 Tukaj vzelo, da imamo prišličenje z vseh prišličenje v bozonu, v grozpitejskih režimu, kaj je režim, kaj je vzelo, za vseh bozonu in kondensaciju. Vzelo, da imamo prišličenje z vseh bozonu in kondensaciju, Kod kon prohibizera, kaj ima ili prične potrebnje, ki in tudi bi razvijeli, mobim, ki nema akroskopiknog očoserva vsakrat, način je muza primkre matematik sem pa, kaj je tudi početno, je začeli zničen, vetko na naši zato razkud. normalizacija gamma-1 vs. 1, in zelo se, da je system, kako je gamma-1, kako je in eigenvalj, kako je 1. A korrespondenja eigenfuncija je konvencija, kako je. V matematiklu perspektiv, čalengu izgleda nekaj režim, nekaj klas interakcija tukaj, ki ima dokončnjenfar postavljenja bozanja in konvencija, izglasovu osnej za mikroskop cruisejsta. Ča doğru se vseč je pozdravil na vseči ekulibrami propiti v svojo režime. Vsečkaj se verjene in, če kaj to uppravili ods子 trafiničke teori, ko je v k Christiansi izglasovani trg, za izglasovani stačni tukaj, bozanja in konvencija, ko je je vizia za teori v bozanjni, Im bari z ničisto težko v počku v Sobolji, da se našlerosni tukaj se vse nistice distractimi, kjer je vsezvima, kofacezm, tak pa tezvim tukaj, ..as tukaj ne curved o prendim. Tukaj imamo tudi po opravitičnih svačegi, z katerimi in podokonenami. Vsezvima, ki vsezvim na to, da izgleda ta podokonen mene v nema savak fakečakov, kabilAbinalne drepte v spremu, zazaj da vedemo, kaj je poslutaj defineh neskodovniki priklad u postavičnev, do čutke in z осetitega. Kaj po pomečenju, je učil nadkratite pri pomečenju. V tem matematici, spremu po vsebenju vsebene, načinjate z analogimi vologijami, ki, as we heard in material lecture, is provided by this many particle Hamiltonian, acting on the subspace of L2R3n of function, which are symmetric under the exchange of particles, bosons in fact. And you recognize here we have the one particle, part of the Hamiltonian, the kinetic energy, the trapping potential and the two-body interaction. And the scattering length is a parameter, effective parameter associated to this interaction, which is defined as follows. Take the zero energy scattering function for your potential with boundary condition that f goes to infinity, then the scattering length is proportional to the integral of the potential times the scattering function. And it's very easy to see what this means for short-range potential, because of course then we can solve the equation outside the range of the potential, and this is the expression of our scattering function. And you see that a appears in the asymptotic behavior of a large distance. From a physical point of view, the scattering length, sorry, not scattering function, but the scattering length a is a sort of effective range of the interaction. Namely, two particle, far apart, they see each other as they wear hard sphere of radius a rather than the range of the potential. So it's an effective range of the interaction. And what we would like to, what I would like to discuss today is how we can recover that the thermodynamic function of bosans and condensin only depends on the interaction through the scattering length in a particular scalar regime, which is the graspy tijesky regime that I am going now to define. In this regime we consider m bosans confined in a region of order one and the potential now is rescaled in such a way. Now you see that the range of the potential is much, is very small, is n to the minus one much smaller than the typical distance among the particles, which is n to the minus one third in this regime. It also just using the scattering equation and rescaling, you see that the scattering length of this potential here is nothing but the scattering length of the potential divided by n. So the peculiarity of this graspy tijesky regime is that we have a range of the interaction and the scattering length which are on the same range n to the minus one much smaller than the average distance but intensity of the interaction is very strong. So we are really describing short range and intense interaction and let me comment a bit more on this. So which are the typical interactions that we are describing in such regime? Well, if you look which is the probability of interaction with a given target particle well, this is extremely small, is n to the minus two. But then the typical intensity of the interaction is very small, n to the two. So this means that the force in this scaling, the force on a single particle coming from all the other is order one. So this makes the contribution from the potential energy be over the n, rather than n squared as one would guess in a different scale. Ok, so this is the graspy tijesky regime and I'm very grateful for the very peering result in this contest. So the force that one understood that the scaling of the graspy tijesky regime could indeed be a good approximation for the, or a good description, for the experiments in cold atoms that were done in the first experiment with Boseinstein condensation for cold atoms in 1995. Yeah, I believe we hope this to, yeah, Leib sidinger Hinverson that realize that this regime could in fact be useful to understand or to describe Boseinstein condensate. And this is the peering result in 2000 take the or many body of the Newtonian then the ground set energy for particle in the limit L to infinity is just described by the minimizer of a one particle energy functional. And now you see that here the potential enters in fact only with its scattering length. And moreover there is condensation for the ground state function namely the one particle density matrix associated to it converges in trace norm to the projector onto the minimizer on the graspy tijesky energy function and in fact this result is even more general not only hold for ground state far for any approximate