1 Introduction↩︎

Coherent states are quantum-mechanical states whose dynamics resemble the behaviour of classical oscillators [1], [2]. The mathematical treatment of coherent states has opened a number of interesting new directions in mathematical physics [3], [4]. In particular, there has been a long standing interest in describing coherent states related to exactly solvable potentials obtained via the method of supersymmetric quantum mechanics (SUSYQM) [5], [6] Recently, such constructions have been applied to rational extensions of various exactly solvable potentials [7] – the latter are closely related to exceptional operators and exceptional orthogonal polynomials [8], [9]. The key methodology is showing that these systems have non-trivial algebras of ladder operators[10]; the coherent states may then be defined as eigenstates of lowering/annihilator operators. A recent article has extended the ladder-operator approach to non-rational extensions of solvable potentials [11].

In this article, we focus on the supersymmetric partners of the harmonic oscillator consisting of a modification of the quadratic potential by a rational function that vanishes at infinity — hence the name rational extension. The rational extensions of the harmonic oscillator are known to possess non-trivial algebras of ladder operators. This observation has been successfully exploited in the study superintegrable systems [10], [12] and rational solutions of Painlevé equations [13], [14].

Recent works established the bispectral character of rational extensions [15] and characterized their algebra of ladder operators using Maya diagrams [16]. In the present article, we combine these two approaches to show that all such rational extensions admit a natural notion of a coherent state as a joint eigenvalue of the commutative subalgebra of lowering/annihilator ladder operators — we name these objects extended coherent states (ECS). The mathematical description of the ECS involves a certain reduction of the \(\tau\)-function for the rational solutions of the KP equation, as is is fully explained in [15], which allows a convenient description utilizing a technique called a Miwa shift; as shown below in equation ?? . As an immediate corollary we obtain the result that the ECS asymptotically saturate the Heisenberg position-momentum uncertainty bound as do the the canonical coherent states (CCS) of the harmonic oscillator [17] first described by Schrödinger.

This article is organized as follows. Section 2 gathers the necessary background on Maya diagrams and related notions in combinatorics and integrable systems. Section 3 reviews Hermite polynomials, the harmonic oscillator, the CCS, and rational extensions. The key result here is the Miwa-shift formula ?? for the generating function of the bound states. Finally, Section 4 introduces the ECS as a modification of the above generating function, defines the annihilator algebra, establishes the joint eigenvalue properties 47 48 , and demonstrates the asymptotic minimization of uncertainty. The section concludes with an explicit example.

2 Preliminaries↩︎

2.1 Partitions and Maya diagrams↩︎

A partition of a natural number \(N\in \mathbb{N}_0\) is a non-increasing integer sequence \(\{ \lambda_i \}_{i\ge 1}\) such that \(|\lambda| := \sum_i \lambda_i = N\). Implicit in this definition is the assumption that \(\lambda_i=0\) for \(i\) sufficiently large. The length \(\ell\) of \(\lambda\) is the number of non-zero elements of the sequence. The Young diagram corresponding to \(\lambda\) is an irregular tableaux consisting of \(\lambda_i\) cells in rows \(i=1,\ldots, \ell\). Formally, \[Y_\lambda = \{ (i,j) \in \mathbb{N}^2\colon 1\le i \le \ell,\; 1\le j \le \lambda_i \},\] Note that, unlike the usual convention, we place the longest row of the Young diagram at the bottom.

The hook \[\operatorname{Hk}_\lambda(i,j) = \{ (i,k)\in Y_\lambda \colon j\le k \} \cup \{ (k,j)\in Y_\lambda \colon i\le k \}\] is the set of cells connecting cell \((i,j)\) to the rim of the diagram. The hooklength \(\operatorname{hk}_\lambda(i,j)\) is the cardinality of hook \((i,j)\in Y_\lambda\). The number \[\label{eq:dlamdef} d_\lambda = \frac{N!}{\prod_{(i,j)\in Y_\lambda} \operatorname{hk}_\lambda(i,j) }\tag{1}\] counts the number of standard Young tableaux of shape \(\lambda\) and corresponds to the dimension of an irreducible representation of the symmetric group \(\mathfrak{S}_N\).

Closely related to partitions is a concept called a Maya diagram. We say that \(M\subset \mathbb{Z}\) is a Maya diagram if \(M\) contains a finite number of positive integers and excludes a finite number of negative integers. Let \(\mathcal{M}\) denote the set of all Maya diagrams. For \(M\in\mathcal{M}\) and \(n\in\mathbb{Z}\), \(M+n=\{m+n:m\in M\}\) is also a Maya diagram. Thus, \(\mathcal{M}\) admits a natural \(\mathbb{Z}\) action by translations.

