 So let me first thank the organizers and apologize for not having been able to come earlier than today I'm looking for chalk. Why chalk actually And so yes, I will speak about Well, what was the title again? Exactly Exactly so Well Let's start with the following something you've already seen. So we look on we're working in rd Instead of zd, right? So that's our base space and in rd. We pick To make it simple lambda some volume, but to make it simple. Let it be a cube. Okay, pick some cube in rd and On L2 of lambda So this is where our particles will be living, right? We consider a Schrodinger operator h sub omega lambda Which is just minus laplacian plus some potential Restricted to lambda so in mind I have that I actually have a random potential. That's what the omega stands for Living on all of rd, but I just consider its restriction on some cube within rd, right? So on rd. I have h omega, which is minus laplacian plus v omega You could consider zd, right in the same way and if you are more comfortable with discrete operators you can do the same thing with discrete operators and Well, that's a single particle, right? This is one particle in the box lambda But of course as you all know In nature particles do not come alone, right? You cannot or you can but it's hard work isolate a particle But it's even very difficult if you think of electrons to isolate them So usually what you should look at is many particles So you have n copies or n particles and To make things simple we assume that these particles of course they live in the same Space and we assume that they are identical, right? in lambda So the natural Hamiltonian these particles being identical would be to do the following h omega And Lambda which is just the following you take h omega lambda in the first variable tensor product One in the other variables right plus and you do the same thing for all the particles You put the one here And the one of course stands for the identity on L2 lambda H omega lambda, right? And of course you have n terms in this sum and this is the product of n terms Right and one of the questions is you can define this so this is just here you have a differential Operator, right? So this gives you another differential operator In this time. So if this was an Rd. So here you have d variables Here you have a differential operator in n times d variables and For such a differential operator. Well if you want to define an real operator for this you need to define a domain, right? and Well when comes the domain assuming that your particles are identical, right? There are three standard choices, right? You can assume that essentially they are all the same and No symmetries, right which is called Now I don't remember the name of the German physicist So no symmetries No symmetries then you have that the particles are symmetric So this is called the Bose Einstein Symmetry and or the both symmetry actually Einstein comes up later. This comes both and particles are anti-symmetric Symmetric and this is the Fermi sometimes the rack Symmetry what does it mean? It means that if you look at your functions psi So these are the functions on which you want your operator to act so they belong to the tensor product of L2 of lambda Which is the same thing as? Right or identically can be identified with L2 over lambda to the end You actually are looking at totally symmetric eigenfunct or totally symmetric states Which means that H of X sigma is Going to be psi sorry at H of X sigma is going to be psi at X for any permutation of the particles And where X is X1 Xn and X Well, let me call it sigma of X Sigma of X is just X sigma of 1 X sigma of n Right So that's the symmetry the particles in this case are called bosons Right, that's the both symmetry and the other one is the Fermi symmetry is just that Psi of sigma of X is Minus one to the psi So this is just a signature of psi of sigma sorry signature sigma time psi of X and this of course also for all Okay, and the case so To define an operator properly right outside putting boundary conditions on Lambda to the end if you want to consider a physically realistic realistic system You also need to choose which symmetry class you're looking at right, so I'll be working For or in the last setting right we'll be looking at fermions looking so at anti-symmetric States So and we will well actually I give different names where I call it This way so my Hilbert space actually the basic Hilbert space. I'm looking at h n will be just the That's the definition of this space right L2 of Lambda yeah, the lambda is actually not very well chosen, but okay I leave it the way it is Which is just the psi in L2 under to the n Anti-symmetric okay, and Now the lambda with the n above it. It's just a cube. It's just it's just yeah, exactly Oh this one that's the anti-symmetric tensor product, so it's just a notation. That's definition of this symbol The definition of this symbol is just a psi is L2 functions over this cube Right of large dimension that satisfy this anti-symmetry right that's the definition now so what you have now and Well to make things simple. I will not I'll consider simple boundary conditions right so I need to define an operator I Will do the following now. I do I take the domain of the operator Well actually I can make the domain and then I'll take a closure of the operator on this domain It's just going to be the following I'm going to I'm going to consider h omega n Lambda on Well c0 infinity of lambda to the n intersected with h n Okay, that's my domain and actually the operator I'm looking at so this is a non-negative operator Right, actually it's low bounded. Let's assume to make things simple to make things simple. I assume that my V omega Is going to be larger than some constant? Omega almost