{(0)}_{1})!}\dfrac{\partial^{|{\boldsymbol{\alpha}}^{(0)}_{1}|}P}{\partial\boldsymbol{x}^{{\boldsymbol{\alpha}}^{(0)}_{1}}}(\boldsymbol{x}),\dfrac{1}{({\boldsymbol{\alpha}}^{(1)}_{1})!}\dfrac{\partial^{|{\boldsymbol{\alpha}}^{(1)}_{1}|}P}{\partial\boldsymbol{x}^{{\boldsymbol{\alpha}}^{(1)}_{1}}}(\boldsymbol{x}),\dots,\dfrac{1}{({\boldsymbol{\alpha}}^{(k)}_{1})!}\dfrac{\partial^{|{\boldsymbol{\alpha}}^{(k)}_{1}|}P}{\partial\boldsymbol{x}^{{\boldsymbol{\alpha}}^{(k)}_{1}}}(\boldsymbol{x}),\dots,\dfrac{1}{\boldsymbol{\beta}!}\dfrac{\partial^{|\boldsymbol{\beta}|}P}{\partial\boldsymbol{x}^{\boldsymbol{\beta}}}(\boldsymbol{x})\end{bmatrix}.$

This a row vector with $N_{k-1}+m$ components. Recall that the dimension of
the linear space of multivariate polynomials of degree less or equal to $k$ is
$N_{k}=\binom{D+k}{D}$.

  3. (3)

Given the set, we define a matrix collecting the corresponding jets at each
puncture $\boldsymbol{x}_{i}$

$\displaystyle\mathcal{J}_{P}(S)=\begin{bmatrix}\mathcal{J}_{P}^{\boldsymbol{\beta}_{1}}(\boldsymbol{x}_{1}),\dots,\mathcal{J}_{P}^{\boldsymbol{\beta}_{q}}(\boldsymbol{x}_{q})\end{bmatrix}.$

This a row vector with $N_{S}:=\sum\limits_{i=1}^{q}(N_{k_{i}-1}+m_{i})$
components.

  4. (4)

We consider the block antidiagonal matrices
$\xi^{(i)}\in\mathbb{C}^{(N_{k_{i}-1}+m_{i})\times(N_{k_{i}-1}+m_{i})}$,
$i\in\\{1,\dots,q\\}$, with coefficients

$\displaystyle(\xi^{(i)})_{{\boldsymbol{\alpha}},\boldsymbol{\beta}}:=\xi_{i,{\boldsymbol{\alpha}}+\boldsymbol{\beta}}$

and the matrix

$\displaystyle\varXi:=\operatorname{diag}(\xi^{(1)},\dots,\xi^{(q)})\in\mathbb{C}^{N_{S}\times
N_{S}}$

  5. (5)

The Christoffel–Darboux jet is given in terms of product of truncations

$\displaystyle\mathcal{K}_{n-1}(S)=\big{(}\mathcal{J}_{P^{[n]}}(S)\big{)}^{\top}(H^{[n]})^{-1}\mathcal{J}_{P^{[n]}}(S)\in\mathbb{C}^{N_{S}\times
N_{S}}.$

Notice that the truncation $P^{[n]}(\boldsymbol{x})$ is a vector of
polynomials and, therefore, $\mathcal{J}_{P^{[n]}}(S)$ is a $N_{n-1}\times
N_{S}$ complex matrix.

######  Theorem A.1 (0D-Uvarov multipolar perturbation).

Given a discrete additive perturbation of the form

$\displaystyle\hat{u}=u+\sum_{i=1}^{q}\sum_{{\boldsymbol{\alpha}}\leq\boldsymbol{\beta}_{i}}\frac{(-1)^{|{\boldsymbol{\alpha}}|}}{{\boldsymbol{\alpha}}!}\xi_{i,{\boldsymbol{\alpha}}}\delta^{({\boldsymbol{\alpha}})}(\boldsymbol{x}-\boldsymbol{x}_{i}),$

the new MVOPR and quasi-tau matrices are given by the following quasi-
determinantal expressions

$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})$
$\displaystyle=\Theta_{*}\left(\begin{array}[]{c|c}I_{N_{S}}+\varXi\mathcal{K}_{n-1}(S)&\varXi\big{(}\mathcal{J}_{K_{n-1}(\cdot,\boldsymbol{x})}(S)\big{)}^{\top}\\\
\hline\cr\mathcal{J}_{P_{[n]}}(S)&P_{[n]}(\boldsymbol{x})\end{array}\right),$
$\displaystyle\hat{H}_{[n]}$
$\displaystyle=\Theta_{*}\left(\begin{array}[]{c|c}I_{N_{S}}+\varXi\mathcal{K}_{n-1}(S)&-\varXi\big{(}\mathcal{J}_{P_{[n]}}(S)\big{)}^{\top}\\\
\hline\cr\mathcal{J}_{P_{[n]}}(S)&H_{[n]}\end{array}\right).$