minimizer so on any state who has the property that is energy for particle is converging to the ground state energy. And let me use this result to explain to you a striking property of this graspy tijesky regime which also contains all this obvious difficulty or many of these difficulty the role of correlation in fact. In fact the result on the graspy energy that was obtained shows us very clearly that even though this system exhibit condensation the ground state energy of the system actually is condensation in l2 norm is it not true that the many particle wave function is just a product of one particle function and it's a very easy computation to see that because if you just compute energy of a factorite state well you realize immediately that from the potential here in the limit for large n this behaves like a delta function and so you get something which has the correct form that you expect for the ground state energy in the limit, the ground state energy for particle in the limit but with the pre-factor which is the integral of the potential and not the scattering length which is much smaller and the reason why if you use a factorite state we don't get the correct result that in fit the many body wave function has a more complicated structure than being a factorite state in particular is two particle density matrix has this form so this is the two particle density matrix so it's not only the projection onto the condensate but has this correlation structure appearing this is the solution of the scattering equation for the new potential the one, the Graspitejeski potential well now for n large if you are just testing this in a nice function well f is going to 1 so in the limit you really get convergence to the gamma 2 that you expect but when this is evaluated against the very single Graspitejeski potential here is the where the f appears informed of the quartic interaction so the role of correlation is very important in this Graspitejeski regime now let me just for simplicity because then the result that I am going to present a bit more easy to state just consider rather than the general trap a box a box of human heat volume lambda and then our Hamiltonian is the following and I am considering it with periodic boundary condition what I want to tell you is that now in fact we have even more information on the statical properties of both gases in fact after 20 years we are very happy that we could understand a little bit more and in particular we can prove that for any that the Graspitej energy is in fact given by this factor 4 pi a n with an error of order 1 and moreover we have a precise bound on condensation and rate for condensation so this 1 minus the expectation of gamma 1 over the condensate state this is just the fraction of particle outside the condensate so this guy here is the number of particles outside the condensate and we can prove that in this Graspitejeski limit this is bounded uniformly in n this is exactly what one expects in a perfect both science condensate that a macroscopic number of particles goes into the condensate and then you have only bounded number of particles outside the condensate in year 5 the condensate wave function is just 1 because we are into the box so the minimizer of the Graspitejeski function is just the constant wave function and one can also understand a bit more what is constant here is order 1 and sorry very important all that has been also standing to trap very recently presenting the result for the box but everything can be also extended to the more general case of the trapping potential so and one can even a bit resolve better the low energy excitation spectrum and so derive an expression for the Graspitejt energy how to error terms which are a smaller order with n and this expression here in orange is in fact what is predicted in the physical literature to be appeared by this famous Bogoljupov theory and one can also obtain information on the spectrum of excitation below low energy excitation and once more you obtain that excitation said that the eigenvalues have this form so we have the sum of the number of particles with momentum p and this dispersion relation which is linear for small momenta this being relating to the emergence of super fluidity in those system and what I wanted to stress here is that in this Graspitejski regime we obtain that the all this quantity here and the spectrum of excitation depends on the scattering length in fact arriving to the proof of this result has a long story starting from less singular regime where we learned a lot and the main novelty of the result is in fact this independence on the details of the potential that was not visible in previous less singular regime than the Graspitejski one which are the ideas of the proof that I am going to tell in a moment the ideas is really to provide a rigorous version of Bogoljupov approximation of Bogoljupov theory so I should maybe first tell what Bogoljupov theory is about and let me tell in words then I will be more precise the idea is that Bogoljupov assume condensation we know that there is condensation for free bosons as Mathieu showed so let us assume that if the interaction is sufficiently small this property is still preserved and then expand around the condensate so consider all the situations where all the particles interacting are in the condensate they are two in the condensate and two outside and so on and then neglect the idea of Bogoljupov was neglect the contribution coming from more than two excitation just because there are very few excitation around so I may expect that terms coming from three excitation and one particle in condensate all excitation are much smaller as a very smaller effect on my system then I have something that I can treat because once I have neglected this