ON THE CONNECTION BETWEEN DIRACAND RICATTI EQUATIONSN. C. Bobillo-AresDepartamento de Matemticas. Facultad de Ciencias.Universidad de Oviedo. Calvo Sotelo s.n.33007 Oviedo, SPAIN.andJ. Casahorr�anDepartamento de F��sica. Facultad de Ciencias.Universidad de Oviedo. Calvo Sotelo s.n.33007 Oviedo, SPAIN.FFUOV-97/08AbstractWe analyse the behaviour of the Dirac equation in d = 1 + 1 with Lorentz scalarpotential. As the system is known to provide a physical realization of supersymmetricquantum mechanics, we take advantage of the factorization method in order to enlarge therestricted class of solvable problems. To be precise, it su�ces to integrate a Ricatti equationto construct one-parameter families of solvable potentials. To illustrate the procedure in asimple but relevant context, we resort to a model which has proved useful in showing thephenomenon of fermion number fractionalization.

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1. Introduction.When solving the Dirac equation in d = 1 + 1 with Lorentz scalar potential, theunderlying supersymmetric structure is crucial. As the system provides a physical real-ization of supersymmetric quantum mechanics (susy qm henceforth), the problem reducesitself to a pair of Schr�odinger-like hamiltonians related by means of supersymmetry. Indoing so, both operators share identical energy spectra unless the Dirac equation allowsthe existence of a zero-mode (i.e. a normalizable solution with null energy). Accordingly,we would like to show how the methods used in susy qm to obtain analytical solutionsof the Schr�odinger equation can be extended to the Dirac case. As a relevant example ofthe aforementioned models one can consider the problem of the Dirac hamiltonian with ascalar potential �(x) which characterizes a position dependent band gap. In such a case theDirac equation serves to describe the behaviour of a x-dependent valence and conduction-band edge of semiconductors in the Brillouin zone [1]. On the other hand, the modi�edKorteweg-de Vries (mKdV) equation exhibits a striking relation to the Dirac equation witha time-dependent scalar potential. If �(x; t) represents a solution of the mKdV equation,the spectrum of the (time-dependent) Dirac operator with the scalar potential �(x; t) istime independent. In other words, the Dirac operators at di�erent times are unitarilyequivalent if the potential itself evolves according to the mKdV equation [2].Particular interest acquire however the models in which the Lorentz potential can belooked upon as the kink-like static solution derived from a classical �eld theory. Whengoing to the quantum level, such solutions give rise in general to new sectors beyondthose that are seen in the standard perturbative regime. These classical solutions havereceived the names of solitary waves, energy lumps, kinks or solitons. In addition, theyappear in any number of dimensions: the �4 kink for d = 1+ 1, the Nielsen-Olesen vortexif d = 2 + 1 or the celebrated 't-Hooft-Polyakov monopole in d = 3 + 1. (The bravereader can enjoy the superb description of the subject contained in the book of Rajaraman[3]). Though more massive than the elementary excitations, the soliton-like con�gurationsbecome stable since an in�nite energy barrier separates them from the ordinary sector.The stability is reinforced by the existence of a topological conserved charge which doesnot arise by Noethers theorem from a well-behaved symmetry of the lagrangian, but takesinto account the large distance behaviour of the �eld con�guration (hence the topologicalcharacter of the solutions themselves).On the other hand, dramatic e�ects appear when a Fermi �eld is coupled to thesoliton. Among other things, such models have proved quite useful in the context of thefermion number fractionalization which has been observed in certain class of polymers likepolyacetylene [4]. Describing the soliton by a non-dynamical c-number �eld, the behaviourof the fermions is governed in this �rst quantized approach by a Dirac operator whichincludes the background provided by the scalar �eld. If one considers interactions enjoyingcharge conjugation symmetry, the states of positive and negative energies match betweenthem. In addition, the spectrum can exhibit also self-conjugate zero-energy solutions whichas a last resort assume the responsibility for the fractionalization of the fermion number.These results are nicely described in topological terms: no matter what the details of themodel, the existence of zero-modes is rigorously predicted by general arguments concerning2

