CPT-Frames for PT-symmetric Hamiltonians

Transcription

1 -a fo P-ytc Haltoa Hua-X Cao Zh-Hua Guo Zhg-L Ch Collg of athatc ad Ifoato Scc Shaax Noal Uvty X'a 76 Cha al: Abtact: P-ytc quatu chac a altatv foulato of quatu chac whch th athatcal axo of Htcty (tapo ad coplx cojugat) placd by th phycally tapat codto of pac-t flcto yty (P-yt A Haltoa H ad to b P-ytc f t cout wth th opato P h y pot of P-ytc quatu thoy to buld a w potv dft poduct o th gv Hlbt pac o that th gv Haltoa Hta wth pct to th w poduct h a of th ot to gv futh athatcal dcuo o th thoy pcally cocpt of P-fa -fa o a Hlbt pac ad fo a Haltoa a popod th xtc ad cotucto a dcud Ky wod: Haltoa P-yty P-fa -fa PACS ub(): w 365G 6Lj Itoducto It wll-ow that a clacal quatu chac yt th t voluto of th yt dcbd by th Schödg quato of a Haltoa H whch a dly-dfd Hta opato th Hlbt pac K L ( ) (wh C ) h Htcty of H u that th pctu of H al It aabl that th Htcty of a Haltoa H ot cay fo th alty of th pctu Bd ad Bottch obvd [] that th alty of pctu of a Haltoa H du to P yty of H Sc P-yty thoy ha ay ally phycal bacgoud ad applcato t ha b wdly dcud ad dvlopd [-6] ad ay cofc o th thoy hav b hld o o ct wo o th topc pla f to [7-5] athatcally th paty P ad th t val a dd th opato dfd a follow ( Pf )( f ( ( f )( f ( f L ( ) () wh th ba a th coplx cojugat ad dot th al fld o coplx fld C h caocal poduct o th Hlbt pac L ( ) gv by f g f g f ( g( Claly P la ad at-la (cojugat la) atfyg P I P P () A Haltoa H ad to b P -ytc f t cout wth P [ H P ] HP PH (3) Sc ( P ) I w that P -yty of H quvalt to H P : ( P) H ( P) H (4) vy Hta Haltoa ha al pctu but ot vy P -ytc Haltoa

2 -a fo P-ytc Haltoa ha al pctu It wa povd [9] that f H ha ubo P -yty th th gvalu of H a all al h y pot of P-ytc quatu thoy to buld a w potv dft poduct o th gv Hlbt pac o that th gv Haltoa Hta wth pct to th w poduct Hc th Hlbt pac P -ytc quatu chac foulatd a a la vcto pac wth a dyac poduct h a of th ot to gv o futh athatcal dcuo o th thoy pcally cocpt of P-fa -fa o a Hlbt pac ad fo a Haltoa wll b popod ad th xtc ad cotucto of -fa hould b dcud P-a ad P -yty I th cto bad o th Bd da w wll toduc ad dcu th abtact P -fa o a coplx Hlbt pac K ad th P -yty of a opato H a Hlbt pac I what follow w u A to dot th Dac adjot of a opato A hu A Hta f ad oly f A A Dfto Lt P b cotuou opato o a coplx Hlbt pac K uch that () P la ad ot th dtty opato I ad cojugat la; () P I P P h th pa { P calld a P -fa o K A opato H : D( H ) K ad to b P -ytc f t cout wth th opato P I th ca w alo that H ha P -yty Not that H cout wth th opato P a that P ( D( H )) D( H ) ad PHx HPx x D( H ) (5) o a P -fa o K ad a opato H : K D( H ) K df H P PHP h H P -ytc f ad oly f H P H Claly fo vy polyoal p ( wth al coffct th opato p (P) P -ytc o a la ubpac Y of K w u P ( Y K) to dot th t of all P -ytc opato fo Y to K Claly P ( Y K) a al la pac ad P ( K) : P( K K) a al utal algba Lt K b a coplx Hlbt pac wth a othooal ba Df { } P x x x x43 x34 x65 x56 x x th { P a P -fa o K h how that vy Hlbt pac ha alway a P -fa oov df opato P : K K K K a follow: wh x P( x ( y ( x ( x ( x K K x K Claly I K c P I K wh c x x c h { P a P -fa o K K o ay dly dfd la opato H : K D( H ) K put ~ H H H H H ~ ~ ~ h PHP PHP H ad o H ~ ~ ~ P -ytc f ad oly f H H f ad oly f H H A la agut alo vald fo ft doal ca hu fo vy coplx atx c c H [ h j ] th opato H ~ P -ytc o C C

