Supersymmetric Quantum Mechanics

Introduction

The techniques based on the factorization method, which try to identify the class of Hamiltonians whose spectral problem can be algebraically solved, have attracted for years people’s attention. In these lecture notes we will elaborate a brief survey of this subject; since we shall show that the factorization method, supersymmetric quantum mechanics (SUSY QM), and intertwining technique are equivalent procedures for generating new solvable potentials departing from an initial one, these names will be indistinctly used to characterize the technique we are interested in. For a collection of books and review articles concerning factorization method and related subjects the reader can seek -.

I would like to emphasize that these lecture notes on SUSY QM are a sequel of those delivered by myself in 2004 at the Latin-American School of Physics . For people wanting to have a global view of the way our group addresses the subject, and to know more details of our achievements, I recommend to read both, this article and , because they complement to each other.

From a historical viewpoint, it is nowadays accepted that the factorization method was introduced for the first time by Dirac in 1935 in order to derive algebraically the spectrum of the harmonic oscillator . In coordinates such that $\hbar=\omega=m=1$ the oscillator Hamiltonian admits the following factorizations:

From these equations it is straightforward to derive the intertwining relationships,

which lead to the commutators ruling the standard Heisenberg-Weyl algebra:

These expressions and the fact that $H$ is positive definite allow to generate the eigenfunctions $\psi_{n}(x)$ and corresponding eigenvalues $E_{n}=n+1/2,\ n=0,1,\dots$ of $H$.

The second important advance in the subject was done by Schrödinger in 1940, who realized that the factorization method can be as well applied to the radial part of the Coulomb problem . In this case the explicit factorizations in appropriate units read:

As in the previous case, these factorizations imply some intertwining relationships:

In a set of works starting from 1941, Infeld and collaborators put forward the method , culminating their research with a seminal paper in which it was classified a wide set of Schrödinger-type Hamiltonians (indeed four families) solvable through the factorization method . The main tool for performing this classification were the factorizations

which lead straightforwardly to the intertwining relationships

With these ingredients, Infeld and Hull were able to derive elegantly the eigenfunctions and eigenvalues for the hierarchy of Hamiltonians $H_{m}$ . Moreover, the very technique allowed them to determine the several forms of the potentials $V_{m}(x)$ admitting the factorization treatment. In this way, after the idea that the factorization methods was exhausted started to dominate; indeed, at some time it was thought that for a potential $V(x)$ to be solvable through factorization, it just has to appear in the Infeld-Hull classification.

Revealing himself against this idea, in 1984 Mielnik performed a simple generalization of the factorization method . He proposed to look for the most general first-order differential operators $b,b^{+}$ factorizing the harmonic oscillator Hamiltonian in the way:

Mielnik was able to determine the most general form of $\beta(x)$. Moreover, it turns out that the product $b^{+}b$ supplies a new Hamiltonian

Hence, the following intertwining relationships are valid:

which interrelate the eigenfunctions of the harmonic oscillator Hamiltonian $H$ with those of $\widetilde{H}$ and vice versa.

Mielnik’s work was a breakthrough in the development of the factorization method, since it opened new ways to explore exactly solvable potentials in quantum mechanics. In particular, his generalization was quickly applied to the radial part of the Coulomb problem . The important resultant expressions for that system become:

In 1985 Sukumar pushed further Mielnik’s factorization by applying it to general one-dimensional potentials $V(x)$ and arbitrary factorization energies $\epsilon$ in the way

which allow to generate the eigenfunctions and eigenvalues of $\widetilde{H}$ from those of $H$.

Along the years, several particular cases and applications of the generalized factorization of have been performed successfully -. Up to this point, however, the factorization operators, becoming the intertwiners in the equivalent formalism, were differential operators of first order. A natural generalization, posed in 1993 by Andrianov and collaborators , requires that Eq.(22) be valid but $A,A^{+}$ are substituted by differential operators of order greater than one. Through this higher-order case, it is possible to surpass the restriction arising from the first-order method that we can only modify the ground state energy of the initial Hamiltonian. In 1995 Bagrov and Samsonov proposed the same generalization, and they offered an alternative view to this issue . On the other hand, our group arrived to the subject in 1997 -, although there were several previous works related somehow to this generalization -. It is worth to mention explicitly some important contributions that members of our group have made to the factorization method and related subjects.

Generation of SUSY partner of the harmonic oscillator

Construction of coherent states for SUSY partners of the harmonic oscillator and for general one-dimensional Hamiltonians .

Determination of SUSY partners of the radial oscillator potential

Characterization of systems ruled by polynomial Heisenberg algebras and connection with SUSY partners of the oscillator

Analysis of SUSY partners of the radial Coulomb problem

Implementation of SUSY transformations involving complex factorization ‘energies’ -

Study of the confluent algorithm for second-order SUSY transformations and applications

Generation of SUSY partners of the Pöschl-Teller potentials -

Application of SUSY techniques to the Lamé and associated Lamé potentials -

It is important to mention that some other groups have elaborated the same subject from different viewpoints, e.g., $N$-fold supersymmetry by Tanaka and collaborators -, hidden nonlinear supersymmetries by Plyushchay et al -, etc.

Standard supersymmetric quantum mechanics

In this section we are going to show that the factorization method, intertwining technique and supersymmetric quantum mechanics are equivalent procedures for generating new solvable potentials departing from an initial one. The most straightforward way to see this equivalence is starting from the technique which involves just first-order differential intertwining operators.

