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