3 -a fo P-ytc Haltoa 3 t f ad oly f H ytc: H : [ hj] [ hj ] wh I c P I c wth I th ut atx ad c ( x x ( x x Nxt ult gv th pctal popt of a P -ytc opato call that th olvt (H ) ad th pctu (H ) of a dly dfd la opato H : K D( H ) K a dfd by ( H ) { : ( I K H ) B( K)} ( H ) \ ( H ) Popoto Lt { P b a P -fa o K ad H : K D( H ) K b dly dfd If H ha P -yty th () ( H ) ( H ) ; () p( H ) p( H ) Poof call that th olvt ad th pctu of H : K D( H ) K a dfd a follow: ( H ) { C : ( I H ) K( I H ) B( ( I H ) K)} ( H ) C \ ( H ) wh B( ( I H ) K) th t of all boudd la opato fo ( I H ) to K Whl th pot pctu p (H ) of H dfd a p( H ) { C : ( I H ) {}} th t of all gvalu of H Lt (H ) h ( I H ) K( I H ) B( ( I H ) K) Sc I H P ( I H ) P ad P a cojugat la hoooph fo K oto K w that ( I H ) ( I H ) K ad ( I H ) P ( I H ) P B( ( I H ) K) h how that (H ) So ( H ) ( H ) ad th ( H ) ( H ) Alo t follow fo I H P ( I H ) P that ( I H ) ( I H ) Hc p( H ) p( H ) h poof copltd o Popoto w ow that th pctu of P -ytc opato caly al S popoto 3 blow Dfto Lt { P b a P -fa o K ad H : K D( H ) K If H P -ytc ad vy gtat of H alo a gtat of P th w ay that th P -yty of H ubo o H ha ubo P -yty Popoto If H ha ubo P -yty th th gvalu of H a all al Poof Lt a b ay gvalu of H wth gtat f h Hf af hu th P -yty of H pl that HPf PHf apf () Sc th vcto f alo a gtat of P th xt a coplx ubb uch that Pf bf By () w hav abf HPf apf abf Sc b ad f a a h how that a al h poof copltd Gally w fo th poof abov that wh H ha ubo P -yty f a gvalu a of H ha a gtat whch alo a gtat of P th a ut b al Popoto 3 Lt

4 -a fo P-ytc Haltoa 4 a b x x H ( a b c d C) P c d y y h { P a P -fa o C ad th followg cocluo a vald a b () H H : (coplx cojugat) c d t t t () H : ( H ) ( H ) (t dot th tapo of a at; H : H H a b (3) H H H H H H H ( a d ) b d t a b (4) H H H H H b d a b (5) H P H PH HP H b a a b (6) H P H H H ( a ) b a P a b (7) H H H H H ( a b ) b a Cod th followg atc: H H H o Dfto ad Popoto 3 w ow that wth pct to th P -fa { P gv by Popoto 4: () A al ytc atx ot caly P -ytc g H () A P -ytc atx ot caly ytc g H (3) A Hta atx ot caly P -ytc g H 3 xapl Lt { P b th P -fa gv by Popoto ad H () wh a ozo al ub It ay to chc that H P -ytc ad o-hta Wh H ha two al gvalu: co co ( ) wth th copodg gtat wh - Claly P ad o H ha ubo P -yty Wh H ha two o-al gvalu: co hu H ha o ubo P -yty xapl Lt