Suppose we have two Schrödinger Hamiltonians

which are intertwined by first-order differential operators $A_{1},A_{1}^{+}$ in the way

By employing that, at the operator level, it is valid

where $f(x)$ is a multiplicative operator when acting on a generic wavefunction $\psi(x)$, the calculation of each member of the first intertwining relationship in Eq.(24) leads to

By plugging the $V_{1}$ of Eq.(29) in Eq.(30) and integrating the result we get:

This nonlinear first-order differential equation for $\alpha_{1}$ is the well-known Riccati equation. In terms of a seed Schrödinger solution $u^{(0)}$ such that $\alpha_{1}={u^{(0)}}^{\prime}/u^{(0)}$ it turns out that:

which is the initial stationary Schrödinger equation. These equations allow to establish the link between the intertwining technique and the factorization method, since it is valid:

Suppose now that $H_{0}$ is a given solvable Hamiltonian, with known eigenfunctions $\psi_{n}^{(0)}$ and eigenvalues $E_{n}$, namely,

In addition, let us take a nodeless mathematical eigenfunction $u^{(0)}$ associated to $\epsilon\leq E_{0}$ in order to implement the intertwining procedure. The equations (24) and the resulting factorizations (33) indicate that, if $A_{1}^{+}\psi_{n}^{(0)}\neq 0$, then \big{\{}\psi_{n}^{(1)}=A_{1}^{+}\psi_{n}^{(0)}/\sqrt{E_{n}-\epsilon}\big{\}} is an orthonormal set of eigenfunctions of $H_{1}$ with eigenvalues $\{E_{n}\}$. This is a basis if there is not a square-integrable function $\psi_{\epsilon}^{(1)}$ which is orthogonal to the full set. Thus, let us look for $\psi_{\epsilon}^{(1)}$ such that

By solving this first-order differential equation we get:

Since $H_{1}\psi_{\epsilon}^{(1)}=\epsilon\psi_{\epsilon}^{(1)}$ (compare Eq.(33)), then the spectrum of $H_{1}$ depends of the normalizability of $\psi_{\epsilon}^{(1)}$. We can identify three different situations.

(i) Let us choose $\epsilon=E_{0}$ and $u^{(0)}=\psi_{0}^{(0)}$, which is nodeless in the domain of $V_{0}$, $\alpha_{1}={\psi_{0}^{(0)}}^{\prime}/\psi_{0}^{(0)}$. Thus, $V_{1}=V_{0}-\alpha_{1}^{\prime}$ is completely determined, and the corresponding Hamiltonian $H_{1}$ has eigenfunctions and eigenvalues given by

Notice that $E_{0}\not\in$ Sp$(H_{1})$ since $\psi_{\epsilon}^{(1)}\propto 1/u^{(0)}$ is not square-integrable.

(ii) Let now a nodeless seed solution $u^{(0)}$ for $\epsilon<E_{0}$ be chosen so that $\alpha_{1}={u^{(0)}}^{\prime}/u^{(0)}$ is non-singular. Note that $u^{(0)}$ diverges at both ends of the $x$-domain $\Rightarrow\psi_{\epsilon}^{(1)}\propto 1/u^{(0)}$ tends to zero at those ends. Hence, the full set of eigenfunctions and eigenvalues is

(iii) Let us employ a solution $u^{(0)}$ with a node at one of the ends of the $x$-domain for $\epsilon<E_{0}$. Thus, the transformation induced by $\alpha_{1}={u^{(0)}}^{\prime}/u^{(0)}$ is non-singular, $\psi_{\epsilon}^{(1)}\propto 1/u^{(0)}$ diverges at the end where $u^{(0)}$ tends to zero, and then the eigenfunctions and eigenvalues of $H_{1}$ become

This implies that $H_{1}$ and $H_{0}$ are isospectral Hamiltonians.

2 Supersymmetric quantum mechanics

Let us build up now the so-called supersymmetric quantum mechanics, by realizing the Witten supersymmetry algebra with two generators

Notice that there is a linear relationship between $H_{\rm ss}$ and $H^{p}={\rm Diag}\{H_{1},H_{0}\}$:

There are two-fold degenerated levels of $H_{\rm ss}$ [$E_{n}-E_{0}$ with $n=1,2,\dots$ for the case (i) and $E_{n}-\epsilon$ with $n=0,1,2,\dots$ for cases (ii) and (iii)] which have associated the two orthonormal eigenvectors

The ground state energy of $H_{\rm ss}$ in cases (i) and (ii) is non-degenerated (and equal to ), with the corresponding eigenstate being given respectively by

In this case it is said that the supersymmetry is unbroken. On the other hand, in case (iii) the ground state energy ($E_{0}-\epsilon$) is doubly degenerated, and then the supersymmetry is spontaneously broken.

Example: trigonometric Pöschl-Teller potentials

Let us apply the previous technique to the trigonometric Pöschl-Teller potentials :

First of all, we require the general solution of the Schrödinger equation (32) with the $V_{0}(x)$ given in (46), which is given by

where $\mu=\lambda+\nu$. The physical eigenfunctions $\psi_{n}^{(0)}(x)$ satisfy $\psi_{n}^{(0)}(0)=\psi_{n}^{(0)}(\pi/2)=0$. If we ask that $\psi_{n}^{(0)}(0)=0$, it turns out that $B=0$. Moreover, in order to avoid that the divergent behavior at $x=\pi/2$ of the hypergeometric function of the remaining term dominates over the null one induced by $\cos^{\nu}(x)$ for $E>0$ it has to happen that

The corresponding normalized eigenfunctions are given by

It will be important later to know the number of zeros which $u^{(0)}$ has, according to the value taken by $\epsilon$. This information can be obtained by comparing the asymptotic behavior of $u^{(0)}$ at $x\rightarrow 0$ and $x\rightarrow\pi/2$; by picking up $B=1$ and $A=-b/a+q$ in Eq.(47) with

If $\epsilon<E_{0}$, $u^{(0)}$ has either or $1$ nodes for $q>0$ or $q<0$ respectively

In general, if $E_{i-1}<\epsilon<E_{i}$, $u^{(0)}$ will have either $i$ or $i+1$ nodes for $q>0$ or $q<0$ respectively, $i=2,3,\dots$

Concerning the spectral modifications which can be induced through the standard first-order supersymmetric quantum mechanics, it is obtained the following.