###### Proof.

From (A.1) we conclude that

$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})=P_{[n]}(\boldsymbol{x})-\mathcal{J}_{\hat{P}_{[n]}}(S)\varXi\big{(}\mathcal{J}_{K_{n-1}(\cdot,\boldsymbol{x})}(S)\big{)}^{\top}$

and, therefore, we deduce

$\displaystyle\mathcal{J}_{\hat{P}_{[n]}}(S)=\mathcal{J}_{P_{[n]}}(S)-\mathcal{J}_{\hat{P}_{[n]}}(S)\varXi\mathcal{K}_{n-1}(S)$

that is, the unknowns $\mathcal{J}_{\hat{P}_{[n]}}(S)$ satisfy the following
linear system

$\displaystyle\mathcal{J}_{\hat{P}_{[n]}}(S)\big{(}I_{N_{S}}+\varXi\mathcal{K}_{n-1}(S)\big{)}=\mathcal{J}_{P_{[n]}}(S).$

Let us prove that the quasidefiniteness of $u$ implies that
$I_{N_{S}}+\varXi\mathcal{K}_{n-1}(S)$ is not singular, we follow [28]. If we
assume that $I_{N_{s}}+\varXi\mathcal{K}_{n-1}(S)$ is singular there must
exist a non-zero vector $\boldsymbol{C}\in\mathbb{C}^{N_{S}}$ such that
$\big{(}I_{N_{S}}+\varXi\mathcal{K}_{n-1}(S)\big{)}\boldsymbol{C}=0$ and,
consequently, such that $\mathcal{J}_{P_{[n]}}(S)\boldsymbol{C}=0$. Now, let
us observe that from

$\displaystyle
I_{N_{s}}+\varXi\mathcal{K}_{n}(S)=I_{N_{s}}+\varXi\mathcal{K}_{n-1}(S)+\varXi\big{(}\mathcal{J}_{P_{[n]}}(S)\big{)}^{\top}(H_{[n]})^{-1}\mathcal{J}_{P_{[n]}}(S).$

we get $\big{(}I_{N_{s}}+\varXi\mathcal{K}_{n}(S)\big{)}\boldsymbol{C}=0$ and,
consequently, $\mathcal{J}_{P_{[n+1]}}(S)\boldsymbol{C}=0$. By induction, we
deduce that $\mathcal{J}_{P_{[l]}}(S)\boldsymbol{C}=0$, for
$l\in\\{n,n+1,\dots\\}$ ; i.e., the generalized function
$\mathcal{J}(S)\boldsymbol{C}:=\sum\limits_{i=1}^{q}\sum\limits_{{\boldsymbol{\alpha}}\leq\boldsymbol{\beta}_{i}}\dfrac{(-1)^{|{\boldsymbol{\alpha}}|}}{{\boldsymbol{\alpha}}!}C_{i,{\boldsymbol{\alpha}}}\delta^{{\boldsymbol{\alpha}}}(\boldsymbol{x}-\boldsymbol{x}_{i})$
is such that
$\big{\langle}\mathcal{J}(S)\boldsymbol{C},P_{[l]}(\boldsymbol{x})\big{\rangle}=0$,
for $l\in\\{n,n+1,\dots\\}$. Equivalently, that is to say

$\displaystyle\mathcal{J}(S)\boldsymbol{C}$
$\displaystyle\in\Big{(}\\{P_{{\boldsymbol{\alpha}}}\\}_{|{\boldsymbol{\alpha}}|\geq
n}\Big{)}^{\perp}=\big{\\{}\tilde{u}\in\big{(}\mathbb{C}[\boldsymbol{x}]\big{)}^{*}:\langle\tilde{u},P_{\boldsymbol{\alpha}}\rangle=0,\forall{\boldsymbol{\alpha}}\in\mathbb{Z}_{+}^{D}:|{\boldsymbol{\alpha}}|\geq
n\big{\\}}.$