5 -a fo P-ytc Haltoa 5 P a H θ z y x z y x wh ad a a ozo al ub It ay to chc that } { P bco a P -fa o 3 C ad H P -ytc ad o-hta Wh H ha th al gvalu: a ad co co ( ) wth th copodg gtat a ad wh a Claly a a P ad P ad o H ha ubo P -yty Wh H ha a al gvalu a ad two o-al gvalu: co hu H ha o ubo P -yty xapl 3 Lt θ θ θ θ H P u z y x u z y x () Wh a ozo al ub It ay to chc that } { P bco a P -fa o 4 C ad H P -ytc ad o-hta It ca b coputd that wh ) ( H ha fou al gvalu: co co ( ) wth th copodg gtat wh Sc P ) ( H ha ubo P -yty Wh o H ha two o-al gvalu: co o co

6 -a fo P-ytc Haltoa 6 hu H ha o ubo P -yty xapl 4 Lt x x H P (3) y y wh a ozo al ub ( ) uch that th quc { } ad { } a boudd Put H H P P h H P a boudd la opato o th Hlbt pac: ( N) {( x x x ) : x C( ) x } whl a boudd cojugat la opato o th a pac atfyg P I P P HP PH Claly { P bco a P -fa o ad H P -ytc o xapl w that wh ( ) H ha al gvalu: wh co co ad wth th copodg gvalu: wh ( ) wh ( ) ( ) ( ) ( ) ( ) ( ) ( ) Claly a gtat of P fo gvalu Hc H ha ubo P -yty I th ca wh fo o H ha two o-al gvalu: co hfo H ha o ubo P -yty xapl 5 o vy f L ( ) df h P ( Pf )( x f ( x ( f )( x f ( x ( P f )( x f ( x ( P f )( x f ( x I P I P P ( ) P P ad o th pa { P ad

7 -a fo P-ytc Haltoa 7 { P ( ) a all P -fa o L ( ) Claly P : PP P P It ay to that th Haltoa 3 H ( pˆ qˆ ) ( xˆ yˆ ) xˆ yˆ 3 ( ) both P -ytc ad P -ytc but ot P -ytc xapl 6 o vy f L ( 3 ) df ( Pf )( x y z) f ( x y z) ( P f )( x y z) f ( x y z) ( P f )( x y z) ( P f )( x y z) 4 f ( x y z) f ( x y z) ( P f )( x y z) 3 ( f )( x y z) f ( x y z) f ( x y z) h th pa { P ad { P ( 34 ) a all P -fa o L ( 3 ) Claly P PP P 3 P3 P P P P P3 It ay to that th Haltoa H ( pˆ qˆ ˆ ) ( xˆ yˆ zˆ ) xˆ yˆ zˆ ( ) both P -ytc ad P ( 3 ) -ytc but ot -ytc c H P 4 P 4 H 3 -a I th cto bad o th Bd da [9] w wll toduc ad dcu th abtact -fa o a coplx Hlbt pac K ad fo a opato H a Hlbt pac Dfto 3 Lt { P b a P -fa o a Hlbt pac K ad C a la opato o K uch that () PC C I ; () PC potv dft wth pctv to th poduct o K PCx x ( x K) ; PCx x x h th tpl { C P ad to b a -fa o K If addto C cout wth H : K D( H ) K th th tpl { C P ad to b a -fa fo H Popoto 3 Lt { C P b a -fa o K () h foula PC df a potv dft poduct o K K calld a - poduct o K oov th o ducd by th w poduct atf CP PC K () Wth pct to -poduct th adjot opato of a dly dfd la opato A : K D( A) K A ( PC) A ( PC) wh A dot th uual adjot of A wth pct to th ogal poduct o K (3) A ( ) A ( ) wh A : A (calld th tapo of A ) (4) A A PCA A PC ;wh A A A A [ A PC] (5) Wh A A A A [ A ] (6) If { C P a -fa fo a P -ytc opato H : K K ad h ( PC) H ( PC) th h h f ad oly f H ytc ( oly f H -Hta H H Poof () Claly D( A ) D( A) w coput that P 4 H H ) f ad