(a) Deleting the ground state of $H_{0}$. Let us choose $\epsilon=E_{0}$ and

The first-order SUSY partner of $V_{0}$ becomes (see Eq.(29))

Since $\psi_{\epsilon}^{(1)}\propto 1/\psi_{0}^{(0)}$ diverges at $x=0,\pi/2$ $\Rightarrow E_{0}\not\in{\rm Sp}(H_{1})=\{E_{n},n=1,2,\dots\}$. Note that $V_{1}$ can be obtained from $V_{0}$ by changing $\lambda\rightarrow\lambda+1,\ \nu\rightarrow\nu+1$; this property is known nowadays as shape invariance , although it was known since Infeld-Hull work .

(b) Creating a new ground state. Let us take now $\epsilon<E_{0}$ and a nodeless $u^{(0)}$ (Eq.(47) with $B=1$, $A=-b/a+q$, $q>0$). As $u^{(0)}\rightarrow\infty$ when $x\rightarrow 0,\pi/2$ $\Rightarrow$ $\psi_{\epsilon}^{(1)}(0)=\psi_{\epsilon}^{(1)}(\pi/2)=0$, i.e., $\psi_{\epsilon}^{(1)}$ is an eigenfunction of $H_{1}$ with eigenvalue $\epsilon$, and therefore Sp($H_{1}$)=$\{\epsilon,E_{n},n=0,1,\dots\}$. In order to deal with the singularities at $x=0,\pi/2$ induced by $u^{(0)}$ on $V_{1}$, it is expressed $u^{(0)}=\sin^{1-\lambda}(x)\cos^{1-\nu}(x){\rm v},$ where ${\rm v}$ is a nodeless bounded function in $[0,\pi/2]$. Hence

(c) Isospectral potentials. They arise from the previous case (for $\epsilon<E_{0}$) in the limit when $u^{(0)}$ acquires a node at one of the ends of the domain so that $\psi_{\epsilon}^{(1)}$ leaves to be an eigenstate of $H_{1}$. We can take, for example, the solution given in Eq.(47) with $A=1$, $B=0$ $\Rightarrow$ $u^{(0)}(0)=0$. We isolate the singularities induced on $V_{1}$ by expressing $u^{(0)}=\sin^{\lambda}(x)\cos^{1-\nu}(x){\rm v}$, where ${\rm v}$ is a nodeless bounded function in $[0,\pi/2]$. Hence

Notice that now $H_{1}$ and $H_{0}$ are isospectral.

An illustration of the several first-order SUSY partner potentials $V_{1}(x)$ of $V_{0}(x)$ (see Eqs.(52-54)) is given in Figure 1.

Second-order Supersymmetric Quantum Mechanics

The second-order supersymmetric quantum mechanics can be implemented through the iteration of two first-order transformations. However, by means of this approach it is not revealed in full the rich set of spectral manipulations which is offered by the second-order technique. This is avoided if we address the method directly, by assuming that the intertwining operator $B_{2}^{+}$ is of second-order in the derivatives, namely

A calculation similar as for the first-order case leads to the following set of equations :

By decoupling this system, assuming that $V_{0}$ is a given solvable potential, we arrive at:

Notice that $V_{2}$ and $\gamma$ become completely determined once it is found $\eta$; this task is performed by assuming the ansätz

where the functions $\beta$ and $\xi$ are to be determined. Therefore

which is once again a Riccati equation for $\beta$. By assuming that $\beta={u^{(0)}}^{\prime}/u^{(0)}$ it turns out that $u^{(0)}$ satisfies the stationary Schrödinger equation

Since $\xi=\pm\sqrt{c}$, we get in general two different values for $\epsilon$, $\epsilon_{1}\equiv(d+\sqrt{c})/2$, $\epsilon_{2}\equiv(d-\sqrt{c})/2$. We thus arrive at a natural classification of the second-order SUSY transformations, depending on the sign taken by the parameter $c$ .

By making the difference of both equations it is obtained:

Notice that now the new potential $V_{2}$ has no extra singularities with respect to $V_{0}$ if $W(u^{(0)}_{1},u^{(0)}_{2})$ is nodeless (compare Eq.(58)).

The spectrum of $H_{2}$ depends on the normalizability of the two eigenfunctions $\psi^{(2)}_{\epsilon_{1,2}}$ of $H_{2}$ associated to $\epsilon_{1,2}$ which belong as well to the kernel of $B_{2}$,

Their explicit expressions in terms of $u^{(0)}_{1}$ and $u^{(0)}_{2}$ become

We have observed several possible spectral modifications induced on the Pöschl-Teller potentials.