di�erential operators. It may be worth spelling that susy qm represents in its own rightan alternative tool in the analysis of this connection between physics and topology (seefurther on).The organization of the article is as follows. Section 2 takes care of the Dirac equationin d = 1 + 1 with static Lorentz scalar potential. There we carefully describe the way inwhich the Ricatti equation enables us to construct one-parameter families of isospectralLorentz scalar potentials for the Dirac case. To illustrate the procedure in a simple butrelevant context, we resort in section 3 to the problem consisting in the analysis of the Diracequation on the background provided by the kink-like con�guration of the �4 model. (The�rst appendix includes a capsule introduction to susy qm while the second one outlinesthe properties of the eigenfunctions associated with the P�osch-Teller potential).2. Supersymmetric structure of the Dirac equation.Although there are several applications of susy qm in the context of the Dirac equation,we discuss only the underlying supersymmetric structure of such equation in d = 1+1 withstatic Lorentz scalar potential represented by f(x). In doing so, we return to the ideasexposed by Cooper et al. [5] some years ago. The purpose of this section is to show howwe can take advantage of susy qm in order to enlarge in a systematic way the restrictedclass of solvable models. To be precise, it is the general solution of a Ricatti equation thatprovides us with a one-parameter family of potentials which serve to extend the �rst resultsderived from f(x). To begin with, let us give some words about the notation used in thenext two sections. Space-time coordinates are represented by x�(� = 0; 1;x0 = t; x1 = x).As regards the metric tensor we have g�� = g�� , where g00 = �g11 = 1 and g�� = 0 if� 6= �. Moreover @� stands for space-time derivatives @=@x�. When considering a fermion�eld (x; t) moving in the static Lorentz scalar potential f(x), the behaviour of the systemis governed by the Dirac equation[i �@� � f(x)] (x; t) = 0 (2:1)Letting (x; t) = (x) exp (�i!t) (2:2)the proper Dirac equation reduces to 0! (x) + i 1 0(x)� f(x) (x) = 0 (2:3)where the prime denotes as usual derivative with respect to the spatial coordinate. Uponchoosing now 0 = �1 = � 0 11 0� (2:4a) 1 = i�3 = � i 00 �i� (2:4b)3

(x) = � �(x) +(x)� (2:4c)we get the pair of coupled equationsA �(x) = ! +(x) (2:5a)Ay +(x) = ! �(x) (2:5b)where A = ddx + f(x) (2:6a)Ay = � ddx + f(x) (2:6b)It remains to decouple the equations, i.e.AyA �(x) = !2 �(x) (2:7a)AAy +(x) = !2 +(x) (2:7b)which are of the form of susy qm (see appendix A for a brief introduction to the subject),namely � �00(x) + �f2(x)� f 0(x)� �(x) = !2 �(x) (2:8a)� +00(x) + �f2(x) + f 0(x)� +(x) = !2 +(x) (2:8b)where the Lorentz scalar potential f(x) is just the superpotential function. To sum up,we substitute the Dirac equation on behalf of the two Schr�odinger-like equations of agenuine susy qm model. In such a case, the components �(x) and +(x) represent theeigenfunctions of the hamiltonians H� = AyA and H+ = AAy. It is customarily assumedthat whenever the Schr�odinger equations for potentials V� = f2(x) � f 0(x) are solvable,then there always exists a static Lorentz scalar potential f(x) for which the correspondingDirac equation is also exactly solvable. As expected, the magic of supersymmetry impliesthat both hamiltonians H� and H+ become solvable simultaneously since the action of thesupercharges establish the right connection between the eigenfunctions �(x) and +(x).The spectrum of energies is of course degenerate except for the hypothetical zero-mode.At this point we can recall the way in which the topological behaviour of the superpo-tential f(x) dictates the existence (absence) of zero-modes. It is customarily assumed thatsupersymmetry, a new kind of symmetry which puts together both bosonic and fermionicdegrees of freedom, represents one of the most ambitious attempts recently made to achievea trustworthy description of nature. For better or worse, we have no experimental evidence4