8 -a fo P-ytc Haltoa 8 A A PC A K ( PC) PC A A PC So A ( PC) A ( PC) ad () follow Sc A A ad I w A A ad th () pl that A ( PC) A ( PC) ( PC) A ( C P) ( ) A ( ) Cocluo (4) ad (5) ca b obtad by () ad (3) pctvly o (6) w ot that H P H pl that H PHP hu H H f ad oly f H PHP Sc PC ( ) PC P P C C HC CH w gt h ( PC) H ( PC) h ( PC) ( PC) H ( PC) ( PC) ( PC) ( PHP)( PC) Hc h h f ad oly f H PHP f ad oly f H H By () w that H H ( ) H H ( ) H ( ) H ( ) H H h poof copltd Coollay 3 Lt opato P b a q () ad { C P a -fa fo a P -ytc opato H : D( H ) L ( ) h : PC ( ( ( : ) (4) L H -Hta f ad oly f H ytc ( H : H H ) ad that ca H la to th opato h ( PC) H ( PC) wth h h whl dfft gvalu of H hav -othogoal gtat Coollay 3 Lt { C P b a o C wh Df () t (coplx cojugat) C PC ad PC th a potv dft poduct o C () h adjot of a atx (a a opato) A C C : wth pct to K A ( PC) A ( PC) wh A ( A) t t (3) A : A A A ( ) A ( ) (4) A A PCA A PC ;wh A A A A [ A PC] (5) Wh A A A A [ A ] (6) If H (C) ytc ( H H ) ad P -ytc uch that HC CH th H -Hta ad la to Hta atx h ( PC) H ( PC) o dcu th xtc of -fa o L ( ) w d th followg la La 3 Lt f b Lbgu auabl fucto o ( ) wth f( d x h th f ( alot vywh covgt o ad t u fucto Lbgu tgabl o wth ho 3 Lt ( f ( f P b a q () ad C a boudd la opato o L ( )

9 -a fo P-ytc Haltoa 9 dfd by ( C )( x ( dy (5) wh c ( x a tgal l h () PC f ad oly f x x a x y () C I wh (3) If } N f ad oly f ( z) f ( z ( dy L ( ) ad f ( z x z { a othooal ba fo L ( ) wth ( ( ( a z fo all ad all x uch that ad x ( ) ( ) x y th { C P a -fa o L ( ) Poof () By q (5) w coput that L ( ) x ( dy x ( ( PC )( dy ad ( )( x ( dy hu PC f ad oly f x ( dy x ( x f ad oly f () L ( ) x a x y w hav ( C )( x ( dy ( C )( z) dy ( L ( ) a ad o wh f ( z z x Hc ( dy z x ( dy ( f ( z dy z x C I f ad oly f ( C )( z) ( z) L ( ) a z h lat quvalt to ( z) f ( z ( dy L ( ) a z (3) L ( ) hu th codto that La 3 pl that yld that ( ( ( d y ( ( ( d y a x a x x )

10 -a fo P-ytc Haltoa ( C )( x ( dy vald Sc L ( ) w hav PC w that PC potv O th oth had PC L ( PC )( ( ( CP )( ( ( ( ( ( ( ( ( ( ( ( ( dy x ( ( ) ( dy ( ( ( ( ) ( dy ( ( ( ) dy ( ( ( ) dy [ ( ( ] dy ( ( L L ( th pl that PC f ad oly f ( ) L f ad oly f L c pa{ } N L ( ) Now w hav povd that PC potv dft By ug th aupto that ( ( ( w gt that P C P C C P ad o P C P C C P L ( ) hat [ C P] Sc { N a othooal ba fo L ( ) q (6) pl that C fo all L ( ) So C I h poof copltd Nxt tho about th xtc of a -fa fo a ubo P -ytc Haltoa ho 33 Lt P b a q () L ( ) ad H b a la opato o ad hav ubo P -yty who oalzd gtat { } gat h th xt uch that : atfy P ad H fo all If addto th (6) } ( ) : ( x ) ( ( ) (7) () h fucto x ( ( a dfd o () h foula ad auabl ( C )( x ( dy (8)