In the one hand, the orthogonality relations for the MVOPR leads us to
conclude that
$\Big{(}\\{P_{{\boldsymbol{\alpha}}}\\}_{|{\boldsymbol{\alpha}}|\geq
n}\Big{)}^{\perp}=\mathbb{C}\big{\\{}P_{\boldsymbol{\alpha}}^{*}\\}_{|{\boldsymbol{\alpha}}|<n}$,
with the covectors defined by $\langle
P^{*}_{\boldsymbol{\alpha}},P_{\boldsymbol{\beta}}\rangle=\delta_{{\boldsymbol{\alpha}},\boldsymbol{\beta}}$;
thus, $\dim\Big{(}\\{P_{{\boldsymbol{\alpha}}}\\}_{|{\boldsymbol{\alpha}}|\geq
n}\Big{)}^{\perp}=N_{n-1}$. In the other hand, we notice that
$\mathbb{C}\big{\\{}\boldsymbol{x}^{\boldsymbol{\alpha}}u\big{\\}}_{|{\boldsymbol{\alpha}}|<n}\subset\Big{(}\\{P_{{\boldsymbol{\alpha}}}\\}_{|{\boldsymbol{\alpha}}|\geq
n}\Big{)}^{\perp}$, but as
$\dim\mathbb{C}\big{\\{}\boldsymbol{x}^{\boldsymbol{\alpha}}u\big{\\}}_{|{\boldsymbol{\alpha}}|<n}=N_{n-1}$
we deduce $\Big{(}\\{P_{{\boldsymbol{\alpha}}}\\}_{|{\boldsymbol{\alpha}}|\geq
n}\Big{)}^{\perp}=\mathbb{C}\big{\\{}\boldsymbol{x}^{\boldsymbol{\alpha}}u\big{\\}}_{|{\boldsymbol{\alpha}}|<n}$.
Consequently, there exists a non-zero polynomial
$Q(\boldsymbol{x})\in\mathbb{C}[\boldsymbol{x}]$, $\deg Q\leq n-1$, such that
$\mathcal{J}(S)\boldsymbol{C}=Qu$. Finally, let us notice that
$\prod\limits_{i=1}^{q}(\boldsymbol{x}-\boldsymbol{x}_{i})^{\boldsymbol{\beta}_{i}}\in\operatorname{Ker}(\mathcal{J}(S)\boldsymbol{C})$,
so that
$\prod\limits_{i=1}^{q}(\boldsymbol{x}-\boldsymbol{x}_{i})^{\boldsymbol{\beta}_{i}}Q(\boldsymbol{x})u=0$,
and $u$ can not be quasidefinite, in contradiction with the initial
assumptions.

Now, as $I_{N_{s}}+\varXi\mathcal{K}_{n-1}(S)$ is not singular we deduce

$\displaystyle\mathcal{J}_{\hat{P}_{[n]}}(S)=\mathcal{J}_{P_{[n]}}(S)\big{(}I_{N_{s}}+\varXi\mathcal{K}_{n-1}(S)\big{)}^{-1}.$

Thus,

$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})=P_{[n]}(\boldsymbol{x})-\mathcal{J}_{P_{[n]}}(S)\big{(}I_{N_{s}}+\varXi\mathcal{K}_{n-1}(S)\big{)}^{-1}\varXi\big{(}\mathcal{J}_{K_{n-1}(\cdot,\boldsymbol{x})}(S)\big{)}^{\top}$

and the result follows. From (A.2) we have

$\displaystyle\hat{H}_{[n]}$
$\displaystyle=H_{[n]}+\mathcal{J}_{\hat{P}_{[n]}}(S)\varXi\big{(}\mathcal{J}_{P_{[n]}}(S)\big{)}^{\top}$
$\displaystyle=H_{[n]}+\mathcal{J}_{P_{[n]}}(S)\big{(}I_{N_{s}}+\varXi\mathcal{K}_{n-1}(S)\big{)}^{-1}\varXi\big{(}\mathcal{J}_{P_{[n]}}(S)\big{)}^{\top}.$

∎

######  Corollary A.1 (0D-Uvarov mass perturbation).

Given the set of pairs $\\{\boldsymbol{x}_{i},\xi_{i}\\}_{I=1}^{q}$, positions
and masses, and a discrete additive mass perturbation of the form

$\displaystyle\hat{u}=u+\sum_{i=1}^{q}\xi_{i}\delta(\boldsymbol{x}-\boldsymbol{x}_{i}),$

the new MVOPR and quasi-tau matrices are given by the following quasi-
determinantal expressions

