JournalofAppliedMathematicsandPhysics,2019,7,1013-1026
http://www.scirp.org/journal/jamp
ISSNOnline:2327-4379
ISSNPrint:2327-4352
ANewClassofExactlySolvableModelswithin
theSchr¨odingerEquationwithPosition
DependentMass
AnisDhahbi
1
,YassineChargui
1
,AdelTrablesi
2
1
PhysicsDepartment,CollegeofScienceandArtsatArRass,QassimUniversity,Al-Mulida,KSA
2
Universit´e deTunis ElManar, Facult´e desSciences de Tunis,Unit´ede Recherchede Physique Nucl´eaire etdes
HautesEnergies,Tunis,Tunisie
Howtocitethispaper:Dhahbi,A.,
Chargui,Y.andTrablesi,A.(2019)ANew
ClassofExactlySolvableModelswithinthe
Schr¨odingerEquationwithPositionDepen-
dentMass.JournalofAppliedMathematics
andPhysics,7,1013-1026.
https://doi.org/10.4236/jamp.2019.75068
Received:April3,2019
Accepted:May3,2019
Published:May6,2019
Copyright
c
2019byauthor(s)andScientific
Research PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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saeurisusblanditmattis.Maurisutnunc.
Inhachabitasseplateadictumst.Aliquam
egettortor.Quisquedapibuspedeinerat.
Nuncenim.Induinulla,commodoat,con-
sectetuernec,malesuadanec,elit.Aliquam
ornaretelluseuurna.Sednecmetus.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integer metus.Morbi pulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehicula purus.Fusceacnislaliquam ante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cum sociis natoquepenatibus etmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbi magna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubianostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretiumnonummyfringilla,tortor lacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumacmetus dictumtempus.Nam ut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aenean egestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligula nec orci.Aliquam purus turpis, aliquam
id, ornarevitae,porttitor non,wisi.Maecenas
luctusportalorem.Donec vitaeligulaeuante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclaciniaauctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuada egestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis,felis.Morbinulla nibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinar blandit.Nuncsitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel,metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac,mauris.Maecenaset elit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat,wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiamfermentum convallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarculibero,malesuadavitae,ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisse utpurussed sem
tempor rhoncus.Ut quamdui, fringilla at,dic-
tum eget,ultricies quis,quam.Etiam sem est,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sedscelerisque sus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareantenuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisiultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummy eget,ligula.
Crasegestasipsumanisl.Vivamus variusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes, nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisicommodofelis, utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisque eratelit, viverrasit amet, sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputateelit.Pellentesque
sem.Praesent eusapien.Duis elitmagna,ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimetcommodomalesuada,tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultriciesaccumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eusapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisseconsequat.Mauris adui
et tortormattispretium.Sed nulla metus,vo-
lutpatid, aliquameget, ullamcorperut, ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Ut blanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretiumultricies.Fusce
temporpellentesqueodio.Vivamus auguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duis enim.Nulla interdum, nunc eusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inporta congue,velit leo facilisisodio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,vel posuere nibhlacus necnibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulum libero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla, lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sitameteuismod lectusfeliset
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pedecursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,variuseu,commodo sed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis,sed pellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuervariusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velit at elit vehicula porttitor.Phasel-
lusnislmi,sagittisac,pulvinarid,gravida sit
amet,erat.Vestibulumest.Loremipsumdo-
lorsitamet,consectetueradipiscingelit.Cur-
abitur idsem elementum leo rutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreeteget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Mauris enimsem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisque lacus.Donec etipsum sit
amet elitnonummy aliquet.Sed viverranisl at
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tus sitamet purusfaucibus vehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac,nunc.Maurismalesuadamalesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin,ultriciesid,libero.Phasellus bibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.In hachabitasse plateadictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristiquequis,felis.Vestibulumante ip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris portadui,vitae aliquamarcuturpis ac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellusaliquam, tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementum dui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamus pedearcu, euismod ac,tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovel nibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna,blanditnec, iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringilla turpisnon arcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbiipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuadavestibu-
lum.Quisquesollicitudinipsumvelsem.Nul-
laenim.Proin nonummyfelisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum, justoat dignissim
auctor,wisi odiolobortisarcu,sedsollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nulla quammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudin hen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in,dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper,dui ut