We will refer to the equivalence class \(M/\mathbb{Z}\) as an unlabelled Maya diagram. Intuitively, an unlabelled Maya is a horizontal sequence of filled \(\tikz{\fill circle (3pt)}\) and empty \(\tikz{\draw circle (3pt)}\) states beginning with an infinite segment of \(\tikz{\fill circle (3pt)}\) and terminating with an infinite segment of \(\tikz{\draw circle (3pt)}\). Here, \(\tikz{\fill circle (3pt)}\) in position \(m\) is taken to indicate membership \(m\in M\). A choice of origin serves to convert an unlabelled Maya diagram to a subset of \(\mathbb{Z}\). The index of a Maya diagram \(M\in \mathcal{M}\) is the integer \[\sigma_M:= \#\{ m\in M \colon m\geq 0\}- \#\{ m\notin M \colon m<0 \};\] i.e., the difference between the number of \(\tikz{\fill circle (3pt)}\) to the right of the origin and the number of \(\tikz{\draw circle (3pt)}\) to the left of the origin. Evidently, \(\sigma_{M+n} = \sigma_M+n\).

There is a natural bijection between the set of partitions and the set of unlabelled Maya diagrams. To visualize this bijection, represent a filled state with a unit downward arrow and an empty state with a unit right arrow. As can be seen in Figure 2, the resulting “bent” Maya diagram traces out the boundary of the Young diagram of the corresponding partition \(\lambda\); see [18] for more details.

The bijection may also be described as \(\lambda\to M_{\lambda}/\mathbb{Z}\) where \(\lambda\) is a partition and \[\label{eq:Mlam} M_{\lambda}= \{ \lambda_i-i \}_{i\in \mathbb{N}}\tag{2}\] The bijection claim is justified by showing that \(\sigma_{M_{\lambda}}=0\) and that every equivalence class in \(\mathcal{M}/\mathbb{Z}\) contains a unique Maya diagram with zero index.

The flip \(f_{k}\) at position \(k\in\mathbb{Z}\) is the involution \(f_k:\mathcal{M}\rightarrow\mathcal{M}\) defined by \[f_k:M\mapsto \begin{cases} M\cup\{k\},& k\notin M\\ M\setminus\{k\},&k\in M \end{cases}.\] In the event that \(k\notin M\), the flip \(f_k\) is said to act on \(M\) by a state-deleting transformation \(\tikz{\draw circle (3pt)}\rightarrow\tikz{\fill circle (3pt)}\), while in the opposite scenario (\(k\in M\)), it is said to act by a state-adding transformation \(\tikz{\fill circle (3pt)}\rightarrow\tikz{\draw circle (3pt)}\).

Let \(\mathcal{Z}\) denote the set of all finite subsets of \(\mathbb{Z}\). For a finite set of integers \(K=\{ k_1,\ldots, k_p \}\in \mathcal{Z}\) we define the corresponding multi-flip to be the transformation \(f_K:\mathcal{M}\rightarrow\mathcal{M}\) defined according to \[f_K(M)=(f_{k_1}\circ\dotsm\circ f_{k_p})(M).\]

Observe that multi-flips are also involutions. This means that Maya diagrams together with multifips have the natural structure of a complete graph \((\mathcal{M},\mathcal{Z})\). The edge connecting \(M_1, M_2\in \mathcal{M}\) is the unique multi-flip \(f_K\) such that \(f_K(M_1)=M_2\) and \(f_K(M_2)=M_1\). The corresponding \(K\in \mathcal{Z}\) is given as the symmetric set difference \[\label{edge} K=M_1\ominus M_2=M_2\ominus M_1\tag{3}\]

Since \((\mathcal{M},\mathcal{Z})\) is a complete graph, we can define a bijection \(\mathcal{Z}\rightarrow \mathcal{M}\) given by \(K\mapsto f_K({M_0})\), where \({M_0}= \{ m \in \mathbb{Z}\colon m<0 \}\) denotes the trivial Maya diagram, and where \(K=M\ominus M_0\) is the index set of the Maya diagram \(M\).

The hooklength formula 1 can be re-expressed in terms of a Maya diagrams and index sets as follows. Let \(\lambda\) be a partition of length \(\ell\). Define \(K_\lambda\) to be the index set of \(M_{\lambda}+\ell\). Then, \(k_i= \lambda_i-i+\ell,\; i=1,2,\ldots,\ell\) is the decreasing enumeration of \(K_\lambda\), and \[\frac{N!}{d_\lambda}= \prod_{i,j} \operatorname{hk}_\lambda(i,j) = \frac{\prod_i k_i!}{\prod_{i<j}(k_i-k_j)}.\]