$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})=\Theta_{*}\begin{bmatrix}1+\xi_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})&\dots&K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&\xi_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x})\\\
K_{n-1}(\boldsymbol{x}_{2},\boldsymbol{x}_{1})&1+\xi_{2}K_{n-1}(\boldsymbol{x}_{2},\boldsymbol{x}_{2})&\dots&K_{n-1}(\boldsymbol{x}_{2},\boldsymbol{x}_{q})&\xi_{2}K_{n-1}(\boldsymbol{x}_{2},\boldsymbol{x})\\\
\vdots&&\ddots&&\vdots\\\
K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&K_{n-1}(\boldsymbol{x}_{q}\boldsymbol{x}_{2})&\dots&1+\xi_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&\xi_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x})\\\
P_{[n]}(\boldsymbol{x}_{1})&P_{[n]}(\boldsymbol{x}_{2})&\dots&P_{[n]}(\boldsymbol{x}_{q})&P_{[n]}(\boldsymbol{x})\end{bmatrix},$
$\displaystyle\hat{H}_{[n]}=\Theta_{*}\begin{bmatrix}1+\xi_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})&\dots&K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&-\xi_{1}\big{(}P_{[n]}(\boldsymbol{x}_{1})\big{)}^{\top}\\\
K_{n-1}(\boldsymbol{x}_{2},\boldsymbol{x}_{1})&1+\xi_{2}K_{n-1}(\boldsymbol{x}_{2},\boldsymbol{x}_{2})&\dots&K_{n-1}(\boldsymbol{x}_{2},\boldsymbol{x}_{q})&-\xi_{2}\big{(}P_{[n]}(\boldsymbol{x}_{2})\big{)}^{\top}\\\
\vdots&&\ddots&&\vdots\\\
K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{2})&\dots&1+\xi_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&-\xi_{q}\big{(}P_{[n]}(\boldsymbol{x}_{q})\big{)}^{\top}\\\
P_{[n]}(\boldsymbol{x}_{1})&P_{[n]}(\boldsymbol{x}_{2})&\dots&P_{[n]}(\boldsymbol{x}_{q})&H_{[n]}\end{bmatrix}.$

###### Proof.

We take
$S=S_{0}=\big{\\{}\boldsymbol{x}_{i},\boldsymbol{\beta}_{i}\big{\\}}_{i=1}^{q}$
with $\boldsymbol{\beta}_{i}={\boldsymbol{\alpha}}^{(0)}_{1}$, for
$i\in\\{1,\dots,q\\}$, i.e., we have no derivatives in the delta functions, we
have $N_{S}=q$ and

$\displaystyle\mathcal{J}_{P_{[n]}}(S_{0})$
$\displaystyle=\begin{bmatrix}P_{[n]}(\boldsymbol{x}_{1}),\dots,P_{[n]}(\boldsymbol{x}_{q})\end{bmatrix},$
$\displaystyle\xi^{(i)}$ $\displaystyle=\xi_{i}\in\mathbb{C},$
$\displaystyle\varXi$
$\displaystyle=\operatorname{diag}(\xi_{1},\dots,\xi_{q})$
$\displaystyle\mathcal{K}_{n-1}(S_{0})$
$\displaystyle=\begin{bmatrix}P^{[n]}(\boldsymbol{x}_{1}),\dots,P^{[n]}(\boldsymbol{x}_{q})\end{bmatrix}^{\top}\big{(}H^{[n]}\big{)}^{-1}\begin{bmatrix}P^{[n]}(\boldsymbol{x}_{1}),\dots,P^{[n]}(\boldsymbol{x}_{q})\end{bmatrix}=\begin{bmatrix}K_{n-1}(\boldsymbol{x}_{i},\boldsymbol{x}_{j})\end{bmatrix},$
$\displaystyle\mathcal{J}_{K_{n-1}(\cdot,\boldsymbol{x})}(S_{0})$
$\displaystyle=\begin{bmatrix}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}),\dots,K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x})\end{bmatrix}$

and the result follows. ∎

This result was discussed by Uvarov in §2 of [75] and its multivariate
extension was presented in [27].

We now illustrate the general 0D-Uvarov transformations formulæ by considering
the addition of first order derivatives, in _physical language_ the addition
of dipoles instead of masses. We consider $D$-dimensional gradient operator
$\nabla=\begin{bmatrix}\dfrac{\partial}{\partial
x_{1}},\dots,\dfrac{\partial}{\partial x_{D}}\end{bmatrix}$. In terns of it,
for each vector
$\boldsymbol{\xi}=(\xi_{1},\dots,\xi_{D})^{\top}\in\mathbb{C}^{D}$ we have the
normal derivative
$\nabla_{\boldsymbol{\xi}}=\sum\limits_{a=1}^{D}\xi_{a}\dfrac{\partial}{\partial
x_{a}}$. Finally, given a function
$K(\boldsymbol{x},\boldsymbol{y}):\mathbb{R}^{D}\times\mathbb{R}^{D}\to\mathbb{C}$
we denote by