contribution I have really an explicit model where I can compute everything well surprise I compute and I see that the energy that comes out the spectrum that comes out do depend on the potential but Bogoljupov knew very well the reason why this was the case he had neglected a lot of terms that maybe he was not allowed to neglect but at the very hand he just realized that some of the expression in the expression for the energy there was like a serious expansion for the scattering length this is the so-called Bohr series for A where you write that A P A is just the integral of the potential plus errors that you can write explicitly that are order A over R naught so if this is small this is a reasonable expansion but notice that in the regime we are considering in this Grossi-Pitajeski regime this is really order one so for us this expansion would not hold at all this makes the Grossi-Pitajeski regime very complicated but anyway with this consideration Bogoljupov managed to obtain back the scattering length appearing in the in the ground set energy spectrum and which is the line of our proof well first of all we show in fact that condensation of course in this Grossi-Pitajeski regime so we have a priori energy bounce on the number of particle of excitation and on the energy of this excitation then as in Bogoljupov theory we expand around the condensate but now we do not neglect this cubic and quartic term we take into account the energy coming from these terms and once we do this we obtain that our system can be expressed once more in terms of a quadratic Hamiltonian but now this quadratic Hamiltonian has been renormalized so now the coefficient of this Hamiltonian are not depending anymore on the potential but only on the scattering length of the interaction so in effect the scattering length appears naturally in the result as an effect of this renormalization and this was not absolutely our idea in the sense in the physical community immediately after the work of Bogoljupov there were several attempts of trying to do perturbation theory around Mogoljupov model with renormalization group technique of the same type that you have seen in Malfred direction yesterday and try to figure out which were the relevant diagrams giving this renormalization effect they say that this grime are Lander diagrams so very similar to those that Malfred described even though in a different language here what is that controlling the errors so really showing that all the other diagrams you can throw how this was not an easy task and in fact we can do it here just because in this Graspita eski scaling we have something that help us namely that the parameter of our expansion rho a cubed if you wonder why it's it well this is the parameter in fact appearing I I'm not sure that I can show here anyway so the parameter that it's appearing in our expansion is of order n to the minus 2 in this Graspita eski regime because this scattering length is very small and this helps us in controlling the error terms but the idea of doing such a procedure was days back already to the very first work after Bogoljubov teori ok and which are the key aspect in our proof well identification of the relevant length scales first of all or energy scales according to how you see it so the length scale are the range of the potential which is very small and what is called the hailing length of the interaction which is of order 1 and then the idea of modeling this correlation as you have seen already in the discussion that I made on the paper by Lib siding well we model this correlation using some suitable operators and then functional estimate ok so if there are no questions so far maybe I try to give an idea of really how you can reach these goals from a mathematical point of view and to do so I need to give you a very quick introduction of what is a representation which is very useful when we deal with many particle system which is the Fox space the Fox space is just the space where you describe your quantum system when you want to deal with a grand canonical setting where the number of particles is not fixed so just take the direct sum of all n particles and this is the Fox space so now a vector in the Fox space is just a collection of vectors with any possible sorry a collection of wave function with any possible number of particles and now that you can have several sorry now that you can change the number of particles in your system it is also convenient to introduce operators that do change this number of particles and in particular here I am defining the creation of A star p and Np in this way so A star p is just creating a wave function with momenta p and then it just symmetries over all the particles because we are dealing with bosons and annihilation operator is just integrating out one variable against this Aepx and it is just a computation checking that this commutation and annihilation operator can be applied by this canonical commutation radiation and now any operator in the Fox space can be very easily expressed in term of this creation annihilation operator let me convince you of that first of all the number of particle operator if I want to count the number of particles in my system we are on the torus so I am doing everything in the momenta space because we are in the torus we can use as the basis so to each particle we can associate is momentum so if we want to count the number of particles just sum over all momenta the number of particles with p and this is just counting the number of particles because I first destroy a particle then I create one so I am counting the number of particles and very similarly the Miltonian has this form so here I am just saying that to add the number of I count the number of particles with momenta p and I associate the energy p