dignissim auctor, orciliberofringilla massa, b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tincidunt fermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netuset malesuadafames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.In lectuswisi, elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipitsitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringillatempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitant morbi tristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnasedenim.Donec egeterosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpat faucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulum ante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semperat, metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputatepurus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,nonporttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vellectusutme-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatis vitae,vulputatea,pharetra sit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid,mi.Nullamnecnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nunc nulla pulvinarjusto,aeuismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdum sapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus,eusodales magnarisusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursusarcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donec feliselit,dignissim sed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et, commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectusneque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In eget risusluctus wisiconva-
llis pulvinar.Vivamus sapien risus,tempor in,
viverra in, aliquetpellentesque, eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisquedignissim,lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligulasuscipiturna,vitaeportaodioer-
at quissapien.Proin luctusleo iderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursussed,
justo.Phasellus nec lorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fusce vehicula,tortor etgravidaporttitor, me-
tusnibhconguelorem,uttempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit,imperdiet vitae,faucibusnec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbi fringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisse ornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeros elit,gravidaac,pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehiculaat, conguesit amet,
sem.Sedsagittis pedea nisl.Sedtincidunt o-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nullaettellusnonfelis imperdi-
etmattis.Aliquameratvolutpat.
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ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallis hendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes, nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsitametnibhnecpurusdignissimmat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpisdiam feugiat ar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulum elit.Nam ligulaarcu, volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisusatmetuspulvinarmalesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
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luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nunc bibendumaugue luctus
diam.Etiam nec sem.Sed eros turpis, facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajusto vehiculadiam,eu condimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
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Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
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eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitametest.Vivamuspulvinarleonon
sapien.Aliquameratvolutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet,mattis suscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibhullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamusloremdui, commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisquevestibulum pulvinar di-
am.Phasellustincidunt,leovitaetristiquefa-
cilisis,ipsumwisiinterdum sem,dapibussem-
pernulla velitvel lectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
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sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
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eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim,sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitae dolorsollicitudin adipiscing.
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eratid,tristiqueeu,enim.Vestibulumante
ipsum primisin faucibusorciluctus etultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere,augueerosporttitorlec-
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etnulla.
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ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
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ametlorempharetralobortis.
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diam.Etiamultricies,dolorsitametfermen-
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elit.Quisquevenenatislobortisaugue.Sed
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pededuimalesuadamauris,velultricesurna
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lis.Vivamuspulvinar.Aliquameratvolutpat.
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nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
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sit ametmauris.Cras commodo, urna ut port-
titor venenatis, arcu metussodales risus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultrices felis erat at lectus.In est massa, male-
suadanon,suscipitat,ullamcorpereu,elit.
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tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibhpretiumtellus,atdignissimleoeros
egetpede.Nullavenenatiseleifenderos.Ae-
nean ut odio dignissim augue rutrum faucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamus vehicula leoajusto.Quisquenecau-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenimsitametrisus.Curabitur vitaevelit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
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Fusceeunulla utnulla interdum consectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
Thestudyofphysicalsystemsendowedwithaposition-dependent
mass(PDM)remainsafundamentalissueofquantummechanics.
Inthispaperweuseanewapproach,recentlydevelopedbyus
forbuildingthequantumkineticenergyoperator(KEO)within
theSchr¨odingerequation,inordertoconstructanewclassofex-
actlysolvablemodelswithapositionvaryingmass,presentinga
harmonic-oscillator-likespectrum.Todosoweutilizetheformalis-
mofsupersymmetricquantummechanics(SUSYQM)alongwith
theshapeinvariancecondition.Recentoutcomesofnon-Hermitian
quantummechanicsarealsotakenintoaccount.