for such a novel symmetry being exactly realized in physics so that one of the most pursuedgoals in the last two decades has been to describe a suitable mechanism by means of whichsupersymmetry shows up itself spontaneously broken. As susy qm constitutes an excellentlaboratory to analyse the hypothetical spontaneous breaking phenomenon, Witten [6] in-troduced an order parameter (henceforth the Witten index �) which in general serves toshed light on this subject. As a matter of fact the aforementioned index � reads [6]�(�) = Tr [exp (��H�)� exp (��H+)] (2:9)where � stands for a regularization parameter necessary when considering susy qm withscattering states. In doing so, �(�) should represent a measurement of the di�erence of �(x) and +(x) modes all of them with zero-energy since for positive energies duplicationoccurs between the two parts of the model. In order not to clutter the article, we simplypoint out that when taking into account the subtleties associated with the counting ofstates in the continuous part of the spectrum the Witten index �(�) satis�es the equation[7] d�(�)d� = 1p4�� �f+ exp (��f2+)� f� exp (��f2�)� (2:10)where f� represent the asymptotic values of the superpotential f(x) as x! �1. Althoughsometimes cumbersome �nal indices are obtained if any of the two limits f� or f+ becomesnull, it is the case that whenever we observe a change of sign when going from f� to f+,the integration of (2.10) gives�(�) = � �12�(f+p�) + 12�(f�p�)� (2:11)being � the Fresnel probability function. Now it su�ces the limit � ! 1 to reveal thetopological roots of the model: the Witten index amounts to �1 so that the asymptoticbehaviour of f(x) dictates the existence (absence) of the non-degenerate zero-mode (seeappendix A). From a qualitative point of view, this result can also be derived from (A.11):a normalizable zero-energy eigenfunction exists for the superpotentials just exhibiting theright behaviour at in�nity. In summary, the odd-like superpotentials yields a zero-modewhile the even-like ones are not capable of producing such a state.However, the above discussion is only part of the whole story. Once we assumethe existence of a solvable quantum mechanical problem for the pair of one-dimensionalSchr�odinger-like equations written in (2.8), di�erent procedures can be pro�tably used forgenerating one-parameter families of models which share exactly the same set of eigen-values. The reasoning becomes much more accesible when resorting to susy qm in thespirit of the factorization schemes exposed by Mielnik [8]. When making an attempt inorder to extend the formalism to the Dirac case, we start from a couple of susy partnershamiltonians H� and H+ like the ones written in (2.7) and (2.8). To �x the ideas, let ustake f+ > 0 and f� < 0 so that a well-behaved zero-mode exists associated with H�. Indoing so, the energy spectrum of H+ is positive whereas H� starts from E = 0. Being �o(x) precisely the zero-energy eigenfunction of H�, the static Lorentz scalar potential5

f(x) appears as the logarithmic derivative of such a state, i.e. f(x) = � 0�o(x)= �o(x).At this stage, we ask ourselves whether the factorization H+ = AAy is unique or can bemore general than usually realized. Writing now a new factorization likeH+ = BBy (2:12)for B = ddx + F (x) (2:13a)By = � ddx + F (x) (2:13b)the function F (x) must satisfy the Ricatti-like equationF 0(x) + F 2(x) = V+(x) (2:14)Whenever we know a particular solution of (2.14) (the potential f(x) itself!), thegeneral solution can be obtained by putting [9]F (x) = f(x) + �(x) (2:15)so that �0(x) + �2(x) + 2f(x) �(x) = 0 (2:16)Now it proves convenient the introduction of �(x) = 1=�(x) to transform (2.16) intoa �rst-order di�erential equation like��0(x) + 2f(x) �(x) + 1 = 0 (2:17)whose general solution corresponds to�(x) = exp�Z x 2f(t) dt� ��+ Z x exp��Z t 2f(z) dz� dt� (2:18)where in principle � is a real number. Bearing in mind the relation between f(x) and thezero-mode 0�o(x), we get for the function F (x) the useful expressionF (x) = f(x) + 2�o(x)��+ R x 2�o(t) dt� (2:19)whenever the choice for � avoids any singularity along the real axis. To sum up, we gofrom the small potential f(x) to the large potential represented by F (x) just by solving inthe standard way a Ricatti di�erential equation. In doing so, we can take advantage againof the commutation formula of susy qm to get a family of hamiltonians ~H, i.e.~H = ByB (2:20)6