squared and the interaction can seems a bit more mysterious but it's not I'm just representing the interaction in this way I have two incoming particles one with momenta p one with momenta q they interact let me represent this weekly line just the interaction and then we have two upcoming particles we are creating two particles they now have momenta p plus r u minus r so this is just a way to represent our two bad interaction just as a scattering process involving two particles that are disappearing and two new particles that are appearing and of course there is momentum conservation in this process ok so this is the tool that we use but here comes a very nice idea of Levin-Nam-Serfati-Solovi why don't you use this fox space rather than to describe the fact that particles can vary just to describe the fact that excitation can vary so let me consider a system where the number of particles is fixed is capital N but we know that excitation can go from 0 to capital N in fact so are not fixed so just use this fox space to describe excitation say it better in this way then there is a unique composition of your many particle wave function in terms of with this factor so you have a term where all particles are in the condensate and you have no excitation then N minus particle in the condensate one excitation and so on and this position exist and is unique if these alpha j's are j particle wave function in any variable so this splitting here just defines a unitary map from your initial space of n particle to a fox space here where now this alpha 0, alpha 1 so on represent excitations so here you are just saying well my many particle states is described by is excitations so I am removing in a sense the condensate wave function from the excitation with respect to it so once more if you now ask which is the number operator on this fox space or why I am using this notation well this is just the less equal capital N stands for the fact that here I cannot have more than capital N excitation clearly and the plus is to recall our self that this alpha j are orthogonal to the condensate in particular this means that all moment so now the number of particles in this fox space this just counts the number of particles of the condensate and in particular if you take the factorized state this is the state with 0 excitation now let's take our fox space Hamiltonian this is just a Hamiltonian written in the fox space and apply this map U this map U factorize out the condensate in particular the action of this map U is the following U N is there P at 0 for example where I am considering here P different from 0 so it just leave invariant particle with momenta different from 0 and substitute particle in the condensate with in fact the number of particles of the condensate recall this is the number of excitation minus of course the contrary total number of particles minus number of particles in the condensate so and this but you use this map in fact this is just a rigorous version of the the first step in Bogolibov approximation it allows you to focus on the condensate and you extract from here several contribution this one is the contribution that comes where all these operators act on momentum 0 so you just get the integral of the potential then you have terms which are quadratic in this annihilation or creation operator cubic and quartic and now remember that Bogolibov approximation just corresponded to neglect this L3 and L4 but now we want to keep track of them and the remark is that notice if you now take this Hamiltonian on the vacuum so on the state with no excitation so this means that these terms here are just 0 what we get again and times the integral of the potential so the wrong answer but we already know, right because the vacuum state was the factorized state that we also considered before and we know that the factorized state is not a good approximation to the ground state of the system in this Rospitalijski regime ok we need to take into account for the energy of correlation this is the message and now how do we do to do this we use suitable unitary operators acting in this fox space so this is the Hamiltonian just lift at the level of this excitation so now it became an Hamiltonian from this fox space of excitation to fox space of excitation and now we conjugate we conjugate this with suitable unitary operators I will explain in a moment how these operators are done but just I want to stress the fact that the kernel here of this operator this eta is related to the scattering equation it's just a modified version of the scattering equation because we don't want the scattering equation in the full space, we are on a box so we should just consider the problem on the box so that's why it's like modified with respect to scattering equation you see before but it shares the same property so take this scattering function and then build this eta that is 1-f multiplied by hand just to have something of order 1 so the eta goes as 1 over x for large x so you see that this eta is 0 as long particle are not correlated anymore ok, so the choice of the kernel is related to the scattering equation and we need to put several operators two operators, T and S that I will discuss but if you use these operators and look now at this new Hamiltonian that you just built in this way well it has very nice expression I don't pretend you to follow all the details here I'm not defined yet this B operator I will do in a moment just think to that as usual creation and elation operator but the point is that you don't see in this expression beside here the potential appearing any longer everywhere in the constant term in the quadratic terms in the cubic term only the scattering length appears so this is the renormalization I was referring to and the potential is not