Keywords
Schr¨odingerEquation,PositionDependentMass,KineticEnergy
Operator,SolvableModels,SupersymmetricQuantumMechanics,
ShapeInvariance
1.Introduction
Theideaofnon-relativisticorrelativisticparticlewhosemassisa
functionofspacebecomesanintensivefieldofresearchforthepast
fewdecades.Indeed,therearemanyproblemsinphysics,chemistry,
biologyandeveninmedicinewheretheevolutionofaphenomenon
can be assimilated by a Schr¨odinger, Klein-Gordon or even Dirac type
equation,relativetoaparticleofvariablemassinspace.Especial-
ly,inthecaseofsolid-statephysics,themovementofaparticlein
aperiodicpotentialrepresentingthecrystallinelattice,isassimilated
tothemovementofafreeparticlewithaneffectivemass,whichde-
pendsessentiallyonthecharacteristicsofthelattice.Ifthesample
DOI:10.4236/jamp.2019.75068May6,20191013JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
iscomposedofseveralpartsrepresentingdifferentmaterials,called
heterostructures,themasswilltakedifferentvaluesineachpartof
thestructure.Accordingly,PositionDependentMass(PDM)model-
s[1–12]weresuccessfullyappliedinthedeterminationofelectronic
propertiesofsemiconductors[12–16],aswellasforthedescriptionof
thepropertiesofquantumdots[17]andnuclearclusters[18,19].The
PDMissuehasalsoattractedanincreasinginterestinmolecularand
atomicphysics[20,21],densityfunctionaltheory[2,5,6,22,23],and
even the theory ofgeneral relativity, wheresystems with PDMare de-
scribedwithanotheroriginofcreation,differentfromthecondensed
matterone[24].Hence,quantummechanicstheoristscontinuetoin-
vestigatethesubjectofPDMHamiltoniansasafundamentalissue
which isstillfarfrombeingentirely resolved.Inaddition totheexact
solvability[25–32],thePDMproblemraisesseveralmajorconceptual
questions,like theboundaryandcontinuity conditions[11,12,33],the
Galileaninvarianceofthetheory[14,33]andtheimplementationof
path-integraltechnique[34,35].
Whiletheinterest insearching exactlysolvablequantum PDMsys-
temsisstillincreasing[25,36],themostinterestingquestionremains
howtoorderthemassoperatorwithrespecttothemomentumop-
eratorwhenitcomestobuildingtheKEOoftheHamiltonian.The
dilemmadoesnotseemobviouswhenthemassisconstant,butwhen
thislatterbecomesafunctionoftheparticleposition,thebewilder-
mentappearsinthenon-commutativitybetweenthemassandthe
momentumoperators.Inotherwords,whenaparticleisendowed
withavariablemass(position-dependentmass),thislatterwillnot
commutewiththeparticle’smomentumoperatoranymore.Thus,
theHermiticity conditionoftheHamiltonian,seemstobeinadequate
todefinitelyspecifyauniqueformfortheKineticEnergyOperator
(KEO)withaposition-dependenteffectivemass.Suchadilemma
bringsusbacktorethinkaboutthegeneralrulehastobefollowedin
associating operatorsin quantum-mechanicalwith respectto theclas-
sicalquantities(seeforexampleRef.[37]).Forthispurpose,many
criteriahavebeenadoptedinordertodefinetherightformofthe
KEO, such asthecurrent-density conservation andtheagreement be-
tweentheoreticalandexperimentalresults,butnoneoftheproposed
forms ofKEO’shasbeen hithertoabletoprovide a clear-cutanswerto
thisquestion.Inthisregard,Ref.[38]canbereviewedtocheckmore
detailsoutaboutthatissue.Moreover,manycontributionstothis
subjectarealsoworthyofnote,such asBenDanielandDuke[7],Zhu
andKroemer[11],DutraandAlmeida[31],GoraandWilliams[10],
VonRoos[14],Trabelsietal.[38]andMorrowandBrownstein[12].