(a) Deleting 2 neighbor levels. By taking as seeds two physical eigenfunctions of $H_{0}$ associated to a pair of neighbor eigenvalues, namely $\epsilon_{1}=E_{i},$ $\epsilon_{2}=E_{i-1}$, $u_{1}^{(0)}=\psi_{i}^{(0)},\ u_{2}^{(0)}=\psi_{i-1}^{(0)}$ it is straightforward to show that

is a nodeless bounded funcion in $[0,\pi/2]$. Hence,

Notice that the two mathematical eigenfunctions $\psi_{\epsilon_{1}}^{(2)},\ \psi_{\epsilon_{2}}^{(2)}$ of $H_{2}$ associated to $\epsilon_{1}=E_{i},\ \epsilon_{2}=E_{i-1}$ do not vanish at $0,\pi/2$,

Thus, $E_{i-1},E_{i}\not\in$ Sp$(H_{2})=\{E_{0},\dots E_{i-2},E_{i+1},\dots\}$, i.e., we have deleted the levels $E_{i-1},E_{i}$ of $H_{0}$ in order to generate $V_{2}(x)$.

(b) Creating two new levels. Let us choose now $E_{i-1}<\epsilon_{2}<\epsilon_{1}<E_{i},i=0,1,2,\dots$, $E_{-1}\equiv-\infty$, with the corresponding seeds $u^{(0)}$ as given in Eq.(47) with $B_{1,2}=1,\ A_{1,2}=-b_{1,2}/a_{1,2}+q_{1,2}$, $q_{2}<0$, $q_{1}>0$. This choice ensures that $u_{2}^{(0)}$ and $u_{1}^{(0)}$ have $i+1$ and $i$ alternating nodes, which produces a nodeless Wronskian . The divergence of the $u^{(0)}$solutions for $x\rightarrow 0,\pi/2$ is isolated by expressing

${\rm v}_{1,2}$ being bounded for $x\in[0,\pi/2]$, ${\rm v}_{1,2}(0)\neq 0\neq{\rm v}_{1,2}(\pi/2)$. Since the second term of the Taylor series expansion of ${\rm v}_{1,2}$ is proportional to $\sin^{2}(x)$ $\Rightarrow$ ${\rm v}_{1,2}^{\prime}$ tends to zero as $\sin(x)$ for $x\rightarrow 0$ and as $\cos(x)$ for $x\rightarrow\pi/2$. Thus,

where ${\cal W}=W({\rm v}_{1},{\rm v}_{2})/[\sin(x)\cos(x)]$ is nodeless bounded in $[0,\pi/2]$. Hence

Since $\lim_{x\rightarrow 0,\frac{\pi}{2}}\psi_{\epsilon_{1,2}}^{(2)}=0$ $\Rightarrow$ Sp$(H_{2})=\{E_{0},\dots,E_{i-1},\epsilon_{2},\epsilon_{1},E_{i},\dots\}$, i.e., we have created two new levels $\epsilon_{1},\epsilon_{2}$ between the originally neighbor ones $E_{i-1},E_{i}$ for generating $V_{2}(x)$.

It is worth to mention some other spectral modifications which can be straightforwardly implemented :

An illustration of the potentials $V_{2}(x)$ induced by some of the previously discussed spectral modifications is given in Figure 2.

2 (ii) Confluent case with c=0𝑐0c=0

and adjusting as well $w_{0}$, where $[x_{l},x_{r}]$ is the $x$-domain of the involved problem, $x_{0}\in[x_{l},x_{r}]$. There is one eigenfunction, $\psi^{(2)}_{\epsilon}\propto u^{(0)}/w$, of $H_{2}$ associated to $\epsilon$ belonging as well to the kernel of $B_{2}$, which implies that ${\rm Sp}(H_{2})$ depends of the normalizability of $\psi^{(2)}_{\epsilon}$.

For the Pöschl-Teller potentials, several possibilities for modifying the initial spectrum have been observed.

where ${\rm v}$ is bounded for $x\in[0,\pi/2]$, ${\rm v}(0)\neq 0\neq{\rm v}(\pi/2)$. By calculating the involved integral with $x_{0}=0$ it is obtained:

Notice that $w$ is nodeless in $[0,\pi/2]$ for $w_{0}>0$ and it has one node for $w_{0}<0$. By choosing now the $w$ of Eq.(85) with $w_{0}>0$, we have to isolate anyways a divergence of kind $\cos^{3-2\nu}(x)$ arising for $x\rightarrow\pi/2$ through the factorization

${\cal W}$ being a nodeless bounded function for $x\in[0,\pi/2]$. From Eq.(58) we finally get

Since $\lim_{x\rightarrow 0,\frac{\pi}{2}}\psi_{\epsilon}^{(2)}=0$, it turns out that Sp$(H_{2})=\{\epsilon,E_{n},n=0,1,\dots\}$. It is important to remark that the procedure can be implemented for solutions associated to $\epsilon>E_{0}$, i.e., by means of the confluent second-order transformation one is able to modify the levels above the ground state energy of $H_{0}$

Some additional possibilities of spectral manipulation are now just pointed out .

The potentials $V_{2}(x)$ induced by some of the mentioned spectral modifications are illustrated in Figure 3.

3 (iii) Complex case with c<0𝑐0c<0

In both cases we get that ${\rm Sp}(H_{2})={\rm Sp}(H_{0})$.

The potentials $V_{0}$ and $V_{2}$ turn out to be isospectral. An illustration of $V_{0}(x)$ and the $V_{2}(x)$ of Eq.(93) is given in Figure 4.

A general conclusion, which can be inferred of the previously discussed first and second-order SUSY QM but it is valid for an arbitrary order, is that the spectra of the initial and new Hamiltonians differ little. Thus, one would expect that some properties which are somehow related to the corresponding spectral problems would be essentially the same. We will see next that this is true, in particular, for the algebraic structure ruling the two involved Hamiltonians.