2.2 Vertex operators and Schur functions↩︎

For \(k\in \mathbb{N}_0\), define the ordinary Bell polynomials \(B_k(t_1,\ldots, t_k) \in \mathbb{Q}[t_1,\ldots, t_k]\) as the coefficients of the power generating function \[\label{eq:Bgf} \exp\left(\sum_{k=1}^\infty t_k z^k\right) = \sum_{k=0}^\infty B_k(t_1,\ldots, t_k) z^k,\tag{4}\] where \({\boldsymbol{t}}=(t_1,t_2,\ldots)\). The multinomial formula implies that \[\label{eq:Bksum} \begin{align} B_k(t_1,\ldots, t_k) &= \sum_{ \Vert \mu\Vert=k} \frac{t^{\mu_1}_1}{\mu_1!} \frac{t^{\mu_2}_2}{\mu_2!} \cdots \frac{t^{\mu_{k}}_{k}}{\mu_{k}!},\qquad \Vert \mu \Vert = \mu_1 + 2\mu_2 + \cdots + {k} \mu_{k}\\ &= \frac{t_1^k}{k!} + \frac{t_1^{k-2}t_2}{(k-2)!} + \cdots + t_{k-1} t_1 + t_k. \end{align}\tag{5}\] For a partition \(\lambda\) of \(N\), define the Schur function \(S_\lambda(t_1,\ldots, t_N)\in \mathbb{Q}[t_1,\ldots, t_N]\) to be the multivariate polynomial \[\label{eq:Slamdef} S_\lambda= \det(B_{m_i+j})_{i,j=1}^\ell, \quad m_i = \lambda_i-i,\tag{6}\] where \(B_k=0\) when \(k<0\). Moreover, since \[\partial_{t_i} B_j(t_1,\ldots, t_j) = B_{j-i}(t_1,\ldots, t_{j-i}),\quad j\geq i,\] we may re-express 6 in terms of a Wronskian determinant, \[\label{eq:Slamwronsk} S_\lambda= \operatorname{Wr}[B_{m_\ell+\ell},\ldots, B_{m_1+\ell}],\tag{7}\] where the Wronskian is taken with respect to \(t_1\).

Let \(\mathbf{X}_m= \mathbf{X}_m({\boldsymbol{t}},\partial_{\boldsymbol{t}}),\; m\in \mathbb{Z}\), be the operators defined by the generating function \[\label{eq:bVdef} \begin{align} \mathbf{V}({\boldsymbol{t}},\partial_{\boldsymbol{t}},z) &=\exp\left(\sum_{k=1}^\infty t_k z^k \right)\exp\left(\sum_{j=1}^\infty - j^{-1}\partial_{t_j} z^{-j} \right)\\ &= \sum_{m=-\infty}^\infty \mathbf{X}_m({\boldsymbol{t}},\partial_{\boldsymbol{t}}) z^m. \end{align}\tag{8}\] Expanding the above formulas gives \[\begin{align} \tag{9} \mathbf{X}_m &= \sum_{j=0}^\infty B_{j+m}( t_1,\ldots, t_k) B_j\left( \partial_{t_1},\ldots, j^{-1} \partial_{t_j}\right),\; m\ge 0;\\ \tag{10} \mathbf{X}_m &= \sum_{j=0}^\infty B_{j}( t_1,\ldots, t_k) B_{j-m}\left( -\partial_{t_1},\ldots, - j^{-1} \partial_{t_j}\right),\; m<0. \end{align}\] It can be shown that the above operators obey the fundamental relation \[\begin{align} \label{eq:XmXn} &\mathbf{X}_m \mathbf{X}_n + \mathbf{X}_{n-1} \mathbf{X}_{m+1} = 0. \end{align}\tag{11}\] Despite the fact that the \(\mathbf{X}_m({\boldsymbol{t}},\partial_{\boldsymbol{t}})\) are differential operators involving infinitely many variables, they have a well-defined action on polynomials. In particular, when applied to Schur functions, they function as multi-variable raising operators.

The proof of 11 –?? can be found in [19]. As an immediate corollary we obtain the following two results [15].

By construction, the action of \(\mathbf{V}({\boldsymbol{t}},z)\) on a polynomial \(P({\boldsymbol{t}})\in \mathbb{C}[t_1,\ldots, t_n]\) is \[\label{eq:bVP} \mathbf{V}({\boldsymbol{t}},z) P({\boldsymbol{t}}) = \exp\left(\sum_{k=1}^\infty t_k z^k \right)P\!\left(t_1- z^{-1}, t_2 - \frac{z^{-2}}{2}, \ldots, t_n - \frac{z^{-n}}{n}\right).\tag{12}\] Proposition 2 allows the action of \(\mathbf{V}({\boldsymbol{t}},z)\) on a Schur polynomial to be conveniently written in terms of the “insertion” procedure \(m\triangleright\lambda\):