Therein,several formsofthe KEOareshown to beusefulin lotofap-
plications,asforexampleLi-Kuhn[8],BenDanielandDuke[2],and
soon.However,thecommondenominatorofalltheseresearchesis
the portrayal of the KEO’sasdifferent orderingsofpowers of the mass
andthemomentumoperators.
Butontheotherhand,itwasprovedbyC.Benderworks[39–47]
that there is no physical reason to restrict quantum mechanics to Her-
mitianHamiltonianonly.Although,thisworkproposestoexplore
otheruncustomarywaystobuildthequantumHamiltonianKEOof
aPDMsystemwithmoresimplicityandgenerality.Infact,wehave
recently developed a novel approach for constructing the quantum op-
erator associated to the non-relativistic classical kinetic energy, within
theSchr¨odingerequation[48].Theideabehindliesonthefactthat
DOI:10.4236/jamp.2019.750681014JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
thecoefficient-functionsofthemomentumintheKEO,maytake,a
priori, an arbitrary form.As a matter of fact, inthe literature[49–52],
these functionshavebeenalways assumed to bepowers of thevarying
mass.Moreover, incompliancewithrecentoutcomesofnon-Hermitian
quantum mechanics, therequirement thattheHamiltonian be Hermi-
tian hasbeen dropped in favorof thepseudo-Hermiticitycondition.In
this work, we intend to use this method to generate some new solvable
modelswithPDM,onthebasisoftheformalismofsupersymmetric
quantummechanics(SUSYQM)andthenotionofshapeinvariance.
Thepaperisorganizedasfollows:inSection1,weoutlinethenew
wayweshallusetoapproachtheproblemofaquantumHamiltonian
withPDM.InSection2,weemploytheSUSYapproachrelativeto
thisHamiltonianalongwiththeshapeinvariancecondition,inorder
togenerateaclassofexactlyPDMsolvablemodelswithaharmonic-
oscillator(HO)-like spectrum.Inthefinalsectionwe giveourconclu-
sion.
2.OutlinesoftheNewApproachto
PDMSE
In Ref [48], we have addressed the problem of a one-dimensional PDM
Schr¨odingerequation,bysuggestingaKEO
ˆ
Thavingtheform:
ˆ
T=
1
2
αxx) ˆx)(1)
where α,βand γarearbitrary realfunctions oftheposition operator,
andnotnecessarypowersofthemassfunction,asitisassumedin
theliteraturesofar.Butofcourse,thesefunctionshavetoverifythe
condition:
α(x)β(x)γ(x) = 1/m(x)(2)
wherem(x)istheposition-dependentmassoftheconsideredsystem.
Then,givenarealpotentialinteractionVx),theresultingHamil-
tonianisingeneralnotHermitian.Asamatteroffact,ithasbeen
shownthatthereisnophysicalreasontorestrictquantummechanic-
stoHermitianHamiltonianonly[53–56].Indeed,aquasi-Hermitian
Hamiltonianmaybringsbacktoarealspectrum.Asprovenin[56],
suchanassumptionrequiresthatthereexistsaHermitian,positive-
definiteandboundedoperatorη,allowingtowrite:
ˆ
H
= η
ˆ
1