$\displaystyle K^{(\nabla,0)}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=\begin{bmatrix}\dfrac{\partial K}{\partial
x_{1}}(\boldsymbol{x},\boldsymbol{y}),\dots,\dfrac{\partial K}{\partial
x_{D}}(\boldsymbol{x},\boldsymbol{y})\end{bmatrix}^{\top},$ $\displaystyle
K^{(0,\nabla)}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=\begin{bmatrix}\dfrac{\partial K}{\partial
y_{1}}(\boldsymbol{x},\boldsymbol{y}),\dots,\dfrac{\partial K}{\partial
y_{D}}(\boldsymbol{x},\boldsymbol{y})\end{bmatrix},$ $\displaystyle
K^{(\nabla,\nabla)}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=\begin{bmatrix}\dfrac{\partial^{2}K}{\partial x_{1}\partial
y_{1}}(\boldsymbol{x},\boldsymbol{y})&\dots&\dfrac{\partial^{2}K}{\partial
x_{1}\partial y_{D}}(\boldsymbol{x},\boldsymbol{y})\\\ \vdots&&\vdots\\\
\dfrac{\partial^{2}K}{\partial x_{D}\partial
y_{1}}(\boldsymbol{x},\boldsymbol{y})&\dots&\dfrac{\partial^{2}K}{\partial
x_{D}\partial y_{D}}(\boldsymbol{x},\boldsymbol{y})\end{bmatrix},$
$\displaystyle K^{(\boldsymbol{\xi},0)}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=\sum_{a=1}^{D}\xi_{a}\frac{\partial K}{\partial
x_{a}}(\boldsymbol{x},\boldsymbol{y}),$ $\displaystyle
K^{(\boldsymbol{\xi},\nabla)}(\boldsymbol{x},\boldsymbol{y})$
$\displaystyle:=\begin{bmatrix}\sum\limits_{a=1}^{D}\xi_{a}\dfrac{\partial^{2}K}{\partial
x_{a}\partial
y_{1}}(\boldsymbol{x},\boldsymbol{y}),\dots,\sum\limits_{a=1}^{D}\xi_{a}\dfrac{\partial^{2}K}{\partial
x_{a}\partial y_{D}}(\boldsymbol{x},\boldsymbol{y})\end{bmatrix}.$

######  Corollary A.2 (0D-Uvarov dipole perturbation).

Given couples of vectors (positions and strength of the dipoles)
$S_{1}=\\{\boldsymbol{x}_{i},\boldsymbol{\xi}_{i}\\}_{i=1}^{q}$ and a
corresponding discrete additive dipolar perturbation of the form

$\displaystyle\hat{u}=u+\sum_{i=1}^{q}\nabla_{\boldsymbol{\xi}_{i}}\delta(\boldsymbol{x}-\boldsymbol{x}_{i}),$

the new MVOPR and quasi-tau matrices are given by the following quasi-
determinantal expressions

$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})$
$\displaystyle=\Theta_{*}\left[\begin{array}[]{cc|c|cc|c}1+K^{(\boldsymbol{\xi}_{1},0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&K^{(\boldsymbol{\xi}_{1},\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&K^{(\boldsymbol{\xi}_{1},0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{1},\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{1},0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x})\\\
\boldsymbol{\xi}_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&I_{D}+\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&&\boldsymbol{\xi}_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&\boldsymbol{\xi}_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x})\\\
\hline\cr\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&&\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&\vdots\\\
\hline\cr
K^{(\boldsymbol{\xi}_{q},0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&K^{(\boldsymbol{\xi}_{q},\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&1+K^{(\boldsymbol{\xi}_{q},0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{q},\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{q},0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x})\\\
\boldsymbol{\xi}_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&\boldsymbol{\xi}_{q}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&&\boldsymbol{\xi}_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&I_{D}+\boldsymbol{\xi}_{q}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&\boldsymbol{\xi}_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x})\\\
\hline\cr P_{[n]}(\boldsymbol{x}_{1})&\nabla
P_{[n]}(\boldsymbol{x}_{1})&\dots&P_{[n]}(\boldsymbol{x}_{q})&\nabla
P_{[n]}(\boldsymbol{x}_{q})&P_{[n]}(\boldsymbol{x})\end{array}\right],$
$\displaystyle\hat{H}_{[n]}$
$\displaystyle=\Theta_{*}\left[\begin{array}[]{cc|c|cc|c}1+K^{(\boldsymbol{\xi}_{1},0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&K^{(\boldsymbol{\xi}_{1},\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&K^{(\boldsymbol{\xi}_{1},0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{1},\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&-\nabla_{\boldsymbol{\xi_{1}}}P_{[n]}(\boldsymbol{x})\\\
\boldsymbol{\xi}_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&I_{D}+\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&&\boldsymbol{\xi}_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&-\boldsymbol{\xi}_{1}P_{[n]}(\boldsymbol{x}_{1})\\\
\hline\cr\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&&\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&\vdots\\\
\hline\cr
K^{(\boldsymbol{\xi}_{q},0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&K^{(\boldsymbol{\xi}_{q},\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&1+K^{(\boldsymbol{\xi}_{q},0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{q},\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&-\nabla_{\boldsymbol{\xi_{q}}}P_{[n]}(\boldsymbol{x}),\\\
\boldsymbol{\xi}_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&\boldsymbol{\xi}_{q}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&&\boldsymbol{\xi}_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&I_{D}+\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&-\boldsymbol{\xi}_{q}P_{[n]}(\boldsymbol{x}_{q})\\\
\hline\cr P_{[n]}(\boldsymbol{x}_{1})&\nabla
P_{[n]}(\boldsymbol{x}_{1})&\dots&P_{[n]}(\boldsymbol{x}_{q})&\nabla
P_{[n]}(\boldsymbol{x}_{q})&H_{[n]}\end{array}\right].$