11 -a fo P-ytc Haltoa df a la opato C o (3) h tpl { C P a -fa fo th opato H ad th gtat { } N a -othooal Poof Sc a gtat of H th a cotat uch that H h ubo P -yty of H pl that th a cotat c uch that P c h fact that ( P) I pl that c hu th xt uch that Claly a all al : atf P () h codto ad H u ad th ( ) ( ) d d x y x y ( d x h how that ( ( covgt alot vywh o t u fucto c ( x a dfd ad auabl o () Lt h by hu x ( dy w hav ( d x ( ( d c Popoto pl that y ad thfo c ( x ( dy a x h how that th fucto C (8) wll-dfd o Sc w gt that ( C )( pcally wh ( ( ( d y ( ( ( d y a x ( dy (7) u that ( C )( ( x hfo La 3 yld that ( ( ( dy ( ( dy ( ) ( ( ( ( dy a x hu C ( ) ( ) h how that th foula (8) df a la opato C o (3) w coput fo () that C C( ) ( ) C ( ) HC H ( ) ( ) CH ( PC) ( ) P ( ) C Sc pa{ } N w hav C I HC CH ad P C C P Sc w coput that

12 -a fo P-ytc Haltoa PC ( PC )( ( P( ( ( dy ( ( ( dy ( ( dy ( ( hu w coclud that PC potv Lt PC h fo vy w obta that It follow fo P ( ( ( ( ( ( ( ( ( ( ) ( P pa{ } that P h how that { C P N fo H oov c P C () PC ( PC )( ( ( CP )( ( ( C )( ( ( ) ( ( L w hav So th gtat } a -othooal h coplt th poof { I th ft doal ca w hav th followg ho 34 Lt { P b a P -fa o a ft doal Hlbt pac H a la opato o havg ubo P -yty h H ha gtat ( wh d d atfyg P H ( If addto pa{ : d} ad d d d Cc ( ) c ( c c c d ) C (9) th { C P a -fa fo H atfyg ( f ad oly f ( ) : P ( ) ( () Poof Sla to th bgg of th poof of ho 33 w that H ha gtat ( atfyg P H ( Nxt w au that pa{ : d} h th foula df a la opato C o wth ( ( C ) Suppo that { C P a -fa fo H atfyg all d w hav h fo

13 -a fo P-ytc Haltoa 3 P Covly w uppo that () hold Sc pa{ : d} ad P H C () ( P( ) C ( ) PC ( ) ( ) w gt that pl that PCx x HC CH PC ad C I d x x ( ) d d o vy x x q () P x x ( ) ( ) x So PC a potv dft opato o Hc { C P a -fa fo H oov ( ) ( ) PC P ( ) hu h poof copltd a 3 Ud codto of ho 34 f th opato H ytc H : H H th fo th alty of w gt that ( ) P PH H P P H P ( ) Hc ( ) whv Nxt xapl a applcato of ho 34 xapl 3 Lt H P b a xapl wth h H ha gvalu co co wth copodg gtat co co Claly P ad o H ha ubo P -yty Obvouly Put co th } a ba fo { d C atfyg P H P P It follow fo ho 34 that th opato: C : C C C a a a a gv a -fa { C P fo H uch that { } a othooal ba fo th Hlbt pac C Sc C w that C wh PC t A ay coputato how that ta c C () c ta co h jut th C opato obtad [9] Latly c H both ytc ad P -ytc Coollay 3(6) pl that H -Hta: H H ad la to th Hta atx h ( PC) H ( PC) oov ud ba } th t {