(3)
where
ˆ
H
isthestandardHermitianadjoint of
ˆ
H.Hence,considering
anoperatorη(x) = γ(x)(x)allowstofulfillthecondition(3),and
makes
ˆ
Hquasi-Hermitian,providedthatηisboundedandα(x)and
γ(x)havethesamesign.Inthatcase,itwillbe possibletorelatethe
original
ˆ
HtoaHermitiancounterpart,letitbe
ˆ
H
h
,satisfying:
ˆ
H
h
= η
1/2
ˆ
1/2
=
1
2
ωx) ˆx) ˆx)+Vx)(4)
withω=
αγ.Eventually,
ˆ
Hand
ˆ
H
h
areiso-spectral,andtheeigen-
vectors|ψioftheHamiltonian
ˆ
Hwillbeexpressedasafunctionof
thesolutions|ψ
h
ioftheHamiltonian
ˆ
H
h
as:
|ψi= η
1/2
|ψ
h
i(5)
DOI:10.4236/jamp.2019.750681015JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
Atthisregard,theHamiltonian
ˆ
H
h
canbefactorizedusingtwo
intertwiningoperators,
ˆ
Aand
ˆ
A
as
ˆ
H
h
=
ˆ
A
ˆ
A+E
0
(6)
whereE
0
istheground-stateenergyof
ˆ
H
h
and
ˆ
Aand
ˆ
A
aredefined
by
ˆ
A=
1
2
p
βxx)iWx),
ˆ
A
=
1
2
ωx) ˆp
p
βx)+iWx)
(7)
with Wx) the corresponding real superpotential.Thus we can estab-
lishthat
ˆ
H
2
=
ˆ
H
1
+
~
2
2
p
mx)
(
W
0
x)
Wx)
p
mx)
+
2
~
"
˜
Vx)
Wx)
+Wx)
#)
0
(8)
where
ˆ
H
1
and
ˆ
H
2
aretheSUSYpartnerHamiltonians,definedas
ˆ
H
1
=
ˆ
H
h
=
ˆ
A
ˆ
A+E
0
,
ˆ
H
2
=
ˆ
A
ˆ
A
+E
0
(9)
Then, by imposing the integrability condition within the SUSY QM
theory,thatistheshapeinvariance(SI)ofthepotentialV(x),we
showthattheeigenvalueproblemcorrespondingtotheHamiltonian
(4)cansolvedexactlyifthefollowingconditionisfulfilled:
~
2
2
p
m(x)
(
W
0
(x)
W(x)
p
m(x)
+
2
~
"
˜
V(x)
W(x)
+W(x)
#)
0
= F(x)(10)
withF(x)thefunctiongiven,accordingtotheSIcondition,as
F(x) = V(x,f(a
1
))V(x,a
1
)+R(a
2
)(11)
wherea
1
isingeneralthesetsofparametersofthepotentialV(x).
Inaddition,thefunctionβ(x),seenasthefreefunctionoftheHamil-
tonian
ˆ
H
h
,isrelatedtothepotential
˜
Vandthesuperpotentialas
β(x) =
1
W(x)
p
m(x)
exp
(
Z
x
p
2m(y)
~
"
W(y)
˜
V(y)
W(y)
#
dy
)
(12)
where theintegration constant has beenset toone andthis choice has
no restriction.Thus, given a massfunction m(x) and asuperpotential
W(x),Equation(10)canbe usedtofixthepotential V(x),forwhich
thecorrespondingquantumproblemisexactlysolvable.ThenEqua-
tion (12)allows todetermine thecorresponding KEO.Notice thatthe
groundstatewavefunctionψ
0
(x),isdeterminedfromtheequation
ˆ
0
= 0.Thiswouldgive
ψ
0
(x) = C
p
β(x)m(x)exp
(
Z
x
p
2m(y)
~
W(y)dy
)
(13)
withCanormalizationconstant.Andletusrecallthatψ
0
(x)should
benormalizableinordertohaveanunbrokenSUSY[57].Onthe
otherhand,weshouldnotethatEquation(10)isactuallyaRiccati
equationforW(x),soitcanbealwaystransformedtotheasecond
orderlineardifferentialequation.
Inthenextsection,weshallstartfromEquation(10)inorderto
generateaclassofsolvablesystemshavingaHO-likespectrum.