Algebra of the SUSY partner Hamiltonians

Given an initial Hamiltonian $H_{0}$, which is characterized by a certain algebraic structure, it is natural to pose the following questions:

What is the algebra ruling the potentials generated through SUSY QM?

What other properties of $H_{0}$ are inherited by its SUSY partner Hamiltonians?

These questions will be analyzed here through a simple system which algebraic structure generalizes the standard Heisenberg-Weyl (oscillator) algebra. Thus, let us consider a one-dimensional Schrödinger Hamiltonian

whose eigenvectors $|\psi_{n}^{(0)}\rangle$ and eigenvalues $E_{n}$ satisfy (for calculational convenience we switch now to the Dirac bra and ket notation)

It is supposed that there is an analytic dependence between the eigenvalues $E_{n}$ and the index labeling them, namely,

Let us define now the intrinsic algebra of $H_{0}$, by introducing a pair of annihilation and creation operators $a_{0}^{\pm}$ whose action onto the eigenvector of $H_{0}$ is given by

With these definitions it is straightforward to show that:

It is introduced the number operator $N_{0}$ through its action onto the eigenvectors of $H_{0}$:

Now, the intrinsic algebra of the systems is characterized by

It is important to realize that for general $E(n)$ the intrinsic algebra becomes nonlinear (since $f(N_{0})$ is not necessarily a linear function of $N_{0}$). However, this intrinsic nonlinear algebra can be linearized by deforming the corresponding annihilation and creation operators in the way

In particular, for the harmonic oscillator Hamiltonian the intrinsic algebraic structure reduces to the standard Heisenberg-Weyl algebra:

On the other hand, for the trigonometric Pöschl-Teller potentials we recover the su(1,1) algebra:

Let us characterize now the algebraic structure of the SUSY partner Hamiltonians of $H_{0}$ . In order to illustrate the procedure, we restrict ourselves just to a second-order SUSY transformation which creates two new levels at the positions $\epsilon_{1},\ \epsilon_{2}$ for $H_{2}$, namely,

Notice that the eigenstates of $H_{2}$, $|\psi_{n}^{(2)}\rangle$, associated to $E_{n},n=0,1,\dots$ are obtained from $|\psi_{n}^{(0)}\rangle$ and vice versa through the action of the intertwining operators $B_{2}^{+}$, $B_{2}$ in the way:

The eigenstates of $H_{2}$ constitute a complete orthonormal set,

It is important to notice that the two new levels $\epsilon_{1},\epsilon_{2}$ can be placed at positions essentially arbitrary (either both below the initial ground state $E_{0}$ or in between two neighbor physical levels $E_{i-1},E_{i}$). This fact somehow breaks the symmetry defined by $E(n)$. Thus, it should be clear that we need to isolate, in a sense, the two new levels $\epsilon_{1},\epsilon_{2}$. With this aim, it is defined now the number operator $N_{2}$ through the following action onto the eigenstates of $H_{2}$:

Departing now from the operators $a_{0}^{\pm}$ of the intrinsic algebra of $H_{0}$, let us construct those of the natural algebra of $H_{2}$ , which represent a generalization of the operators introduced previously for the SUSY partners of the harmonic oscillator ,

An schematic representation of these operators is given in Figure 5. Their action onto the eigenvectors of $H_{2}$ is given by

Let us build up now the annihilation and creation operators $a_{2}^{\pm}$ which generate the intrinsic algebra of $H_{2}$,

Their action onto the eigenstates of $H_{2}$ reads

Notice that this algebra of $H_{2}$ is the same as the intrinsic algebra of $H_{0}$ when we work on the restriction to the subspace spanned by the eigenvectors $|\psi_{n}^{(2)}\rangle$ associated to the isospectral part of the spectrum.

Let us construct, finally, the annihilation and creation operators $a_{2_{L}}^{\pm}$ of the linear algebra of $H_{2}$,

Their action onto the eigenstates of $H_{2}$ is given by

Once again, this algebra of $H_{2}$ coincides with the corresponding one of $H_{0}$ on the subspace spanned by $\{|\psi_{n}^{(2)}\rangle,n=0,1,\dots\}$.

Coherent states

Once identified the algebraic structures associated to $H_{0}$ and $H_{2}$, it would be important to generate the corresponding coherent states. It is well know that there are several definitions of coherent states ; here we will derive them as eigenstates with complex eigenvalues of the annihilation operators of the systems. Since there are available several annihilation operators for $H_{0}$ and $H_{2}$, we will call them with the same name as the corresponding associated algebra.

Since for $H_{0}$ we identified two different algebraic structures, intrinsic and linear one, we will look for the two corresponding families of coherent states.

The intrinsic coherent states $|z,\alpha\rangle_{0}$ of $H_{0}$ satisfy:

Expanding $|z,\alpha\rangle_{0}$ in the basis of eigenstates of $H_{0}$,

and substituting in Eq.(127), it is obtained a recurrence relationship for $c_{n}$,

An important property that should be satisfied is the so-called completeness relationship, namely,

where the positive defined measure $d\mu(z)$ can be expressed as

Working in polar coordinates and changing variables $y=|z|^{2}$, it turns out that $\rho(y)$ must satisfy

This is a moment problem depending on $\rho_{n}$, i.e., on the spectral function $E(n)$ . If one is able to solve Eq.(134) for $\rho(y)$, the completeness relation becomes valid.