3 The harmonic oscillator and its rational extensions↩︎

3.1 Hermite polynomials↩︎

Hermite polynomials \(\{H_n(x)\}_{n\in \mathbb{N}_0}\) are classical orthogonal polynomials that satisfy the second-order eigenvalue equation \[\label{eq:hermde} y''-2xy' = 2n y,\quad y= H_n(x),\tag{13}\] and the orthogonality relation \[\label{eq:hortho} \int_{\mathbb{R}} H_m(x) H_n(x) e^{-x^2} dx = \sqrt{\pi}\, 2^n n! \delta_{n,m}.\tag{14}\] The above is equivalent to the 3-term recurrence relation \[\label{eq:h3term} H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x),\quad H_0(x)=1\tag{15}\] The generating function for the Hermite polynomials is \[\begin{align} \label{eq:hermgf} e^{xz - \tfrac14 z^2} &= \sum_{n=0}^\infty H_n(x) \frac{z^n}{2^nn!}, \end{align}\tag{16}\] which can be readily established by observing that \[\label{eq:Dx2Dz} (\partial_x+ 2\partial_z) \left(e^{xz - \tfrac14z^2- x^2}\right)= (\partial_x+ 2\partial_z) e^{-(x-z/2)^2} = 0,\tag{17}\] and by applying the well-known Rodrigues formula \[\label{eq:HnRodrigues} H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2},\; n\in \mathbb{N}_0\tag{18}\]

Comparison of 16 with 4 shows that the Hermite polynomials are specializations of Bell polynomials: \[\begin{align} H_n(x) &= n! 2^n B_n(x,-\tfrac14,0,\ldots) .\\ \end{align}\] Applying 5 then gives the well-known formula \[H_n(x) = \sum_{j=0}^{\lfloor n/2\rfloor} (-1)^j \frac{n!}{(n-2j)! j!} (2x)^{n-2j}.\]

In the sequel, we will also make use of the conjugate Hermite polynomials: \[\label{eq:tHndef} \tilde{H}_n(x) = n! 2^n B_n(x,\tfrac14,0,\ldots) =i^{-n} H_n(i x),\quad n\in \mathbb{N}_0\tag{19}\]

3.2 The canonical Hamiltonian and coherent state↩︎

Write \(p= i \partial_x\), so that \[T(x,\partial_x) = p^2+x^2=- \partial_x^2 + x^2\] is the Hamiltonian of the quantum harmonic oscillator. We say that a function \(\psi(z)\) is quasi-rational if its log-derivative, \(\psi'(z)/\psi(z)\), is a rational function. The quasi-rational eigenfunctions of \(T\) are the Hermite functions \[\label{eq:psindef} \psi_n(x)=\begin{cases} e^{-\tfrac{x^2}{2}}H_n(x), & n\geq 0\\ e^{\tfrac{x^2}{2}}\tilde{H}_{-n-1}(x), & n<0. \end{cases},\tag{20}\] We now show that the \(\psi_n,\; n\ge 0\), represent the bound states of the harmonic oscillator, while the \(\psi_n,n<0\) represent virtual states. Multiplication of 16 by \(e^{-x^2/2}\) yields the generating function for the bound states: \[\label{eq:psingf} \Psi_0(x,z) := e^{-\tfrac12(x-z)^2+\tfrac14z^2}= \sum_{n=0}^\infty \psi_n(x) \frac{z^n}{2^n n!}.\tag{21}\] By a direct calculation, we have \[\label{eq:TPsi0} T(x,\partial_x) \Psi_0(x,z) = (2 z \partial_z+1)\Psi_0(x,z).\tag{22}\] Applying the above relation to 21 and comparing the coefficients of the resulting power series then returns the desired eigenvalue relation \[\label{eq:Tpsin} T\psi_n = (2n+1) \psi_n,\quad n\in \mathbb{Z}.\tag{23}\]

The classical ladder operators \[\label{eq:L43idef} \begin{align} L_{\mp}(x,\partial_x) := \partial_x \pm x \end{align}\tag{24}\] satisfy the intertwining relations \[T L_- = L_- (T-2),\qquad T L_+ = L_+(T+2).\] An immediate consequence are the lowering and raising relations: \[\label{eq:lrrel} L_- \psi_n = \begin{cases} 2n \psi_{n-1}, &n \ge 0\\ \psi_{n-1}, & n<0 \end{cases}\qquad L_+ \psi_n = \begin{cases} \psi_{n+1},& n> 0\\ 2(n+1) \psi_{n+1} & n\le 0 \end{cases}\tag{25}\]

Now define the canonical coherent state (CCS) to be \[\label{eq:Phi0def} \Phi_0(x,t;\alpha) := e^{-it} \Psi_0(x,\alpha e^{-2it}).\tag{26}\] The change of variable \(z=\alpha e^{-2it}\) transforms 17 and 22 into \[\begin{align} L_- \Phi_0 &= \alpha e^{-2it} \Phi_0\\ T \Phi_0 &= i \partial_t\Phi_0. \end{align}\] Thus, the CCS is an eigenfunction of \(L_{-}\) and an exact solution to the time-dependent Schrödinger equation.