###### Proof.

In this case we take
$S=S_{1}=\big{\\{}\boldsymbol{x}_{i},\boldsymbol{\beta}_{i}\big{\\}}_{i=1}^{q}$
with $\boldsymbol{\beta}_{i}={\boldsymbol{\alpha}}^{(1)}_{D}$, for
$i\in\\{1,\dots,q\\}$, and choose

$\displaystyle\xi_{i,{\boldsymbol{\alpha}}^{(0)}_{1}}$ $\displaystyle=0,$
$\displaystyle\boldsymbol{\xi}_{i}$
$\displaystyle=\begin{bmatrix}\xi_{i,{\boldsymbol{\alpha}}^{(1)}_{1}}\\\
\vdots\\\
\xi_{i,{\boldsymbol{\alpha}}^{(1)}_{D}}\end{bmatrix}\in\mathbb{C}^{D}.$

we have $N_{S}=q(D+1)$ and

$\displaystyle\mathcal{J}_{P_{[n]}}(S_{1})$
$\displaystyle=\begin{bmatrix}P_{[n]}(\boldsymbol{x}_{1}),\nabla
P_{[n]}(\boldsymbol{x}_{1}),\dots,P_{[n]}(\boldsymbol{x}_{q}),\nabla
P_{[n]}(\boldsymbol{x}_{q})\end{bmatrix},$ $\displaystyle\xi^{(i)}$
$\displaystyle=\begin{bmatrix}0&(\boldsymbol{\xi}_{i})^{\top}\\\
\boldsymbol{\xi}_{i}&0_{D}\end{bmatrix}\in\mathbb{C}^{(D+1)\times(D+1)},$
$\displaystyle\varXi$
$\displaystyle=\operatorname{diag}(\xi^{(1)},\dots,\xi^{(q)})\in\mathbb{C}^{q(D+1)\times
q(D+1)},$ $\displaystyle\mathcal{J}_{K_{n-1}(\cdot,\boldsymbol{x})}(S_{1})$
$\displaystyle=\begin{bmatrix}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}),\big{(}K^{(\nabla,0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x})\big{)}^{\top},\dots,K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}),\big{(}K^{(\nabla,0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x})\big{)}^{\top}\end{bmatrix},$
$\displaystyle\mathcal{K}_{n-1}(S_{1})$
$\displaystyle=\begin{multlined}\begin{bmatrix}P^{[n]}(\boldsymbol{x}_{1}),\nabla
P^{[n]}(\boldsymbol{x}_{1}),\dots,P^{[n]}(\boldsymbol{x}_{q}),\nabla
P^{[n]}(\boldsymbol{x}_{q})\end{bmatrix}^{\top}\big{(}H^{[n]}\big{)}^{-1}\\\
\begin{bmatrix}P^{[n]}(\boldsymbol{x}_{1}),\nabla
P^{[n]}(\boldsymbol{x}_{1}),\dots,P^{[n]}(\boldsymbol{x}_{q}),\nabla
P^{[n]}(\boldsymbol{x}_{q})\end{bmatrix}\end{multlined}\begin{bmatrix}P^{[n]}(\boldsymbol{x}_{1}),\nabla
P^{[n]}(\boldsymbol{x}_{1}),\dots,P^{[n]}(\boldsymbol{x}_{q}),\nabla
P^{[n]}(\boldsymbol{x}_{q})\end{bmatrix}^{\top}\big{(}H^{[n]}\big{)}^{-1}\\\
\begin{bmatrix}P^{[n]}(\boldsymbol{x}_{1}),\nabla
P^{[n]}(\boldsymbol{x}_{1}),\dots,P^{[n]}(\boldsymbol{x}_{q}),\nabla
P^{[n]}(\boldsymbol{x}_{q})\end{bmatrix}$
$\displaystyle=\left[\begin{array}[]{cc|c|cc}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})\\\
K^{(\nabla,0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&K^{(\nabla,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&&K^{(\nabla,0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&K^{(\nabla,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})\\\
\hline\cr\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&&\lx@intercol\hfil\vdots\hfil\lx@intercol\\\
\hline\cr
K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})\\\
K^{(\nabla,0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&K^{(\nabla,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&&K^{(\nabla,0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&K^{(\nabla,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})\end{array}\right].$