Once Eq.(132) is satisfied, an arbitrary quantum state can be expressed in terms of coherent states. In particular, an intrinsic coherent state $|z^{\prime},\alpha\rangle_{0}$ admits the decomposition,

where the reproducing kernel ${}_{0}\langle z,\alpha|z^{\prime},\alpha\rangle_{0}$ is given by

As $|z=0,\alpha\rangle_{0}=|\psi_{0}^{(0)}\rangle$, it turns out that the eigenvalue $z=0$ is non-degenerate. If the system is initially in an intrinsic coherent state $|z,\alpha\rangle_{0}$, then it will be at any time in an intrinsic coherent state,

where $U_{0}(t)=\exp(-iH_{0}t)$ is the evolution operator of the system.

Now, let us look for the linear coherent states such that

A procedure similar to the previous one leads straightforwardly to

which, up to the phases, has the form of the standard coherent states. Now the completeness relationship is automatically satisfied,

Therefore, an arbitrary linear coherent state $|z^{\prime},\alpha\rangle_{0_{L}}$ is expressed in terms of linear coherent states,

Notice that, once again, the only eigenstate of $H_{0}$ which is also a linear coherent state is the ground state, $|z=0,\alpha\rangle_{0_{L}}=|\psi_{0}^{(0)}\rangle$. Moreover, since $[a_{0_{L}}^{-},a_{0_{L}}^{+}]=1$ then the linear coherent states also result from the action of the displacement operator $D_{L}(z)$ onto the ground state $|\psi_{0}^{(0)}\rangle$:

Finally, an initial linear coherent state evolves in time as a linear coherent state, namely,

Let us remember that we have analyzed three different algebraic structures for $H_{2}$, namely, natural, intrinsic and linear ones; thus three families of coherent states will be next constructed.

The natural coherent states $|z,\alpha\rangle_{2_{N}}$ of $H_{2}$ are defined by

A procedure similar to the previously used leads to

Notice that now the completeness relationship has to be modified to include the projector onto the subspace spanned by the eigenvectors of $H_{2}$ associated to the two isolated levels $\epsilon_{1},\epsilon_{2}$, namely,

It is clear that the moment problem characterized by Eq.(150) is more complicated than the one associated to Eq.(134) (compare the moments given by Eqs.(131) and (147)). Since $B_{2}|\psi_{\epsilon_{i}}^{(2)}\rangle=a^{-}_{2_{N}}|\psi_{\epsilon_{i}}^{(2)}\rangle=0,\ i=1,2$ and $a^{-}_{2_{N}}|\psi_{0}^{(2)}\rangle=0$, then the eigenvalue $z=0$ of $a_{2_{N}}^{-}$ is 3-fold degenerate. Finally, as for the coherent states of $H_{0}$, it turns out that the natural coherent states $|z,\alpha\rangle_{2_{N}}$ evolve coherently with $t$,

where $U_{2}(t)=\exp(-iH_{2}t)$ is the evolution operator of the system characterized by $H_{2}$.

The modified completeness relationship reads now

where $d\mu(z)$ is the same as for $H_{0}$. Since $a_{2}^{-}|\psi_{\epsilon_{i}}^{(2)}\rangle=0,\ i=1,2$ and $|z=0,\alpha\rangle_{2}=|\psi_{0}^{(2)}\rangle$, the eigenvalue $z=0$ of $a_{2}^{-}$ turns out to be $3$-fold degenerate. As for the natural coherent states of $H_{2}$, the intrinsic coherent states $|z,\alpha\rangle_{2}$ evolve in time in a coherent way.

The completeness relationship is expressed as

Once again, the eigenvalue $z=0$ of $a_{2_{L}}^{-}$ is $3$-fold degenerate. Moreover, the linear coherent states $|z,\alpha\rangle_{2_{L}}$ appear from the action of the displacement operator $D_{2_{L}}(z)$ onto the extremal state $|\psi_{0}^{(2)}\rangle$:

As in the previous cases, the linear coherent states of $H_{2}$ evolve in time coherently.

Let us end up these two sections, about the algebraic structure of the SUSY partner Hamiltonians and the corresponding coherent states, with the following conclusions.

The intrinsic and linear algebraic structures of $H_{0}$ are inherited by $H_{2}$ on the subspace associated to the levels of the isospectral part of the spectrum

This reflects as well in the corresponding coherent states of $H_{2}$, linear and intrinsic ones, whose expressions on the same subspace are equal to the corresponding ones of $H_{0}$

We generalized successfully the construction of the natural algebra of $H_{2}$ when the initial potential $V_{0}(x)$ is more general than the harmonic oscillator

SUSY partners of periodic potentials

The exactly solvable models are important since the relevant physical information is encoded in a few expressions. Moreover, they are ideal either to test the accuracy (and convergence) of numerical techniques, or to implement some analytic approximate methods, as perturbation theory. Unfortunately, the number of exactly solvable periodic potentials is small, so it is important to enlarge this class. As we saw in the previous sections, a simple method to do the job is supersymmetric quantum mechanics. Before applying the technique to periodic potentials, however, let us review first the way in which it is classified the energy axis in allowed and forbidden energy bands.