3.3 Hermite Pseudo-Wronskians↩︎

Let \(M\in \mathcal{M}\) be a Maya diagram and \(K=\{k_1,\ldots, k_p\}\) the corresponding index set arranged in increasing order \(k_1<\cdots k_q<0 \le k_{q+1} < \cdots k_p\). Define the pseudo-Wronskian \[\label{eq:pWdef} H_M= \det \begin{vmatrix} \tilde{H}_{-k_1-1} & \tilde{H}_{-k_1} & \ldots & \tilde{H}_{-k_1+p-1}\\ \vdots & \vdots & \ddots & \vdots\\ \tilde{H}_{-k_q-1} & \tilde{H}_{-k_q} & \ldots & \tilde{H}_{-k_q+p-1}\\ H_{k_{q+1}} & D_x H_{k_{q+1}} & \ldots & D_x^{p-1}H_{k_{q+1}}\\ \vdots & \vdots & \ddots & \vdots\\ H_{k_p} & D_x H_{k_p} & \ldots & D_x^{p-1}H_{k_p} \end{vmatrix}.\tag{27}\] One can show [18] that the normalized polynomial \[\label{eq:hHdef} \hat{H}_M=\frac{(-1)^{(p-q)q}H_M}{\prod_{i<j\le q}2(k_j-k_i)\prod_{q+1\le i<j}2(k_j-k_i)}\tag{28}\] is translation-invariant \[\label{eq:HM43n} \hat{H}_M=\hat{H}_{M+n},\quad n\in\mathbb{Z}.\tag{29}\] and hence may be regarded as a function of the corresponding partition \(\lambda\). Moreover, in [15], it was shown that the normalized Hermite pseudo-Wronskian 28 has the following expression in terms of Schur functions: \[\label{eq:HMSlam} \hat{H}_M(x) = \frac{2^{N}N!}{d_\lambda} \, S_\lambda(x,-\tfrac14,0,\ldots ),\tag{30}\] where \(N=|\lambda|\) and where \(d_\lambda\) is the combinatorial factor defined in 1 .

3.4 Rational Extensions of the Harmonic Oscillator↩︎

Let \(M\in \mathcal{M}\) be a Maya diagram. The pseudo-Wronskian defined in 27 can now be expressed [18] simply as \[\label{eq:HM} H_M(x)=e^{\sigma_M\tfrac{x^2}{2}}\operatorname{Wr}[\psi_{k_1}(x),\dots,\psi_{k_p}(x)],\tag{31}\] where \(\psi_n(x),\; n\in \mathbb{Z}\), are the quasi-rational eigenfunctions 20 , and where \(\sigma_M\) is the index of \(M\). The potential \[\begin{align} \label{rationalext} U_M(x) &=x^2-2\frac{d^2}{dx^2}\log\operatorname{Wr}[\psi_{k_1},\dots,\psi_{k_p}]\\ \nonumber &=x^2+2\left(\frac{H'_M}{H_M}\right)^2-\frac{2H_M^{''}}{H_M}-2\sigma_M \end{align}\tag{32}\] is a rational extension of the harmonic oscillator potential, so called because the terms following the \(x^2\) in 32 are all rational. The corresponding Hamiltonian \[T_M:=-\frac{d^2}{dx^2}+U_M\] is exactly solvable [8] with eigenfunctions \[\begin{align} \label{eigenstates} \psi_{M,m}= e^{\epsilon_M(m)\tfrac{x^2}{2}}\frac{\hat{H}_{M,m}}{\hat{H}_M},\quad \epsilon_M(m) = \begin{cases} -1 & \text{ if } m\notin M\\ +1 & \text{ if } m\in M\\ \end{cases},\quad m\in \mathbb{Z} \end{align}\tag{33}\] where \((M,m):=f_m(M)\). The eigenvalue relation is \[\label{eq:TMpsiMn} T_M\psi_{M,m}=(2m+1)\psi_{M,m},\quad m\in \mathbb{Z}.\tag{34}\] The numerators \(\hat{H}_{M,m}(x),\; m\notin M\) are known as exceptional Hermite polynomials [8]. Relation 29 implies that \(T_M\) and the corresponding eigenfunctions are translation covariant: \[T_{M+n}=T_M+2n,\quad \psi_{M+n,m+n}= \psi_{M,n} ,\; n\in\mathbb{Z}\] Thus, the unlabelled Maya diagram is a representation of the spectrum, with \[\label{eq:IMdef} I_M := \mathbb{Z}\setminus M = (\mathbb{Z}\setminus M_{\lambda}) + \sigma_M,\tag{35}\] serving as the index set for the bound states (the ones with label \(\tikz{\draw circle (3pt)}\)).

As regards regularity, it should be noted that by the Krein-Adler theorem [8], \(H_M\) has no real zeros if and only if all finite \(\tikz{\fill circle (3pt)}\) segments of \(M\) have even size. It is precisely for such \(M\) that \(T_M\) corresponds to a self-adjoint operator and that the eigenfunctions \(\psi_{M,m},\; m\in I_M\) are square-integrable. If this condition fails, then one still has orthogonality and self-adjointness if one deforms the contour of integration away from the singularities [20]. However, in the presence of singularities in \(U_M\), the resulting inner product is no longer positive-definite, but rather has a finite signature.

The generating function for the bound states of a rational extension can be given using the Miwa shift formula 12 .

Observe that if \(M={M_0}\) is the trivial Maya diagram, then ?? reduces to the classical generating function shown in 21 .