renormalized but for our theorem I didn't stress sorry so our theorem holds only for positive potential so in fact we are allowed to throw it for a lower bound and so we managed to write this you excitation Hamiltonian as this main part where only the scattering length appears beside for the positive interaction plus in error terms which is small and inbounded in terms of the other guy appearing so in particular with this we can really prove that this excitation Hamiltonian is bounded from below by the ground set energy at leading order plus the number of excitation so from that we can obtain the uniform bound on the bound of excitation that I was mentioning in my theorem and I just wanted to show you how these operators are made I have a question so do you have the term which is cubic so it is ba star a and you have also the scattering length there in fact because I introduce also this but that would in principle not be necessary no it would be necessary to show condensation for large potential so if you have a small potential you can avoid this so small potential namely a small factor in front of the potential but if you have large potential you need also to renormalize this cubic term otherwise the only thing that you can prove is that this cubic term is bounded with some potential energy plus number of particles but you don't have any small factor in front so you cannot absorb the potential and you will bound the term it's not leading order no exactly but in a preview version of the paper we didn't need to renormalize this so maybe and now let me take some time to explain you how these unitary operators are done because they have a very clear physical meaning this is just what I want you to stress here so first of all I have to tell you what these ba star operator are so these ba star operator are in fact if we look at their action back on the original L2 and space where it's more simple to see this operator ba star are just annihilating a condensate particle and creating an excitation and then are renormalized because we know that we have a lot of condensate particles and similarly ba is creating an excitation and creating an excitation and creating a condensate so this ba star operator takes into account process where you have a particle in the condensate that becomes excited or vice versa so if you look at this first operator here, this with two ba stars well this is just describing a process where two particles in the condensate becomes excited or vice versa so you have the scattering in this picture that I have I have the scattering between two particles with momentum zero then create two excitation or vice versa two excitation with momentum p and momentum p are annihilated and two particles with momentum zero are created and this is done with a kernel which is this eta p which is related to the scattering equation and this cubic here is just the natural generalization of this when one of the particle is no more in zero momenta but this is momenta v so rather than having zero zero p minus p you have v and p minus v so just you are also considering this scattering of particles where you have one particle in the condensate which is in this b high hidden in this b very low momenta and two excitation with high momenta so it's just a generalization of the first transformation that we have seen here and I'm cheating a little bit because I'm not telling you that we also need to put some cutoff in this exchange momenta so we really have to define which is high energy particle and which are low energy high energy particle are particle with an energy around n n is the inverse range of the potential and low energy particle are energy which have an energy which is one over the inverse scaling length of the of the boseisen condensate is of order one this alien length is the length where the alien length is the length where you see the change in the dispersion relations so we saw that the spectrum boseisen condensate had this form so particle with momenta p have an energy ep4 plus 16pa p2 so the region the alien length is where the kinetic energy p2 is equal to this this linear part so really this alien length is scale that defined so one over rho n which is the alien length and which defines the low momenta is this one so sorry maybe I become a bit more precise so in our momentum space we have an element energy scale which is n inverse range of the potential here we have this alien length and so for us these are high momenta and these are low momenta and the low momenta are defined by the moment where so here if you have a quadratic dispersion relations so bosons behaves as free particles while here bosons behaves so linearly in p and let me also remind that here we are in this finite box so we have a gap this makes the Grospitajeski case simpler than the so we don't have any infrared divergences we only have to cure the ultraviolet divergences in the language that Manfred was using yesterday so this was just a comment supposed to be a comment for this remark that the cutoff really play a crucial role so identifying the correct energy scale really plays a crucial role in this problem summarizing so we verify the prediction of Bogolubov for the system of interacting boson despite the fact that this prediction were based on approximation that are not verified in this Grospitajeski regime for example we don't have this bone series for the scattering equation or we cannot throw out the cubic and quartic tab that Bogolubov throw out in his reasoning and we recover the independence of all quifysical quantities on the scattering lens so the dependence only on the scattering lens so independence on the detail of the interaction and the idea of the techniques of the proof is only based on using clever unitary operators that allow on