whose close relation with H� can be explained as follows. Except for the zero-energyeigenstate, the potential ~V (x) included in ~H, namely~V (x) = V+(x)� 2 ddx "f(x) + 2�o(x)��+ R x 2�o(t) dt�# (2:21)becomes isospectral with respect to the V+(x) itself. As anticipated in the second section,the zero-modes are found from �rst-order di�erential operators so that in our case it su�cesto consider � ddx + F (x)� ~ o(x) = 0 (2:22)to obtain ~ o(x) � �o(x) exp��Z x �(t) dt� (2:23)In this regards, the procedure may be understood as a renormalization of the zero-mode, leading to a one-parameter of hamiltonians ~H isospectral with the original H�.Returning to the Dirac equation itself, the situation is both surprising and unexpectedlysubtle. First of all, we start from a static Lorentz scalar potential f(x) which yieldssolvable Schr�odinger-like potentials V�(x) = f2(x) � f 0(x). Afterwards, we enlarge in asystematic way the restricted class of such solvable models just by going from the smallpotential f(x) to the large potential F (x) through the right combination of the commutationformula of susy qm and the general theory of Ricatti equations. In addition, the downcomponents of the spinor remain unchanged whereas the transformation �(x) ! ~ (x)must be carried out as far as the upper components are concerned. In any case, we avoidthe di�culties associated with the handling of the potential ~V (x) since the ~ (x) themselvescan be obtained from +(x) by the action of the operator By.3. Illustration: kinks in the presence of Fermi �elds.To illustrate the above formalism in a simple but physically relevant scenario, weresort now to the Dirac equation in the background provided by the kink-like con�gura-tion of the �4 model. Such system has proved quite useful in the context of the so-calledfractionalization of the fermion number which can be observed in polymers like the poly-acetylene [4]. By any standard, the search for classical solutions represents nowadays oneof the most proli�c areas in quantum �eld theory. The interest is mainly motivated by thebelief that such classical con�gurations, which of course make stationary the action, mayshed light on the properties of the system at issue. In this section we restrict ourselvesto non-dissipative con�gurations with �nite energy because they constitute the naturalcandidates to describe new sectors of the model beyond the perturbative regime. On theother hand, dramatic e�ects like the fractionalization of the fermion number appear whena Dirac �eld is introduced to the kink itself. At this stage, it should be interesting to reviewthe pioneering model of Jackiw and Rebbi [10] in the light of the formalism exposed in theprevious section. For reference, we recall that the behaviour of the system is governed bythe Lagrangian density 7