4 Extended coherent states↩︎

4.1 Ladder Operators↩︎

Let \(T,A\) be differential operator. We say that \(A\) is a ladder operator for \(T\) if \[\label{eq:ladderdef} [A,T] = \lambda A\tag{36}\] for some constant \(\lambda\). As a consequence of the definition, \(A\) acts on eigenfunctions of \(T\) by a spectral shift \(\lambda\), possibly annihilating finitely many eigenfunctions. More generally, we say that \(A\) intertwines \(T_1, T_2\) if \(AT_1=T_2A\). Thus, 36 is a special case of an intertwining relation with \(T_1=T,\; T_2=T+\lambda\).

In [16] it was shown that, within the class of rational extensions, the basic intertwiner between \(T_{M_1},T_{M_2},\; M_1,M_2\in \mathcal{M}\) takes the form \[\label{eq:AMKdef} A_{M_1,K}[y]=\frac{\operatorname{Wr}[\psi_{M,k_1},\dots,\psi_{M,k_p},y]}{\operatorname{Wr}[\psi_{M,k_1},\dots,\psi_{M,k_p}]},\tag{37}\] where \(K= M_1 \ominus M_2=\{k_1,\ldots, k_p\}\) is the index set of the corresponding multi-flip \(f_K\) that connects \(M_1 \to M_2\), and where the \(\psi_{M,m},\; m\in \mathbb{Z}\) are the quasi-rational eigenfunctions of \(T_M\) defined in 33 . One can show that \(A_{M,K}\) is a monic differential operator of order \(p\) such that \[A_{M_1,K} T_{M_1} = T_{M_2} A_{M_1,K}.\] Operators \(T_{M}\) and intertwiners \(A_{M,K}\) have the abstract structure of a category [16] because intertwiners \(A_{M_1,K_1}\) and \(A_{M_2,K_2}\) where \(K_1 = M_2\ominus M_1\) obey the following composition relation \[\label{comp} A_{M_2,K_2}\circ A_{M_1,K_1}=A_{M_1,K_{1}\ominus K_2}\circ p_{K_1,K_2}(T_M),\tag{38}\] where \[\begin{align} p_{K_1,K_2}(m) &= \prod_{k\in K_1\cap K_2} (2k+1-m). \end{align}\] Since \(T_{M+n}=T_M+2n\), the above intertwiners are also translation-invariant: \[\label{translation} A_{M+n,K+n}=A_{M,K},\quad n\in\mathbb{Z}.\tag{39}\] This allows us to consider a quotient category whose objects are rational extensions modulo spectral shifts, and where the ladder operators are precisely the endomorphisms. For the details, see Section 4 of [16].

For a Maya diagram \(M\in \mathcal{M}\) and an integer \(n\in \mathbb{Z}\), let \[\label{eq:LMndef} L_{M,n}:= A_{M,(M+n) \ominus M}.\tag{40}\] By Theorem 4.1 of [16], \[\label{ladder} L_{M,n}T_M=(T_M+2n)L_{M,n}.\tag{41}\] Thus, \(L_{M,n}\) is a ladder operator for the rational extension \(T_M\). The action of \(L_{M,n}\) is that of a lowering or raising operator according to \[L_n[\psi_{M,k}] = C_{M,n,k} \psi_{M,k-n},\quad k\notin M,\] where \(C_{M,n,k}\) is zero if \(\psi_{M,k-n}\) is not a bound state, i.e., if \(k-n\in M\). Otherwise, \(C_{M,n,k}\) is a rational number whose explicit form is given in [16]. In general, the ladder operators \(L_{M,q},\; q\in \mathbb{Z}\) do not commute. However, as we now show, there is a natural subalgebra generated by lowering operators of certain critical degrees \(q\) that does commute.

4.2 The annihilator algebra↩︎

We say that a \(q\)th order ladder operator is an annihilator, if its kernel is spanned by \(q\) bound states. The annihilator algebra of a rational extension is more complicated than in the canonical case, where the annihilator algebra is generated \(L_-=\partial_x+x\). For a non-empty partition, the analogous operators generate a non-trivial algebra of commuting operators with a structure determined by the combinatorics of the corresponding Maya diagram, as we now show.

For \(q\in \mathbb{N}\), we say that a Maya diagram \(M\in \mathcal{M}\) is a \(q\)-core if \(M\subset M+q\). We say that \(q\in \mathbb{N}\) is a critical degree of a Maya diagram \(M\in \mathcal{M}\) if \(M\) is a \(q\)-core. Observe that if \(q\) is a critical degree of \(M\), then \(q\) is a critical degree of \(M+n\) for every \(n\in \mathbb{N}\). Thus, the \(q\)-core property is an attribute of an unlabelled Maya diagram. The set of unlabelled Maya diagrams is naturally bijective to the set of partitions, and so we use \(D_\lambda\), where \(\lambda\) is the partition corresponding to \(M\), to denote the set of all critical degrees. This definition is consistent with the definition of the \(q\)-core partition used in combinatorics; see [21] for more details.