DOI:10.4236/jamp.2019.750681016JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
3.AClassofPDMSystemswitha
HO-LikeSpectrum
Inthissection,weshallbeinterestedintheclassofsystemsexhibit-
ingaHO-likespectrum.Suchaclasscorrespondstothefollowing
commutationrelation:
h
ˆ
A,
ˆ
A
i
= κ(14)
withκarealpositiveconstant.Inthatcasethespectrumofthe
Hamiltonian
ˆ
H
h
isgivenas
E
n
=
κn+E
0
,n= 0,1,2,...(15)
Ontheotherhand,accordingtoEquation(10),thisrelationmeans
thatthepotential
˜
V(x)canbewrittenasafunctionofthemassand
thesuperpotentialas:
~
2
2
p
m(x)
(
W
0
(x)
W(x)
p
m(x)
+
2
~
"
˜
V(x)
W(x)
+W(x)
#)
0
= κ(16)
orequivalently
˜
V(x) =
~
p
2m(x)
W
0
(x)W
2
(x)+
κW(x)
~
Z
x
p
2m(y)dy(17)
Furthermore,itisalwayspossibletoreexpressthesuperpotential
throughanewfunctionUas:
W(x) =
1
2~
Z
x
p
m(y)[κ+U(y)]dy(18)
InsertingthisfrominEquation(17)wecanreadilyshowthatthe
potential
˜
V(x)takestheform
˜
V(x) =
1
2
[κ+U(x)]+
1
2~
2
(
κ
2
Z
x
p
m(y)dy
2
Z
x
p
m(y)U(y)dy
2
)
(19)
Now,thisequationwillbeusedtogeneratesomeconcretemodels
exhibitingaHO-likespectrum.
Case1:m(x) = m
0
1+ax
2
Withm
0
aconstantmassandaaparameterwhichcharacterizes
thedeviationfromtheusualHO.Thisformofthemasshasbeen
extensivelystudied inthe literature, asit isrelevantforthedescription
oftheGaAs/Al
x
Ga
1x
A
s
system[58].Nowbychoosing
U(x) =
κax
2
1+ax
2
(20)
wefind
V(x) =
κax
2
2(1+ax
2
)
+
κ
2
2~
2
a
x
p
1+ax
2
ax
(21)
where we haveomitted theconstantterm ofthe potential.This poten-
tialisroughlyadouble-wellpotential,thusitmayhaveapplications
DOI:10.4236/jamp.2019.750681017JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
in molecular physics, for instance.Thevariations of the potential (21)
areshown inFigure1forsomeillustrative valuesofthefreepotential
parameter.Anotherinterestingpotentialcanbeobtainedbytaking
U(x) =
κ/2
1+ax
2
(22)
Thisgives
V(x) =
κ
2(1+ax
2
)
+
κ
2
8~
2
x
2
1+ax
2
+
2x
a
p
1+ax
2
ax
(23)
ThelastpotentialissketchedinFigure2fortwoillustrativevalues
ofitsfreeparameter.
Figure1.Thepotential(21)inunitsofκ/2,asafunctionof
axfor
κ/~
2
a=0.05(thickline),κ/~
2
a=0.2(thinline)andκ/~
2
a=1(dashed
line).
Figure2.Thepotential(23)inunitsofκ/2,asafunctionof
axfor
κ/~
2
a=0.02(thickline)andκ/~
2
a=0.5(thinline).
DOI:10.4236/jamp.2019.750681018JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
Case2:m(x) = m
0
/
1+ax
2
witha>0
Herem
0
isaconstantmass.ThisformrecallstheMathews-
Lakshmanan(ML)oscillator,amodelwhichisattractingconsider-
ableinterestrecentyears(seeforinstance[59,60]).Thenbyputting
U(x) = 0,weconstructthefollowingpotential.
V(x) =
m
0
κ
2
2a~
2
2
ax
(24)
Forax1thispotentialbehaveslikeaharmonicpotential.Thus
itmaybeusedinphysicalsituationswhenadditionalnonharmonic
termsshouldbeincluded.Thevariationsofthepotential(21)are
represented inFigure 3 fortwo illustrative values of itsfree parameter.
Figure3.Thepotential(24)inunitsofκ,asafunctionof
axfor
m
0
κ
a~
2
=1(thickline)and
m
0
κ
a~
2
=2.4(thinline).