L = 12 @��@��� 12 ��2 � 1�2 + i � �@� � 2� � (3:1)where as usual �(x) and (x) stand for the boson and the fermion �elds respectively.Notice that for the sake of simplicity the coupling constants have been absorbed in the�elds themselves. Unless otherwise noted, it su�ces here to discuss a �rst quantizedapproach where the fermion moves in the non-dynamical c-number �eld provided by thestatic kink �k(x). In doing so, the Dirac equation derived from (3.1) readsi �@� (x; t)� 2�k(x) (x; t) = 0 (3:2)which is of the form written in (2.1). In other words, the kink-like pro�le represents inthis case the static Lorentz scalar potential. At this stage, it is customarily assumed theexistence of the well-known kink of the �4 model (see reference [3] for details), namely�k(x) = tanhx (3:3)a topological con�guration which smoothly interpolates between the constant con�gura-tions �� = �1 and �+ = 1. On comparing now with the general formalism of the secondsection, we face a standard Dirac problem withf(x) = 2 tanhx (3:4)whose properties (spectrum and eigenfunctions) are well studied (see appendix B). Forreference, we simply point out that in this case we haveV�(x) = 4� 6cosh2 x (3:5a)V+(x) = 4� 2cosh2 x (3:5b)while the zero-mode �o(x) reads �o(x) � 1cosh2 x (3:6)Now we can write thatF (x) = 2 tanhx+ cosh�4 x��+ tanhx� 13 tanh3 x� (3:7)so that the choice j�j > 2=3 assures the regular character of the large potential F (x) alongthe real axis. As regards the one-parameter family of potentials ~V (x) we have~V (x) = 4� 6cosh2 x � 2 ddx " cosh�4 x��+ tanhx� 13 tanh3 x�# (3:8)while the renormalization of the zero-mode translates into8

~ o(x) � cosh�2 x exp�"Z x cosh�4 x��+ tanhx� 13 tanh3 x� dt# (3:9)The procedure closes itself when writing the spinor solutions of the large potentialF (x) of (3.7) in terms of the components +(x) and ~ (x) according to the general methodexposed in section 2. 4. Conclusions.In this article we have considered in detail the procedure for obtaining exact energyeigenvalues and eigenfunctions for the Dirac equation involving a Lorentz scalar potentialin d = 1 + 1. The generic model is useful in di�erent branches of physics, ranging fromcondensed matter to nonlinear equations. For instance, it serves to study the fermion num-ber fractionalization observed in certain class of polymers like polyacetylene, the positiondependent band gap in semiconductors or the theory of solitons of the modi�ed Korteweg-de Vries equation. Taking into account the way in which the system itself provides uswith a realization of supersymmetric quantum mechanics, we face ultimately a pair ofSchr�odinger-like hamiltonians. As expected, the two aforementioned operators share theenergy spectra except for the hypothetical existence of a zero-mode, i.e. a normalizablesolution with E = 0.In addition, the so-called topological techniques have proved quite useful in modernphysics. When we need to analyze the spectrum of a well-behaved linear di�erential opera-tor, the zero-modes are the most crucial ones for applications. To determine the propertiesof these modes we can resort to the results contained in the index theorems: no matterwhat the local details of the model at issue, the existence of zero-modes depends on theglobal behaviour of the system (hence the topological character of the procedure itself).Taking advantage of the structure associated with susy qm, it su�ces to consider a Ricattiequation in order to enlarge the restricted class of problems whose eigenvalues and eigen-functions can be presented in closed form. To sum up, whenever we have a pair of solvableSchr�odinger-like equations then there also exists a one-parameter family of Lorentz scalarpotentials for which the Dirac equation is exactly solvable.

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Appendix AIn this appendix we present a brief introduction of one-dimensional susy qm, as tai-lored to our needs. In its most simple formulation the model deals with a time-independenthamiltonian matrix Hs of the formHs = �H� 00 H+ � (A:1)which contains both one-particle hamiltonians H� and H+. This operator Hs is part of agraded algebra (superalgebra in the jargon of physics) whose closure is achieved by meansof the right mixing of commutation and anti-commutation relations. At this point, it iscustomarily assumed the existence of the so-called superpotential function W (x) whichgives rise to the couple of �rst-order di�erential operators A, Ay given byA = ddx +W (x) (A:2a)Ay = � ddx +W (x) (A:2b)Now we can write the supercharges Q, Qy, i.e.Q = � 0 0A 0� (A:3a)Qy = � 0 Ay0 0 � (A:3b)which together with the hamiltonian Hs constitute the superalgebra of susy qm, namelyHs = fQ;Qyg (A:4a)[Hs; Q] = �Hs; Qy� = 0 (A:4b)fQ;Qg = fQy; Qyg = 0 (A:4c)In more physical terms susy qm can be understood as a matrix hamiltonian Hs actingon two-component wave-functions (x) like (x) = � �(x) +(x)� (A:5)where it should be emphasized that H+ is obtained from H� just by reversing the orderof the operators A and Ay. In other wordsH� = AyA (A:6a)10