A \(q\in \mathbb{N}\) fails to be in \(D_\lambda\) if and only if there exists an \(m\in M\) and a \(k\in I_M\) such that \(q=m-k\). The smallest empty position on a Maya diagram occurs at position \(m_{\ell+1}+1 = \sigma_M-\ell\), while the largest occupied position occurs at \(m_1= \lambda_1-1+\sigma_M\). It then follows that \[\label{eq:qcdef} q_c := m_1 - (\sigma_M-\ell)+1 = \lambda_1+\ell\tag{42}\] is a threshold critical degree, in the sense that \(q\in D_\lambda\) for all \(q\ge q_c\) and \(q_c-1\notin D_\lambda\). See Figure 3 for an example.

Let \(K_q = (M+q)\ominus M,\; q\in \mathbb{Z}\) so that, by 40 and 37 , the kernel of \(L_{M,q}\) is spanned by \(\psi_{M,k},\; k\in K_q\). By 33 , \(L_{M,q},\; q\in \mathbb{Z}\) is an annihilator if and only if \(K_q\subset I_M\), if and only if \(K_q = (M+q)\setminus M\), and if and only if \(M\subset M+q\). Therefore, \(L_{M,q}\) is an annihilator if and only if \(q\in D_\lambda\) is a critical degree.

By Theorem 6.1 of [15], for every critical degree \(q\in D_\lambda\), we have \[\label{eq:LqPsiz} L_{M,q}(x,\partial_x) \Psi_\lambda(x,z) = z^q \Psi_\lambda(x,z).\tag{43}\] In other words, the generating function ?? is a joint eigenfunction of the annihilators. Let \(R_\lambda=\operatorname{span}\{ z^q \colon q\in D_\lambda\}\), and observe that if \(q_1, q_2\in D_\lambda\), then \(q_1+q_2\in D_\lambda\) also. It follows that \(R_\lambda\) is closed with respect to multiplication; i.e\(.\) \(R_\lambda\) is a commutative algebra. Also note that composition of annihilation operators on the left of 43 is equivalent to multiplication of eigenvalues on the right. It follows that the annihilators commute, and that \(R_\lambda\) is isomorphic to the annihilator algebra associated with the rational extension \(T_M\).

Relations ?? and 43 entail the following action of the annihilators on the bound states: \[\label{eq:Lqpsim} L_{M,q}(x,\partial_x) \psi_{M,m}(x) = 2^q \gamma_{M,q}(m)\psi_{M,m-q}(x),\tag{44}\] where \(m\in I_M, q\in D_\lambda\), and where \[\gamma_{M,q}(m) = \prod_{k\in K_q} (m-k).\] Note that \(\gamma_{M,q}(m)=0\) when \(\psi_{M,m}\) is a bound state, but \(\psi_{M,m-q}\) is not.

4.3 Definition of the extended coherent states↩︎

We now construct the ECS corresponding to a rational extension \(T_M,\; M\in \mathcal{M}\). We proceed, as in the canonical case, by constructing the coherent state in terms of the generating function. In [15], it was shown that, in terms of the generating function \(\Psi_\lambda(x,z)\), the eigenvalue relation 34 is equivalent to \[\label{eq:TMpsilam} T_M(x,\partial_x) \Psi_\lambda(x,z) = (z\partial_z + 1 + 2\sigma_M) \Psi_\lambda(x,z).\tag{45}\] Using the same change of variables as in 26 , let us therefore set \[\label{eq:Philamdef} \Phi_\lambda(x,t;\alpha) = e^{-(1+2\sigma_M)it}\Psi_\lambda(x,\alpha e^{-2 it }).\tag{46}\] Then by construction, \(\Phi_\lambda(x,t)\) is an exact solution of the time-dependent Schrödinger equation corresponding to the rational extension \(T_M\): \[\label{eq:PhilamTM} i \partial_t \Phi_\lambda(x,t) = T_M(x,\partial_x) \Phi_\lambda(x,t) .\tag{47}\] Applying the same change of variables to 43 , we obtain the annihilator eigenrelation \[\label{eq:LqPhial} L_{M,q}(x,\partial_x) \Phi_\lambda(x,t;\alpha) = \alpha^q e^{-2iqt} \Phi_\lambda(x,z).\tag{48}\] Hence, \(\Phi_\lambda(x,t;\alpha)\) is a joint eigenfunction of the annihilator algebra and satisfies the definition of a coherent state in the sense of Barut-Girardello [3].