Therefore, we compute

$I_{N_{S_{1}}}+\varXi\mathcal{K}_{n-1}(S_{1})\\\
=\left[\begin{array}[]{cc|c|cc}1+K^{(\boldsymbol{\xi}_{1},0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&K^{(\boldsymbol{\xi}_{1},\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&K^{(\boldsymbol{\xi}_{1},0)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{1},\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})\\\
\boldsymbol{\xi}_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&I_{D}+\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{1})&&\boldsymbol{\xi}_{1}K_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})&\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{1},\boldsymbol{x}_{q})\\\
\hline\cr\lx@intercol\hfil\vdots\hfil\lx@intercol\vrule\lx@intercol&&\lx@intercol\hfil\vdots\hfil\lx@intercol\\\
\hline\cr
K^{(\boldsymbol{\xi}_{q},0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&K^{(\boldsymbol{\xi}_{q},\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&\hbox{\multirowsetup
\ldots}&1+K^{(\boldsymbol{\xi}_{q},0)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&K^{(\boldsymbol{\xi}_{q},\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})\\\
\boldsymbol{\xi}_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&\boldsymbol{\xi}_{q}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{1})&&\boldsymbol{\xi}_{q}K_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})&I_{D}+\boldsymbol{\xi}_{1}K^{(0,\nabla)}_{n-1}(\boldsymbol{x}_{q},\boldsymbol{x}_{q})\end{array}\right],$

and the result follows.

∎

###  A.2. 1D-Uvarov perturbations and Fredholm integral equations

We have discussed 0-dimensional additive perturbations of $D$-dimensional
generalized functions in full generality. However, we reckon that this is a
very limited analysis, as in this multivariate context much more general
perturbations do exist, as is illustrated by (2.3). We now discuss a very
particular example, adding a 1D massive string. For this aim we assume that we
have a parametrized curve, i.e. a smooth map from the interval
$I\subset\mathbb{R}$ to $\mathbb{R}^{D}$:

$\displaystyle\boldsymbol{\gamma}:I\to\mathbb{R}^{D},$

as well as a weight function $w:I\to\mathbb{C}$. Then, the linear functional
$v$ is

$\displaystyle\langle
v,P\rangle=\int_{I}P(\boldsymbol{\gamma}(t))w(t)\operatorname{d}t.$

Recalling (A.1) we can write

(A.3)
$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})=P_{[n]}(\boldsymbol{x})-\int_{I}\hat{P}_{[n]}\big{(}\boldsymbol{\gamma}(s)\big{)}K_{n-1}(\boldsymbol{\gamma}(s),\boldsymbol{x})w(s)\operatorname{d}s.$

Now, let us remark one of the basic ideas in the proof of Theorem A.1. First,
one uses (A.1) and then evaluates on the support of the distribution. In that
case, we evaluated again at the points where the delta functions and its
derivatives where supported. In this case, we should evaluate it again at the
curve $\boldsymbol{\gamma}$.

######  Definition A.2.

We introduce some notation

$\displaystyle\hat{\pi}_{[n]}(t)$
$\displaystyle:=\hat{P}_{[n]}(\boldsymbol{\gamma}(t)),$
$\displaystyle\pi_{[n]}(t)$ $\displaystyle:=P_{[n]}(\boldsymbol{\gamma}(t)),$
$\displaystyle\kappa_{n-1}(t,s)$
$\displaystyle:=K_{n-1}\big{(}\boldsymbol{\gamma}(t),\boldsymbol{\gamma}(s)\big{)}w(s).$

Then, (A.3) implies the following integral Fredholm equation

(A.4)
$\displaystyle\hat{\pi}_{[n]}(t)=\pi_{[n]}(t)-\int_{I}\hat{\pi}_{[n]}(s)\kappa_{n-1}(s,t)\operatorname{d}s.$

This integral equation, having as integral kernel $\kappa_{n-1}(t,s)$ a
separable one, can be solved explicitly. In fact,

######  Proposition A.2.

The solution of the separable Fredholm equation (A.4) can be expressed as a
last quasi-determinant as follows