H+ = AAy (A:6b)so that the two one-particle hamiltonians H� and H+ readH� = � d2dx2 + V�(x) (A:7)for V�(x) =W 2(x)�W 0(x) (A:8)where the prime denotes derivative with respect to the spatial coordinate. It is often thecase to refer to the pair of potentials V�(x) and V+(x) as supersymmetric partners inaccordance with the standard susy qm language. To �nish this brief exposition, let usoutline the main properties of our model as far as the standard realization of susy qm isconcerned. First of all, the energy eigenvalues of bothH� and H+ are positive semi-de�nitesince the eigenkets �(x) and +(x) verify in fact the relationsh �jH�j +i = kAj �ik2 (A:9a)h +jH+j �i = kAyj �ik2 (A:9b)where kj �ik represents the norm of the eigenstate j �i. As regards the hypotheticalzero-energy solutions �o(x) or +o(x) we have thatA �o(x) = 0 (A:10a)Ay +o(x) = 0 (A:10b)In doing so, they can be obtained by solving �rst-order di�erential equations so thatwe write the formal solutions �o(x) � exp�� Z xW (y) dy� (A:11)As any well respected normalizable zero-energy eigenfunction requires boundary con-ditions of the type �o ! 0 as x! �1, the meaning of (A.11) is clear: the zero-mode, ifany, must be non-degenerate. The above argument illuminates the framework in which su-persymmetry itself appears either unbroken or spontaneously broken. To be more speci�c,if susy remains exact the energy of the non-degenerate ground-state is zero while the spon-taneous breaking phenomenon translates into twofold ground-state. On the other hand,all other energy eigenvalues appear duplicated. When considering the eigenket �(x) withenergy E�, i.e. H� �(x) = E� �(x) (A:12)11

the action of the operator A on the �(x) itself provides an eigenfunction of the partnerhamiltonian H+. In other wordsH+ [A �(x)] = AAyA �(x) = E� [A �(x)] (A:13)Just by reversing the order of the operators, we start now from the eigenkets +(x),namely H+ +(x) = E+ +(x) (A:14)thus obtaining as expected eigenfunctions of H� through the action of Ay, i.e.H� �Ay +(x)� = AyAAy +(x) = E+ �Ay +(x)� (A:15)To sum up, except for the hypothetical zero-modes all other energy eigenvalues appearduplicated when the whole model is studied. Further on, are the �rst-order di�erentialoperators A, Ay that establish the connection between the two sectors of the system. How-ever, the above structure exhibits important advantages if we study the problem from apractical point of view. If we resort for instance to a computational technique to estimatethe energy spectrum, the original hamiltonian can be substituted on behalf of the isospec-tral one to improve the properties of convergence. The same argument can be applied ofcourse when considering variational methods.

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Appendix BFor reference, we outline in this appendix the mean features concerning the P�osch-Teller potentials which arise in the analysis of the Dirac equation on non-trivial back-grounds like the one associated with the kink of the �4 model. The problem itself can beelegantly solved in a Lie-algebra framework according to the technique described by Frankand Wolf [11], whose notation will be used in the following. To start from scratch, let usconsider the P�osch-Teller hamiltonianHPT = �12 d2dx2 + ccosh2 x + ssinh2 x (B:1)where we choose a system of units in which �h = m = 1. Now it proves convenient theintroduction of the parameters ki (i = 1; 2) given byc = �12 �(2k1 � 1)2 � 14� (B:2a)s = 12 �(2k2 � 1)2 � 14� (B:2b)which in turn enable us to write the energies Ek of the bound states associated with theP�osch-Teller potential of (B:1), namelyEk = � (2k � 1)22 (B:3)where k = kmin; kmin�1; ::: > 1=2 with kmin = k1 � k2. Letting k to be of the formk = 1 + i�2 � � 0 (B:4)the expression of Ek in (B:3) also takes care of the continuous part of the spectrum. Itremains to identify the behaviour of the wave-functions themselves which, as expected,depends on the values of the singularity parameter s in (B:1). At �rst glance the situationis as follows:i) For the bound states we have(k1;k2)k � (coshx)�2k1+3=2 (sinhx)2k2�1=2 F [�; �; ; z] (B:5)where F [�; �; ; z] stand for the 2F1 Gauss hypergeometric function with the identi�cations� = �k1 + k2 + k (B:6a)� = �k1 + k2 � k + 1 (B:6b) = 2k2 (B:6c)13