4.4 Minimized uncertainty↩︎

The canonical coherent state \(\Phi_0(x,t;\alpha)\) saturates the Heisenberg uncertainty relation for position and momentum. Formally, we have \[E(\Delta x)^2 E(\Delta p)^2 = \frac{1}{4}\] where \[\label{eq:canonvars} \begin{align} E(\Delta x)^2 &= \frac{\int_\mathbb{R}x^2 \Phi_0\overline{\Phi_0}dx}{ \int_\mathbb{R}\Phi_0 \overline{\Phi_0} dx} - \left(\frac{\int_\mathbb{R}x\Phi_0 \overline{\Phi_0} dx}{\int_\mathbb{R}\Phi_0 \overline{\Phi_0} dx}\right)^2 \\ E(\Delta p)^2 &= -\frac{\int_\mathbb{R}(\partial_{xx} \Phi_0)\overline{\Phi_0}dx}{ \int_\mathbb{R}\Phi_0 \overline{\Phi_0} dx} - \left(\frac{\int_\mathbb{R}i(\partial_x\Phi_0) \overline{\Phi_0} dx}{ \int_\mathbb{R}\Phi_0 \overline{\Phi_0} dx}\right)^2 \end{align}\tag{49}\]

Without loss of generality, \(M=M_{\lambda}\), whence by ?? and the definition 46 we see that \(\Phi_\lambda\to \Phi_0\) as \(\alpha\to +\infty\). Consequently for an ECS, the minimized uncertainty relation holds asymptotically, in the sense that \[\label{eq:minunlam} E_\lambda(\Delta x)^2 E_\lambda(\Delta p)^2 \to \frac{1}{4},\quad \text{as } \alpha \to + \infty,\tag{50}\] where \(E_\lambda(\Delta x), E_\lambda(\Delta p)\) denote the expectation values of the variances associated with the wave function \(\Phi_\lambda(x,t;\alpha)\). Formally, these are defined in the same way as 49 , but with \(\Phi_\lambda\) in place of \(\Phi_0\).

4.5 Example↩︎

As an example, we construct the coherent state corresponding to the index set \(K=\{2,3\}\). The corresponding Maya diagram, partition, and index are \[M= f_K({M_0}) = \{ \ldots, -2,-1, 2,3 \},\quad \lambda=(2,2),\quad \sigma_M=2,\] while the corresponding rational extension is \[T_M(x,\partial_x) =-\partial_x^2+ \left(x^2+4+\frac{32 x^2}{4 x^4+3}-\frac{384 x^2}{\left(4 x^4+3\right)^2}\right).\] The bound states are indexed by \[I_M = \mathbb{Z}\setminus M = \{ 0,1,4,5,6,\ldots \} ,\] and the bound state with eigenvalue \(2m+1,\; m\in I_M\) given by \[\psi_{M,m} = e^{-\tfrac12x^2} \frac{\hat{H}_{M,m}(x)}{4x^4+3} ,\quad m\in I_M,\] where the corresponding exceptional polynomial is \[\hat{H}_{M,m} = \frac{\operatorname{Wr}(2x^2-1,2x^3-3,H_m)}{4 (m-2)(m-3)},\quad m\ge 0,\; m\neq 2,3.\]

In this case, the Schur function is \[S_\lambda(t_1,t_2,t_3) = \frac{t_1^4}{12}+t_2^2-t_1 t_3.\] Using ?? , the generating function for the bound states is therefore \[\label{eq:psilamex} \Psi_\lambda(x,z) = \left(1- \frac{16x^3}{4x^4+3} z^{-1}+ \frac{12(2x^2+1)}{4x^4+3} z^{-2} \right) e^{-\tfrac12(x-z)^2+\tfrac14z^2}.\tag{51}\]

The set of critical degrees is \(D_\lambda= \{ 4,5,\ldots \}\). Note that there are no critical degrees below the threshold \(q_c=2+2=4\). Figure 3 illustrates the fact that \(q=4\) is a critical degree and that \(q=3\) fails to be a critical degree since \(0+3\in M\) but \(0\notin M\).

The extended coherent state \[\Phi_\lambda(x,t;\alpha) = e^{-5 it} \Psi_\lambda(x,\alpha e^{-2it})\] is an exact solution of the corresponding time-dependent Schrödinger equation 47 . The first 4 annihilators, as defined in 37 , are \(L_{M,q}=A_{M,K_q},\; q\in \{4,5,6,7\}\) with \[K_4 =\{0,1,6,7\},\; K_5=\{0,1,4,7,8\},\; K_6=\{0,1,4,5,8,9\},\; K_7=\{0,1,4,5,6,9,10\}.\] These commuting differential operators generate the annihilator algebra of this rational extension. In each case, one can verify by direct calculation that \(\Phi_\lambda(x,t;\alpha)\) is an eigenfunction of \(L_{M,q}\) with eigenvalue \(\alpha^q e^{-2qi t}\).

The form of 51 makes evident the asymptotic relation \(\Psi_\lambda\to \Psi_0\) as \(\alpha=|z|\to +\infty\). Figure 4 shows the value of the position-momentum uncertainty value \(E_\lambda(\Delta x)^2 E_\lambda(\Delta p)^2\) as a function of time \(t\) for values \(\alpha=4,8,16\). The graphs clearly indicate the corresponding asymptotic minimization of the uncertainty relation as \(\alpha\to +\infty\).

References↩︎