bogo tin hov theory for many body quantum tins
Post on 13-Apr-2022
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Bogo tin hov theory formany
- body quantumTins
Recall :
/ A- thingsHN - Ei - hi tf ?! Vexing)ou l's Inn )
we-kn.owi.at/EN-NlzVlol/eCNexV-gal: determine a) If Lyn ,Hn Yw > e En tk
order one⇒ ( Hn
,Itt - Honed ) j YN) E Canvrih .
to energy \or i - ceo ,yn4o > t 4N
Define :
Un : is IM → FIT.
- §.
he.
( Man
Yn → { no ,.. .
, an } if You . a. 497 a .ge?'II. + an
With N! at 2/3 11 we define1 Lw -
- UntlwuwHn -
- Yen. p' ap.aptffp.g.n.I.lv/a.pwagaqwapwe find
Unaiaolln-
- N - Nt, In
.
- Lai't Ln"tLn"'
r Ln"
Un apt a. Unter' bp't with
inn :c: ::ti: :.im/::i:::i::.::i:.iii:::n::iwith + 'KEVIN ( bpth.ptbph.plbpt-apf-EIUwapafnlffhni.tl/rnEvHlbpIa'vapthc)N Lii! 42N -2814 ang aiaesrap
We knew we have BEC : Mw e l} INN )
( Mn, ¥oap*ap Yn ) E C
If Sw = Un Yn
⇒ Csn ,Nt Sw ) E C=
With some mene Werk : if Mw -
-Rl Hn e EntKNIN
then Sw -- Un Yn is sit .
( Sw ,NE Sw ) E C
Notice :
Ksn ,L "n Sw > I --
⇐ If Your Mnlf a par Aq Sn , agwap fwy← Tn p&g , r
NHI nap .ra , finny"
- I " HI Ragnar full Y"'
⇐ 4N Ahhh.til Sn N'
an"
, Wrt NSW 7
We can also studyHN -
- F! I- Oxit Vexr kill + IN §! V ki -ti )on Ks ( MSN ) .
Mm :
Eng F gig, myEnl 'llNell = I
where
Enid = fax et 't Varlet ttzfvlxylleklllelgliUsing Ty o , we find
7 ten - main,En 1411 e C
a) BEC in minimiter yn of Eu :
I - C Yu , Jn Yn ) E 4Nfor every low - energy Hw e l} CMM .
Let ten = -Or wear a ( Va Hut
' )Eo = lowest er of finD= hu - E . ,
Wix,yl -- llulxlllnlyl thx - y )
E = [ D'" ( Dt 2W ) D"2)
"2
Then :
En -
- N - Eu ( Qu ) - { tr ( Dt W -E ) t 847
Moreover : O ( Hn - EN) n l - ae , -2 ) consists ofeigenvalues of the term
? ni ei +Oh)
where ei are eigenvalues of E ,NIEMISeiriyw ,
Creech - Seirinfer, Lewin- Nam - Safety - Sokvej,Devezinshi - Napierkowski, Pizzo , Rossmann - Petra - Seirigor.
Dynamics : if + Hw,t = NN NINH
Interested in data e- hihiky BEC :
HN e y@ N
HN it I 4t①N
f- Ei , Vla - xp e (Va IHR) ( x . )
→ Et must solve the time - deep . nonlinearhaiku eg :
if tht = - our + ( Va 1914 4T.
It)
Under appropriate assumption V,one
can show :let TN E L's 1MW) he s .
t.
I - C 4 , you 97 F. 0
Let Mme = e-Mntyn .
then
I- L ft , Jn ,
t Yt ) - O
N -soo
where Ut solves H) with Ye..
= 4 .
① this does not imply thatN Ynet - YEN A X o
Can we get a norm - appear'm . ?
Yes , using a similar approach asdiscussed for spectrum - -
-
We cheese NIN -- the Sw where Lsu , Ntsw) EC .
We consider
Mme =e-innit
you = eihntuygwWe defineSmt = Uye Hmt = Uyteittntuygn-
Wn It ; o)
Observe :
ist Wn It ; ol = Ln It) Wn It; o)with Lw Itt ( fist the t ) Uef t ther Hn Uef ]We find :
Lw Irl -- Cn IH t+ fdx Ox at Ox ax t fdx (Valeri) Kl abb : ah
tfdxdyvlx-ylllrlxlfrlytaxaytlkfdxdyvcx-ylllrlxlllrlylab.ioabf t h . c )t y rn fdxdy V Cx -yl 4 rly) bi afax t h . c .
+ 112 N S okay V Ix -yl at ai an ax
We define quadratic generator :
L . HI = CNN t quadratic part of Ln It )The comes p . dynamics :
ih-W.lt ; o ) = La (H W. It ; o)is a time - deep .
B. T.
Then :
www.MSN - Walt:o) 9N =
=- Ww Itis) Wo Csio ) Sw Iii !
= - Sids Ww Itis ) ( hw - hold) -W.Is , ol SN
→ U Wn Ir ; ol Sn - W. Kidsn he unfit
b
k Uq. e-iwt
Uf Sw - W. Kid Sn N e ¥Morally i
n e - inn tues w - Uef W. Hi ol Sw N E FI
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