z = � sinh2 x (B:6d)ii) When considering the scattering states we �nd(k1;k2)k � (coshx)2k1�1=2 (sinhx)2k2�1=2 F [�; �; ; z] (B:7)now with � = k1 + k2 � k (B:8a)� = k1 + k2 + k � 1 (B:8b) = 2k2 (B:8c)z = � sinh2 x (B:8d)With regards to singular potentials like the ones written in (B.1), the careful analysisof the problem can be carried out in accordance with the subtle mathematical tools exposedby Reed and Simon [12]. As far as our model concerns, it might seem plausible to inferthe existence of three well di�erent regions depending on the values of the parameter s,namely:i) If s < �1=8, the model itself becomes physically unrealistic since the spectrum isnot bounded from below.ii) Whenever �1=8 < s < 3=8, the singularity is not strong enough to make the wave-functions vanish at the origin. However, it proves necessary to de�ne a one-parameterfamily of self-adjoint extensions so that a bounded from below spectrum appears once aproper domain is speci�ed. In more physical terms, the wave-functions pass across thesingularity point and the model extends itself along the real axis.iii) Finally, if s > 3=8 the singularity acts as an impenetrable barrier thus dividingthe space into two independent regions, i.e. x > 0 and x < 0, forcing the wave-functionsto vanish at the origin.The above results shed light on the family of susy qm models governed by superpo-tentials like W (x) = ` tanhx; ` = 1; 2; ::: (B:9)which give rise to H� = � d2dx2 + `2 � `(`+ 1)cosh2 x (B:10a)H+ = � d2dx2 + `2 � `(`� 1)cosh2 x (B:10b)In accordance with (B:2) we have for H�14

k1 = 2̀ + 34 (B:11a)k1 = � 2̀ + 14 (B:11b)together with k2 = 34 (B:12a)k2 = 14 (B:12b)so that the discrete energy spectrum readsE�j = `2 � (`� j)2; j = 0; 1; :::; `� 1 (B:13)while the scattering states start at E = `2. With regards to H+ we �ndk1 = 2̀ + 14 (B:14a)k1 = � 2̀ � 34 (B:14b)and again k2 = 34 (B:15a)k2 = 14 (B:15b)with energies E+j = `2 � (`� j)2; j = 1; :::; `� 1 (B:16)The di�erence between both discrete spectra reduces itself to the zero-mode of H�which adopts the form �0(x) � P `̀(tanhx) (B:17)where the P `̀(tanhx) represent associated Legendre polynomials. Bearing in mind thatP `̀(tanhx) � cosh�` x (B:18)the logarithmic derivative of the zero-mode itself provides us with the superpotential func-tions written in (B.8), i.e. 15

W (x) = �P 0 `̀(tanhx)P `̀(tanhx) (B:19)In this case, being s = 0, we get rid of the di�culties associated with the singularbehaviour of the potential. Among other things, the system extends along the real lineand the expected pattern of even and odd wave-functions (notice that we are dealing witheven potential) appears in the end.Acknowledgement.- The authors thank the Comisi�on Interministerial de Ciencia yTecnolog��a for �nancial support.

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