For one-dimensional periodic potentials such that $V_{0}(x+T)=V_{0}(x)$, the Schrödinger equation

can be conveniently expressed in matrix form:

where the $2\times 2$ transfer matrix $b(x)$ is symplectic, which implies that its determinant is equal to $1$. The transfer matrix can be expressed in terms of two real solutions ${\rm v}_{1,2}(x)$ such that ${\rm v}_{1}(0)=1$, ${\rm v}_{1}^{\prime}(0)=0$, ${\rm v}_{2}(0)=0$, ${\rm v}_{2}^{\prime}(0)=1$, as follows

The general behavior of $\psi$, and the kind of spectrum of the Hamiltonian $H_{0}$, depends on the eigenvalues $\beta_{\pm}$ of the Floquet matrix $b(T)$, which in turn are determined by the so-called discriminant $D=D(E)={\rm Tr}[b(T)]$:

By picking up $\Psi(0)$ as one of the eigenvectors of the Floquet matrix $b(T)$ with eigenvalue $\beta$, it turns out that

Moreover, for these vectors in general we have that

If $|D(E)|<2$ for a given $E$, it turns out that

The equation $|D(E)|=2$ defines the band edges, denoted

For these energy values it turns out that $\beta_{+}=\beta_{-}=\pm 1$, the two Bloch functions tend just to one which becomes periodic or antiperiodic. Thus, these energy values belong as well to the spectrum of $H_{0}$. Moreover, the Bloch eigenfunctions $\psi_{j},$ $\psi_{{j^{\prime}}}$ associated to $E_{j},E_{j^{\prime}}$ are real, they have the same number $j$ of nodes, and both are either periodic or antiperiodic

If $|D(E)|>2$ for a given $E$, it turns out that

Then, the Bloch solutions $\psi^{\pm}$ are unbounded either for $x\rightarrow\infty$ or for $x\rightarrow-\infty$ so that they cannot be equipped with a physical interpretation. Therefore, $E$ belongs to a forbidden energy gap

As an illustration, we have plotted in Figure 6 the discriminant $D(E)$ as a function of $E$ for $V_{0}(x)=5\sin^{2}(x)$.

A very special class of periodic systems are given by the Lamé potentials

where ${\rm sn}(x|m)$ is a Jacobi elliptic function. These are doubly periodic special functions defined by

with $x=\int_{0}^{\varphi}\frac{d\theta}{\sqrt{1-m\sin^{2}\theta}}$. The real and imaginary periods are respectively $(4K,2iK^{\prime})$, $(4K,4iK^{\prime})$, $(2K,4iK^{\prime})$, where

The special nature of the Lamé potentials is due to the following properties:

They have $2n+1$ band edges, which define $n+1$ allowed energy bands ($n$ of them finite and $1$ infinite) and $n+1$ energy gaps ($n$ finite and $1$ infinite)

It is possible to determine the general solution of the Schrödinger equation for any real value of the energy parameter

In particular, there exist explicit analytic expressions for the band edge eigenfunctions

Let us illustrate this behavior through the simplest example.

2 The Lamé potential with n=1𝑛1n=1

In this case the Bloch functions for a real arbitrary energy value $\epsilon$ are given by

where $\zeta^{\prime}(z)=-\wp(z),\ [\ln\sigma(z)]^{\prime}=\zeta(z)$, $\omega=K,\ \omega^{\prime}=iK^{\prime}$ and

Note that $\wp$ is the Weierstrass $p$-function, which is related to the Jacobi elliptic function sn through

There are now three band edges, and the explicit expression for the corresponding eigenvalues and eigenfunctions are given by

The energy axis is classified as follows: the energy interval $(-\infty,m/2)$ represents the infinite energy gap (forbidden energies), then it comes the finite energy band $[m/2,1/2]$ (allowed energies), then it arises the finite energy gap $(1/2,1/2+m/2)$, and finally it emerges the infinite energy band $[1/2+m/2,\infty)$.

Let us choose in the first place as seed one real nodeless Bloch function $u^{(0)}$ with $\epsilon\leq E_{0}$. It is straightforward to check that ${u^{(0)}}^{\prime}/u^{(0)}$ is periodic, which implies that $V_{1}(x)$ is also periodic (see Eq.(29)). The SUSY transformation maps bounded (unbounded) eigenfunctions of $H_{0}$ into bounded (unbounded) ones of $H_{1}$. Since for $\epsilon<E_{0}$ the chosen Bloch function $u^{(0)}$ diverges either for $x\rightarrow\infty$ or $x\rightarrow-\infty$ ($\epsilon\not\in{\rm Sp}(H_{0})$), then $1/u^{(0)}$ will diverge at the opposite limits and therefore $\epsilon\not\in{\rm Sp}(H_{1})$. On the other hand, for $\epsilon=E_{0}\in{\rm Sp}(H_{0})$, the corresponding Bloch solution $u^{(0)}$ is nodeless periodic and therefore $1/u^{(0)}$ is also a nodeless periodic eigenfunction of $H_{1}$ with eigenvalue $E_{0}$. In conclusion, for first-order SUSY transformations involving a nodeless Bloch function with $\epsilon\leq E_{0}$ it turns out that Sp($H_{1}$) = Sp($H_{0}$), i.e., the SUSY mapping is isospectral.

In particular, for the Lamé potential with $n=1$ the first-order SUSY transformation which employs the lowest band edge eigenfunction $\psi^{(0)}_{0}(x)={\rm dn}\,x$ leads to

and it is said that $V_{1}(x)$ is self-isospectral to $V_{0}(x)$. This property was discovered, for the first time, by Braden and McFarlane , although the name is due to Dunne and Feinberg . Moreover, by using as seed the Bloch eigenfunction $u^{(0)^{\pm}}(x)$ with $\epsilon<E_{0}$ is is straightforward to show that

where $a$ is related to $\epsilon$ through Eq.(179).

