# Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions

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## Abstract

**:**

**PACS**03.65.Ge; 03.65.Ca

## 1. Introduction

## 2. Conventional SUSY formalism

**Preliminaries and the matrix TDSE.**Let us start by considering two TDSEs in atomic units ($m=1/2,\phantom{\rule{3.33333pt}{0ex}}\hslash =1$), that is,

**The supercharges.**As in the stationary case [4], our goal is to construct a superalgebra with three generators, two of which are called supercharge operators or simply supercharges. These are mutually adjoint matrix operators of the following form

**The intertwining relation and its adjoint.**Note that $\mathsf{\Psi}$ is a solution of our matrix TDSE (5), that is, its first and second component $\psi $ and $\varphi $ solve the TDSEs (1) and (2), respectively. Now, we want this property to be preserved after application of the supercharges, i.e., $L\left(\psi \right)$ and ${L}^{+}\left(\varphi \right)$ are required to be solutions of the TDSEs (2) and (1), respectively. Consequently, L must be an operator that converts solutions of the first TDSE (1) into solutions of the second TDSE (2), and its adjoint ${L}^{+}$ must convert solutions of the second TDSE (2) into solutions of the first TDSE (1). Let us first consider the operator L, which we will determine from the following equation:

**Construction of the superalgebra.**We will need to have another generator besides the supercharges, which we obtain as follows. Consider the operators ${S}_{1}$ and ${S}_{2}$, defined by

## 3. Generalized SUSY formalism

**The generalized TDSE.**Let us consider the following equation, which we will call generalized TDSE:

**Generalized matrix TDSE and supercharges.**Now let us consider another generalized TDSE, which we will relate to its counterpart (26):

**The intertwining relation for $\mathit{L}$.**In order to determine L, we require it to convert solutions of the first TDSE (26) into solutions of the second TDSE (27), and its adjoint ${L}^{+}$ must convert solutions of the second TDSE (27) into solutions of the first TDSE (26). The intertwining relation involving the operator L is given by (9), where the Hamiltonians are taken from (28):

**Resolution of the intertwining relation.**Since there are only three different derivative operators left in our intertwining relation (36), namely, ${\partial}_{xx}$, ${\partial}_{x}$ and the multiplication (derivative of order zero), we obtain three equations. These equations have the following form:

**Potential difference and the operator $\mathit{L}$.**Before we state the operator L in its explicit form, let us find the potential ${V}_{2}$ by solving (38):

**The adjoint operator ${\mathit{L}}^{+}$.**The next task is to find the operator ${L}^{+}$ in the same way as it was just done for L. The intertwining relation to be used is given by the adjoint of (32):

**Construction of the superalgebra.**Since the operators L and ${L}^{+}$ in the generalized case are now determined, at the same time the supercharges Q and ${Q}^{+}$, as given in (31), are determined. As in the coventional case we construct the superalgebra by adding one more generator besides the supercharges, which will be constructed from the following operators ${S}_{1}$ and ${S}_{2}$:

## 4. Reality condition

## 5. SUSY formalism for particular TDSEs

#### 5.1. TDSE for position-dependent mass

**Equation:**

**Relation to generalized TDSE:**

**SUSY operators:**

**Supersymmetric partner potential:**

**Reality condition:**

**Supersymmetric partner potential under reality condition:**

#### 5.2. TDSE with weighted energy

**Equation:**

**Relation to generalized TDSE:**

**SUSY operators:**

**Supersymmetric partner potential:**

**Reality condition:**

**Supersymmetric partner potential under reality condition:**

#### 5.3. TDSE with minimal coupling

**Equation:**

**Relation to generalized TDSE:**

**SUSY operators:**

**Supersymmetric partner potential:**

**Reality condition:**This condition does not apply here, since the potential ${V}_{1}$ as given in (67) is in general not real-valued.

**Supersymmetric partner potential under reality condition:**This condition does not apply here either.

#### 5.4. Conventional TDSE

**Equation:**

**Relation to generalized TDSE:**

**SUSY operators:**

**Supersymmetric partner potential:**

**Reality condition:**

**Supersymmetric partner potential under reality condition:**

## 6. Example: TDSE with weighted energy

## 7. SUSY formalism beyond the TDSE

#### 7.1. The Fokker-Planck equation

**Statement of the problem.**The Fokker-Planck equation (FPE) with constant diffusion and stationary drift ${U}_{1}={U}_{1}\left(x\right)$ has the following form [19]

**Mapping the FPE onto the TDSE.**Following (75), we first need to find an invertible transformation P that takes solutions of the FPE into solutions of the TDSE. Such a transformation is well known [12]: let f be a solution of the FPE (76), then we define the transformation P as

**Application of the SUSY formalism.**Take the solution $\psi $ and an auxiliary solution u of (79), such that $\psi $ and u are linearly independent. Furthermore, we require the function $\psi $ to solve the Schrödinger equation (79) at energy $E=0$, note that $\psi $ and u are allowed to be solutions at different energies E. This is due to the fact that the auxiliary function solves (46), where an arbitrary time-dependent function $C=C\left(t\right)$ can be included. In the time-dependent case this function cancels out, while in the present stationary case it becomes constant and adds to the energy E in equation (79). This is the reason why the solution and its auxiliary counterpart can be taken at different energies. Now we fix an auxiliary function u at energy, say, $\lambda \ne 0$, and apply the operator L to the solution $\psi $ of (79) at $E=0$:

**Mapping the TDSE onto the FPE.**We introduce the inverse transformation ${P}^{-1}$ of P as

**Explicit form of the extended SUSY transformation.**We start with the drift potential ${U}_{2}$, which we are able to write down after combining our previous findings (80) and (82):

**Example: quadratic drift potential.**Let us consider the FPE (76) for the following quadratic drift potential:

#### 7.2. The Burgers equation

**Statement of the problem.**We consider the nonhomogeneous Burgers equation (NBE) in the following form [20]

**Mapping the NPE onto the TDSE.**We must find an invertible mapping P that connects the TDSE with the NBE, as represented by the vertical arrows in figure 1. Such a mapping P is given by the Cole-Hopf transformation:

**Application of the SUSY formalism.**We can transform the solution $\psi $ by means of the operator L, as given in (11):

**Mapping the TDSE onto an NBE.**Inversion of the Cole-Hopf transformation (103) for the solution $\varphi $ of (108) yields

**Explicit form of the extended SUSY transformation.**Let us first take into account that the auxiliary solution u of the TDSE (104) is related to a solution v of the initial NBE (101) via

**Example: linear nonhomogeneity.**We consider the NBE (101) for the following linear nonhomogeneity:

## 8. Concluding remarks

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**MDPI and ACS Style**

Schulze-Halberg, A.; Carballo Jimenez, J.M.
Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions. *Symmetry* **2009**, *1*, 115-144.
https://doi.org/10.3390/sym1020115

**AMA Style**

Schulze-Halberg A, Carballo Jimenez JM.
Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions. *Symmetry*. 2009; 1(2):115-144.
https://doi.org/10.3390/sym1020115

**Chicago/Turabian Style**

Schulze-Halberg, Axel, and Juan M. Carballo Jimenez.
2009. "Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions" *Symmetry* 1, no. 2: 115-144.
https://doi.org/10.3390/sym1020115