LetalsopointoutthatwhenU(x) = 0,Equations(18,19),yield
V(x) = W
2
(x)
κ
2
(25)
andbyvirtueofEquation(12),thisleadstoβ(x) = 1/
p
m(x).Hence,
theresultingKEOcorrespondstoLi-Kuhnordering[8],thatis:
ˆ
T=
1
2
ˆm
1/4
ˆp
1
ˆm
ˆpˆm
1/4
(26)
Case3:m(x) = m
0
e
2ax
/(1+e
ax
)
4
Herem
0
isaconstantmassandaarealparameter.Suchkindof
massdistributionisextensivelyusedinthephysicsofsemiconductor
quantumwell structures[61],andmay haveapplicationsinmolecular
physics[62].Thenbychoosing
U(x) =
b
1+e
ax
(27)
weobtainthefollowingpotential
V(x) =
b
2(1+e
ax
)
+
m
0
2a
2
~
2
(1+e
ax
)
2
"
κ
2
b
2
4(1+e
ax
)
2
#
(28)
DOI:10.4236/jamp.2019.750681019JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
Suchtypeofpotentialisveryrelevantespeciallyforstudiesinmolec-
ularphysics.Inparticularforb=0,i.e.U(x)=0itreducestoa
generalizedWoods-Saxonpotential.Figure4showsthevariationsof
potential(28)forsomeillustrativevaluesofitsfreeparameters.
Figure4.Thepotential(28)inunitsofκ,asafunctionof
axforb/κ=8
and
m
0
κ
a
2
~
2
=1(thickline),b/κ=12and
m
0
κ
a
2
~
2
=1(thinline),b/κ=20
and
m
0
κ
a
2
~
2
=1(dashedline),b/κ=20and
m
0
κ
a
2
~
2
=0.002(dottedline).
Case4:m(x) = m
0
/cosh(ax)
4
Herem
0
isaconstantmassandaarealparameter.Thismass
functiondepictsasolitonicprofileandpresentsaformfamiliarin
effectivemodelsofcondensedmatterandlowenergynuclearphysics
[32].Ithasbeenalsousedtodescribevariablediffractionscoefficient
innonlinearmedia[63].Thenthechoice
U(x) =
b
cosh
2
(ax)
(29)
resultsinthefollowingpotential
Figure5.Thepotential(30)inunitsofκ,asafunctionof
axfor
b/κ=0.2and
m
0
κ
a
2
~
2
=8(thickline),b/κ=52and
m
0
κ
a
2
~
2
=1(thinline),
b/κ=50and
m
0
κ
a
2
~
2
=2(dashedline).
DOI:10.4236/jamp.2019.750681020JournalofAppliedMathematicsandPhysics
A.Dhahbietal.
V(x) =
b
2cosh
2
(ax)
+
m
0
tanh
2
(ax)
2a
2
~
2
κ
2
4b
2
9
4b
2
9cosh
2
(ax)
b
2
9cosh
4
(ax)
(30)
This hyperbolic potentialyields in general a doublewell interaction.
Suchkindofpotentialallowsstudyingsituationsinvolvingbounds-
tates andparticle tunnelingthrougha barrier.Inparticularfor b= 0,
i.e.U(x)=0itreducestoatanh
2
(ax)potential.InFigure5we
representthevariationsofpotential(30)forsomeillustrativevalues
ofitsfreeparameters[64].
4.Conclusion
Tosumup,thisworkpresentsanewclassofexactlysolvablemod-
els,endowedwithaposition-dependentmass,exhibitingaharmonic-
oscillator-likespectrum.Thisclasshasbeengeneratedonthebasis of
anewapproach,recentlydevelopedbyusforthequantumoperator
relativetotheclassicalnon-relativistickineticenergy,andusingthe
formalism of supersymmetricquantum mechanics.The latter involves
factorizing thecorresponding Hamiltonianin termsof intertwiningop-
erators,alongwiththerequirement of theshapeinvariancecondition.
SeveralconcreteexamplesofPDMsystemsbelongingtothisclasswere
discussed.
Acknowledgements
The authorsgratefully acknowledgeQassim University, represented by
theDeanshipofScientificResearch,onthematerialsupportforthis
researchunderthenumber(1671-ALRASSCAC-2016-1-12-S)during
theacademicyear1437AH/2016AD.
ConflictsofInterest
Theauthorsdeclarenoconflictsofinterestregardingthepublication
ofthispaper.
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