one hand to focus on particle on the condensate and on the other hand to model correlation particle and condensate so to and inclusion of this correlation leads to a renormalization of the potential so we renormalize the very singular interactive potential or slowy decaying if we see it in Fourier space with an effective potential which decays on moment of order one and whose integral is in fact related to the scattering lens of V and I do not have time to comment very much on this but with the same ideas you really obtain a norm approximation for the eigenvectors and so you can really go further what I told you, not only compute all observables, depletion of the condensate and whatever several you want pressure, but also study now fluctuations, central limit your enlarge deviation happening on your ground state of observable measured on your ground state so you measure an observable on your ground state you know that in the limit it's given by the expectation of the same observable on both ends and condensate, but now you can really evaluate how these two measure fluctuate, one respect to the other so it's really, this gives really good control on what it's happening in your low energy eigenstate for this system and very, very sketchy, I wanted just to tell you something more so the method I very quickly present you have the nice aspect that they are also very flexible to be adapted to other problems, in particular both ends and condensation as any other phenomenon related to symmetry breaking is very much dependent dimension of the system and in particular in general standing the result to other dimension might be not trivial, so with the same kind of technique, but just adapted to the two dimensional case we were also able to prove analogous result for the two dimensional which is now characterized by these other scaling and the reason for the exponential come from the fact that the scattering length into dimension goes as logarithmic so this is why the scaling is defined like that, but the idea is once more to find the scaling where the effective parameter goes as 1 over n integral Vf goes as 1 over n so also in two dimension you can prove some bounds on the number of particles outside the condensate, almost optimal not really optimal, in fact two dimension is much harder because you can verify the prediction of Bogoljubov theory for this system and you see that here the two dimensional case is even different so that the scattering length does not appear here in this Bogoljubov sum neither in the spectrum of excitation this is come from the fact that two dimension is really different and I want really to quote the previous result where already obtained it by Leib's Heiringer Inverson for the lead and leading order term and that very few is known so far for Bogoljubov theory in two dimension so this was in fact the first result because the other available result was also was only obtained by restricting to a special class of state. Ok, but this was a result for a much more difficult thermodynamic limit, while here we only did with this, Grosz Spetajewski. And finally since Mathieu yesterday spoke about the thermodynamic limit because it's the more challenging one let me also add some recent improvement of this technique to be adapted to the thermodynamic limit and just what I mean with thermodynamic limit where we now have embosons in a box of sides L so this is the Miltonia now the potential is not scaled it's really the fundamental interaction we are not taking any scaling limit and we are considering the regime where number of particles and the volume is sent to infinity with the density kept fixed and in this case very few is known so the currents of condensation is really a main challenge of open problem, is related to the fact that symmetry breaking of continuous symmetry is really very complicated and challenging problem, we have in fact a single model where we can prove bosons and condensation in the thermodynamic limit this is our core boson saddle filling and then we have renormalization group results but that are not conclusive the problem is that for boson you have too many graph this n factorial bounds that we didn't see with the Alfred but are related to so the problem improving boson and condensation thermodynamic limit is related to the prolification of diagrams that you don't know how to control so occurrence of condensation is really a challenging problem but the nice fact is that you are only interested to ground set energy you don't really need that condensation is proven in the wool box going to infinity it is sufficient to show condensation of smaller boxes let's say so a local version of condensation this is sufficient to get the result that you are interested in if you are only interested in thermodynamic function still it's not so trivial so you see that there is a long story a long time passing from the very first result of the ground set energy in the thermodynamic limit these are the result of living order and only very recently the second order lower bound has been obtained and of course the excitation spectrum is still far to be understood but as for the ground set energy is concerned and in general thermodynamic function it's very interesting that one can use some of the ideas that have been developed in this Grospitajeski regime and let me show how and why so let me emphasize that we can now write some scaling limits that interpolates between the Grospitajeski regime that we have seen so far that was with kappa equal to zero and the thermodynamic limit that corresponds to kappa equal to two-third let me convince you that this is in fact the case so take in fact our system so we have a box size one and now the scaling regime corresponds to range of the interaction of sides n minus one plus