$\displaystyle\hat{\pi}_{[n]}(t)=\Theta_{*}\left[\begin{array}[]{ccc|c}H_{[0]}+\int_{I}\pi_{[0]}(s)\big{(}\pi_{[0]}(s)\big{)}^{\top}w(s)\operatorname{d}s&\dots&\int_{I}\pi_{[0]}(s)\big{(}\pi_{[n-1]}(s)\big{)}^{\top}w(s)\operatorname{d}s&\pi_{[0]}(t)\\\
\int_{I}\pi_{[1]}(s)\big{(}\pi_{[0]}(s)\big{)}^{\top}w(s)\operatorname{d}s&\dots&\int_{I}\pi_{[1]}(s)\big{(}\pi_{[n-1]}(s)\big{)}^{\top}w(s)\operatorname{d}s&\pi_{[1]}(t)\\\
\vdots&&\vdots&\vdots\\\
\int_{I}\pi_{[n-1]}(s)\big{(}\pi_{[0]}(s)\big{)}^{\top}w(s)\operatorname{d}s&\dots&H_{[n-1]}+\int_{I}\pi_{[n-1]}(s)\big{(}\pi_{[n-1]}(s)\big{)}^{\top}w(s)\operatorname{d}s&\pi_{[n-1]}(t)\\\\[10.0pt]
\hline\cr\int_{I}\pi_{[n]}(s)\big{(}\pi_{[0]}(s)\big{)}^{\top}w(t)\operatorname{d}s&\dots&\int_{I}\pi_{[n]}(s)\big{(}\pi_{[n-1]}(s)\big{)}^{\top}w(s)\operatorname{d}s&\pi_{[n]}(t)\end{array}\right].$

###### Proof.

The separability of the kernel means that the kernel $\kappa_{n-1}(t,s)$ can
be written

$\displaystyle\kappa_{n-1}(s,t)=\sum_{m=0}^{n-1}\big{(}\pi_{[m]}(s)\big{)}^{\top}\big{(}H_{[m]}\big{)}^{-1}\pi_{[m]}(t).$

Now, with the notation

(A.5) $\displaystyle
C_{[n],[m]}:=\int_{I}\hat{\pi}_{[n]}(s)\big{(}\pi_{[m]}(s)\big{)}^{\top}w(s)\operatorname{d}s$

we can write the Fredholm equation

$\displaystyle\hat{\pi}_{[n]}(t)=\pi_{[n]}(t)-\sum_{m=0}^{n-1}C_{[n],[m]}\big{(}H_{[m]}\big{)}^{-1}\pi_{[m]}(t),$

that can be introduced in (A.5) to get

$\displaystyle C_{[n,[m]}$
$\displaystyle=A_{[n],[m]}-\sum_{l=0}^{n-1}C_{[n],[l]}\big{(}H_{[l]}\big{)}^{-1}A_{[l],[m]},$
$\displaystyle A_{[n],[m]}$
$\displaystyle:=\int_{I}\pi_{[n]}(s)\big{(}\pi_{[m]}(s)\big{)}^{\top}w(s)\operatorname{d}s.$

This linear system for the $C$’s can be written

$\displaystyle\begin{bmatrix}C_{[n],[0]},\dots,C_{[n],[n-1]}\end{bmatrix}=\begin{bmatrix}A_{[n],[0]},\dots,_{[n],[n-1]}\end{bmatrix}-\begin{bmatrix}C_{[n],[0]},\dots,C_{[n],[n-1]}\
\end{bmatrix}\big{(}H^{[n]}\big{)}^{-1}\begin{bmatrix}A_{[0],[0]}&\dots&A_{[0],[n-1]}\\\
\vdots&&\vdots\\\ A_{[n-1],[0]}&\dots&A_{[n-1],[n-1]}\end{bmatrix},$

and the result follows. ∎

######  Proposition A.3.

Given the solution $\hat{\pi}_{[n]}(t)$ to the Fredholm equation (A.4) we find
the perturbed MVOPR and squared norms can be expressed

$\displaystyle\hat{P}_{[n]}(\boldsymbol{x})$
$\displaystyle=P_{[n]}(\boldsymbol{x})-\int_{I}\hat{\pi}_{[n]}(t)K_{n-1}(\boldsymbol{\gamma}(t),\boldsymbol{x})w(t)\operatorname{d}t,$
$\displaystyle\hat{H}_{[n]}$
$\displaystyle=H_{[n]}+\int_{I}\hat{\pi}_{[n]}(t)\big{(}\pi_{[n]}(t)\big{)}^{\top}w(t)\operatorname{d}t.$

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