14 -a fo P-ytc Haltoa 4 opato H C hav th followg atx ptato: h followg ca b chcd aly ho 34 Lt { C P } b a CP -fa o Hlbt pac K fo ad C C C P P P K K K h () { C P a -fa o th Hlbt pac K () If addto { C P } ( ) a CP -fa fo opato H o K pctvly th { C P a -fa fo th opato H : H H o K (3) If H P -ytc ( ) th H : H H P -ytc xapl 3 Lt H H x x P P ( ) y y ta c ta c C C c ta c ta wh a ozo al ub atfyg ( ) Df th followg opato o C C : H H H P P P C C C o xapl ad ho 5 w that { C P a -fa fo H oov H P -ytc Sc H ha fou gvalu: ( a xapl 3) th copodg gtat a a follow: ( a xapl 3) whch a th gtat of P fo gvalu Hc H ha ubo P -yty Claly H ytc ad o -Hta (Coollay 3(6)) Acowldgt h ubjct wa uppotd by th Natoal Natual Scc oud of Cha ( ) th udatal ach oud fo th Ctal Uvt (GK6) ad th Iovato oud fo Gaduat Poga of Shaax Noal Uvty (CXB4) fc [] Bd C Bottch S 998 al pcta o-hta Haltoa havg P-yty Phy v Ltt [] Bd C Body D C Coplx xto of quatu chac Phy v Ltt [3] Bd C Body D C Jo H t al 7 at tha Hta quatu chac Phy v Ltt [4] Bd C Klvay S P al of patcl wth dfft a P-ytc quatu fld thoy Phy v Ltt [5] Guo A Salao G J 9 Obvato of P - ytc bag coplx optcal

15 -a fo P-ytc Haltoa 5 pottal Phy v Ltt [6] ut C a K G Obvato of Paty-t ytc optcal Natu Phyc 9 6 [7] Bd C Jo H 5 Dual P-ytc quatu fld tho Phy Ltt B [8] Bd C Ch J H lto K A 6 P-ytc vu Hta foulato of quatu chac J Phy A [9] Bd C 7 ag of o-hta Haltoa p Pog Phy [] Bd C Hoo D W 8 Cojctu o th aalytcty of P -ytc pottal ad th alty of th pcta J Phy A [] Bd C Body D C Hoo D W 8 Quatu ffct clacal yt havg coplx gy J Phy A [] Bd C Hoo D W 8 xact opctal pa of P-ytc Haltoa J Phy A [3] Bd C Hoo D W g P N t al Pobablty dty th coplx pla A Phy [4] Bd C ah P D P-Syty ad cay ad uffct codto fo th alty of gy gvalu Phy Ltt A [5] oyv N No-Hta quatu chac (Cabdg Uvty P) [6] Cho H Wu J D P-Syty Sctfc J ath -6 [7] Chog Y D G L ad Sto A D P-yty bag ad la-abob od optcal cattg yt Phy v Ltt [8] L Z aza H chlaut t al Udctoal vblty ducd by P-ytc podc tuctu Phy v Ltt 6 39 [9] ah P D ad O B J G Ipact of a global quadatc pottal o galactc otato cuv Phy v Ltt 6 [] g L Ayach Huag J t al Nocpocal lght popagato a lco photoc ccut Scc [] Btt S Dtz B Guth U t al P yty ad potaou yty bag a cowav bllad Phy v Ltt 8 4 [] Ltz G L Cja A t al Pup-ducd xcptoal pot la abov thhold Phy v Ltt [3] Zzyul A Kootop V V Nola od ft-doal P-ytc yt Phy v Ltt [4] aza H Chtodould D N Kova V t al P-ytc albot ffct Phy v Ltt [5] gbg A Bch C A t al Paty-t ythtc photoc lattc Natu