2.2 First-order SUSY using general solutions

Let us choose now a real nodeless linear combination $u^{(0)}$ of the two Bloch functions for $\epsilon<E_{0}$. Although now ${u^{(0)}}^{\prime}/u^{(0)}$ is not strictly periodic, it is asymptotically periodic when $|x|\rightarrow\infty$. This property is transferred as well to the new potential $V_{1}(x)$ which is asymptotically periodic for $|x|\rightarrow\infty$. As in the previous case, the bounded (unbounded) eigenfunctions of $H_{0}$ are mapped into bounded (unbounded) eigenfunctions of $H_{1}$. Since $\lim_{|x|\rightarrow\infty}|u^{(0)}|=\infty$, it turns out that $1/u^{(0)}$ is square-integrable in $(-\infty,\infty)$. We conclude that, when appropriate linear combinations of two Bloch functions of $H_{0}$ associated to $\epsilon<E_{0}$ are used the implement the first-order SUSY transformation, we create a bound state for $H_{1}$, i.e., Sp($H_{1}$)$=\{\epsilon\}\,\cup$ Sp($H_{0}$).

We apply now this kind of transformation to the Lamé potential with $n=1$, using as seed a real nodeless linear combination of the two Bloch solutions $u^{(0)^{\pm}}(x)$ given in Eq.(177) with $\epsilon<E_{0}$ . In this way it is obtained an asympotically periodic potential, as illustrated in Figure 7, where the corresponding bound state of $H_{1}$ is as well shown.

2.3 Second-order SUSY using Bloch functions

Let us apply now a second-order SUSY transformation using two real Bloch solutions $u^{(0)}_{1,2}$ associated to $\epsilon_{1,2}\in(-\infty,E_{0}]$ or $\epsilon_{1,2}\in[E_{j},E_{j^{\prime}}],j=1,2,\dots$ so that $W(u^{(0)}_{1},u^{(0)}_{2})$ is nodeless -. For this choice it turns out that $W^{\prime}(u^{(0)}_{1},u^{(0)}_{2})/W(u^{(0)}_{1},u^{(0)}_{2})$ as well as $V_{2}(x)$ are periodic (see Eqs.(58,68)). The SUSY transformation maps bounded (unbounded) eigenfunctions of $H_{0}$ into bounded (unbounded) ones of $H_{2}$. Thus, $H_{0}$ and $H_{2}$ have the same band structure, namely, Sp($H_{2}$)=Sp($H_{0}$).

For the Lamé potential with $n=1$, if we employ the two band edge eigenfunctions $\psi^{(0)}_{1}(x),\psi^{(0)}_{1^{\prime}}(x)$ bounding the finite gap, we get once again

Moreover, by using as seeds two Bloch functions $u^{(0)^{+}}_{1,2}(x)$ such that $\epsilon_{1,2}\in(E_{1},E_{1^{\prime}})$ it is straightforward to show that

2.4 Second-order SUSY using general solutions

Let us choose now as seeds two real linear combinations of Bloch function for $\epsilon_{1,2}\in(-\infty,E_{0}]$ or $\epsilon_{1,2}\in[E_{j},E_{j^{\prime}}],j=1,2,\dots$ so that $W(u^{(0)}_{1},u^{(0)}_{2})$ is nodeless . Once again, $W^{\prime}(u^{(0)}_{1},u^{(0)}_{2})/W(u^{(0)}_{1},u^{(0)}_{2})$ and consequently $V_{2}(x)$ are asymptotically periodic for $|x|\rightarrow\infty$, the second-order SUSY transformation maps bounded (unbounded) eigenfunctions of $H_{0}$ into bounded (unbounded) ones of $H_{2}$. Since now the two eigenfunctions of $H_{2}$, $u^{(0)}_{2}/W(u^{(0)}_{1},u^{(0)}_{2})$, $u^{(0)}_{1}/W(u^{(0)}_{1},u^{(0)}_{2})$ associated to $\epsilon_{1},\ \epsilon_{2}$, can be made square-integrable for $x\in(-\infty,\infty)$, in turns out that Sp($H_{2}$)$=\{\epsilon_{1},\epsilon_{2}\}\,\cup$ Sp($H_{0}$).

Let us illustrate the procedure for the Lamé potential with $n=1$, using as seeds two real linear combinations $u^{(0)}_{1,2}$ of $u^{(0)^{+}}_{1,2}$ and $u^{(0)^{-}}_{1,2}$ for $\epsilon_{1,2}\in(E_{1},E_{1^{\prime}})$. As in the general case, it is obtained a potential $V_{2}(x)$ which is asymptotically periodic for $|x|\rightarrow\infty$. The corresponding Hamiltonian has two bound states at the positions $\epsilon_{1},\epsilon_{2}$. An illustration of these results is shown in Figure 8.

Our conclusions concerning SUSY transformations applied to periodic potentials are the following.

It was employed successfully the SUSY QM for generating new solvable potentials departing from a periodic initial one

For seed solutions chosen as Bloch functions associated to the band edges or to factorization energies inside the spectral gaps, the new potential is periodic, isospectral to the initial one

For seed solutions which are real linear combinations of Bloch functions, the new potential is asymptotically periodic, and the spectrum of the corresponding Hamiltonian will include a finite number of bound states embedded into the gaps

The technique was clearly illustrated by means of the Lamé potential with $n=1$

Conclusions

Along this paper it has been shown that supersymmetric quantum mechanics is a simple but powerful tool for generating potentials with known spectra departing from a given initial solvable one. Indeed, SUSY QM can be used to implement the spectral manipulation in quantum mechanics

One of our aims when writing this paper is to present the development of the factorization method, in particular the further advances arising after Mielnik’s paper about generalized factorizations

Supersymmetric quantum mechanics has interrelations with several interesting areas of mathematical physics such as solutions of non-linear differential equations , deformation of Lie algebras and coherent states , among others. In my opinion, a further exploration of these interrelations is needed, which represents the future of SUSY QM. We will pursue these issues in the next years with the hope of getting as many interesting results as we were obtaining previously.

Acknowledgments. The author acknowledges Conacyt, project No. 49253-F.

References