k, so larger now rescale variables just to have an interaction of order one as the one you would consider in the thermodynamic limit then of course the size of the box has become n one minus k if you now copy the density well the density is n three k minus two so you see that taking kappa two-third here you get in fact that the density is constant which is in fact the thermodynamic limit so now you say why do you want to write the thermodynamic limit in such a complicated formula well because I want to use the fact that to derive something in the thermodynamic limit I do not really need to work with it kappa two-third maybe I can use some result in between the GP scaling and the thermodynamic limit and this is possible thanks to some very nice localization technique I'm not able now to give all the details but the idea is that some localization results allow to find lower or upper bounds for the ground set energy in the big box in this thermodynamic limit just focusing on what happens in a smaller box of sides L a smaller box where now you look at this kind of a miltonian with kappa that depends on whether you are considering a lower bound or an upper bound and in fact one finds that for a lower bound is sufficient to look to be able to control some miltonian which is a bit more less with the range which is a bit larger than the gross pitajeski scaling so just kappa larger than one is sufficient but therefore a lower bound to get this localization you need to modify the kinetic energy a bit and this introduce a lot of complications and for the upper bound you need kappa larger than one half that is a bit challenging so extending the result to kappa larger than one half requires our work but still is much easier than the thermodynamic limit itself so it was very nice that the technique developed initially for the gross pitajeski regime have now proven are being proven to be successfully also in other scaling and I'm really finished just few conclusions to convince students that some other things needs to be done so this is not at all the end of the story maybe it's the beginning in the sense that here there is just a list of very first question that comes to mind just starting from this result let me stress that all our results only are valid for V in L3 this come from really the way our technique are implemented solution of the scattering equation and would be very much interesting to understand how our core interaction can be treated let me stress that very recently a lower bound, a second order lower bound to the ground set energy of bosons in the thermodynamic limit was obtained for our core interaction these people here but upper bound is still an upper problem then can we extend this result to finite temperature we have here Andreas which is an expert on this and this is certainly a nice question then from the dynamics perfected I didn't have time to explain at all the dynamics but there really is known that the Gross-Pittajesk equation provides a good description for the dynamics of bosons and condensate but much less is known from the fluctuation point of view of the fluctuation in fact the only available result is a much simpler case of the Lorentz gas quantum Lorentz gas with Gross-Pittajesk interaction and just dreaming maybe starting from this we can develop a multi-scale approach that very much in the spirit of renormalization group can give some idea to treat the more challenging problem of bosons and condensation and thanks a lot thanks very much for the nice talk for all the questions, comments yes so in your results about two dimensions in the Gross-Pittajesk can you say a little bit about how this would be related to long range order? Very, very, very good question Manfred, in fact the reason why I would be very interested to understand what happens at positive temperature is really goes in that direction because in fact the only available result in two dimension is also positive temperature in this one if I'm not wrong but they cannot see the costally starless transition and I suspect the reason is that here they restrict to quasi free state so only this bogoljub of transformation and maybe this is why they lose something so I cannot tell much there because it's only a zero temperature result so we have like bosons and condensation and long range order but of course the very nice question would be understanding if we can provide a simplified case where a two-dimensional model of interacting boson where we do not have condensation but we have polynomial and decadent correlation that would be exactly so let's say this is really my motivation for starting the two-dimensional investigation Yes Maybe a remark concerning that Thank you If you look at experiments in the 2D Boso gases you see that you actually have condensation I mean finite size condensation also in 2D and you have it but you have it at the size of the decadent critical temperature so I mean the point is the length scale of the decay is just to long to see it on the small boxes and since GPs are really good approximation for the experiments I guess you would expect that sure No, no, no, you can I mean you can but you see it as boson Then maybe one has to go to so you see now we have really technique to go beyond GP towards the thermodynamic limit so maybe if you cannot see in the GP regime that's very convincing what you told me maybe one there is some hope to see it in some of this other regime analogous of this kappa regime for the 2D case would be different but these analogous which are still as challenging that the thermodynamic limit but maybe already you can see something so this is the second hope so if the Gross-Petajski regime is not enough maybe one can still find the scaling limit which is not as challenging as the thermodynamic one but still relevant for seeing this difference but it's only hope at the moment yeah, thank you More questions? Yes, these two-dimensional considerations this is all for temperature zero This is only for temperature zero positive temperature we have many Wagner and yeah, absolutely yeah, yeah, absolutely just for the audience in fact for two-dimension is really critical for the existence of Bosnian condensation namely, there is no condensation on any positive temperature in two-dimensional in the thermodynamic limit this is just ruled out by a very general theorem which is known as Merbin-Wagner theorem so what you can hope is not proving condensation so you know it does not occur but still you can hope that that the off-diagonal part of the one-particle density matrix that when there is condensation is constant this is an expression what is called this longer-range order maybe it's not constant but the case as a power law so in general where there is no condensation where there is no phase transition let's say where you have a very high temperature this correlation the case exponentially fast in the distance so I hope that you could see a power law decay for the 2D but in fact you are the expert of positive temperature so maybe I'm too naive in expecting that we can prove I just wanted to add a remark which is actually probably well known to many of you the question is about this definition of the concept of Bosnian condensation namely in for the ideal gas for the ideal gas without interaction you can't have condensation at positive temperature in two dimensions in a generalized sense where the occupation so to say of the ground state is not proportional to n but proportional to n divided by log n which is for practical purposes of course almost the same and this is part of an extremely general research for ideal gases at least that it depends on the definition of what you mean so how you let your parameters you can translate this into simply a research where you have condensation in the sense of of the occupation proportional to n but you don't take the usual thermo-denial limit you take a slightly different limit so the lesson is that you have to say always clearly how what are you exactly to talking about this for instance there is a paper from the very early days of this business by Ketterle and Druten where they emphasize this this that they draw sort of curves that look exactly like like what you would expect for the two dimensions so to say in the harmonic for instance you have you don't have conventional condensation in one dimension but you have it in this generalized sense also in one dimension in harmonic but you don't really see it and you get any difference if you look at the numerical I agree with you that from a experimental point you probably determine the limit is not needed because in other setting you still see the same maybe one is thrilled by the fact of proving a result which tells something about symmetry breaking and then in this case you really in trying to understand the thermodynamic limit but thanks for the remark maybe the other setting for practical purposes should be really I've never seriously looked at that so thanks for the remark about the kative condition so there is the decay the causal stress so there is this slam by collision spencer for the classical spin system and in that case the transition is due to a change of behavior of vortices in the system and I was just wondering are there vortices or where are the vortices in very good point so the vortices are in this so if you study the dynamics for example so if you create a condensate in the trap and then you remove the trap and see how the condensate evolves then it evolves according to this graspyta yes equation so cubic nonlinear shreddinger equation and there you can see vortices in fact you but I think we are very far really the behavior of the vortices in this quantum system so as far as I I know so you can let's say from a statical point of view there are several results Jacob is the co-author of several of them and also Nikola here when you put this both sides of condensate maybe in a trap you make the trap rotating and vortices appear and according to the regime you can have giant vortices vortices on a lattice so you have a nice pattern other result in the PD community also study the dynamics of these vortices in paper which are very analogous to the classical one where also Mario was involved but then they can only study the dynamics until vortices are not colliding let's say so for a time which is not so you cannot see collision among the vortices and either in the two-dimensional case the simpler because in the tree with the filaments it's even yeah it's complete so maybe are related it would be amazing to understand something but I think we are quite far also from that point of view so sorry we are still far to understand the vortices themselves how they behave or which is their dynamics maybe we know how they appear but now not how they evolve so put in the connection with this seems quite far but of course of great interest thanks maybe this is just a historic remark the Gross-Betairsti equation was put forward in 1961 independently by Gross and by Betairsti and the title that I remember of this paper is vortices in liquid helium so that's what they were aiming at at that time and it was only then after the experiments mid 90s that Chris Pathick and Gordon Bain they suggested to use the Gross-Betairsti equation for describing these dilute gases in traps so they so to say took the same equation but completely different parameters from the original purpose which was liquid helium and v vortices this is just a historic remark thank you I didn't know that, thanks a lot if there are no questions anymore we can thank Serena