surely right just to make things simple It's not really less necessary, but you can do without but this will simplify the setting a little bit so you can look at the free ricks extension of of H omega n Lambda which will give you I call it actually I call it the same name Right, I won't change the name and you know that it's a lower semi bounded and obviously you have the following bound I have the following bound for the free rick extension, right? The lowest possible energy for each particle is minus c So if I add any of them I get minus c times n. Okay, and what am I interested in? Well, so actually the thing is going to be a Schrodinger operator Right because the only thing you're doing you see here You have a laplacian in the first in the formula of h omega n of lambda. You have a laplacian in each variable Plus some potential the only thing is that for the laplacian doesn't change anything if you some laplacians in each variables You just get a laplacian in the total number of variables, right? That's because laplacian is has separate variables But if you do the potential the potential has a very specific form, right? Because you have potentials that depend on only one set of variables Okay, but it's a Schrodinger operator in the usual sense What is a bit different is that you consider it on the space? Which is not the usual space you're not looking at it on L2 of some cube, right? Because you are adding in the symmetry condition, but except for that you have a standard Schrodinger operator and as such can apply Results known for standard Schrodinger operators Okay, and in particular Right Well, it's easy to see that what you get. Okay, so earlier in here You have something which is lower semi bounded, right? The lower bound depends on that And well you can look at the ground state. So let me call Oh, yeah, let me add one thing. So here I have particles, right? They are each living in their channel Okay, but they don't see each other Because obviously, yeah, maybe I can do the following thing And maybe what I'm going to do is I'm going to just to change name of this thing to incorporate further parameter in my story I'll put a zero here. You're going to see what the zero just to see that this is a free operator zero Zero Okay, and now as I said one of the things one can notice is the following take pick phi J of lambda and Ej of lambda So we are you are looking h omega lambda is Schrodinger operator on the cube lambda, right? My potential is supposed to be let's assume that I assume that it was lower semi bounded Let's assume that it is nice and regular, right? Let's say just bounded or I don't know continuous or something nice, right? Not too nasty then what I know is that I can find I know that the spectrum actually of So this beats the spectrum of h omega lambda is a pure point Right, you're looking at the Schrodinger operator on a finite volume This operator if the potential is not too nasty has compact resolvent Okay, so its spectrum is pure point. So these are just the eigenvalues or eigenfunctions and eigenvalues of H omega lambda, right? Of course, they are ordered So this depends on omega Increasingly and well Low as a bounded by minus C and of course they tend to infinity When J goes to infinity Okay, and the other thing Increasing and the other thing that we know is that What is there? We know that? E zero lambda omega is simple, right? That's a standard result on the reasonable assumptions on V omega I'll be more specific, right? The problem is full generality is quite complicated, but I'll be more specific later on about the model. I choose this is simple and Once you know so this is a spectral deposition the composition of H sub omega on lambda Once you know this you can easily construct the spectral decomposition of H omega Where do I put the indices n? zero lambda Just by doing the following thing Well, you define what is called the date the slater determinants. So pick J1 Jn In Integers, right? So I started zero so n to the n such that J1 less than Less than Jn, right? So you pick a set an ordered set of integers Okay, and you can do the following And do it here you can do the following you can do well this one is a slater determinants Just the following I write in the following J1 exterior phi J2 exterior exterior phi Jn At the point x so remember that x is x1 xn It's nothing but the determinant. So it's one I normalize it one over square root of n factorial and the determinant of the matrix phi Jk xl Jn, sorry k and l running between 1 and n, right? And you take this determinant So I stick with the mutations of the previous speaker and Well, what you can check easily Is that first this is totally anti-symmetric? That's the property of determinant Right and moreover if you compute H omega n zero lambda applied to I write in this form phi j k k going from 1 to n Well, you obtain that this is the sum For j or for k sorry going from 1 to n of what of e j k Super omega lambda Times the same vector times this right that's the first thing you can check So these are eigenvectors of this operator on this space That's the first thing in property And the second thing is if you look at The set of all these anti-symmetric tensor products For j1 J k or j n sorry in n to the n where j1 less than less than jn Well, this is an orthonormal basis Of the exterior tensor product of l2 lambda Right, so I leave this as an exercise to the audience Let's have an exercise to the audience, but it's a simple computation Okay, so what's the nice thing here? Is that well, we know the complete spectral data of our Hamiltonian right, so this gives us The complete spectral decomposition of the full Hamiltonian, right? Well, but of course this is not really what we're interested in because what we see is that all the states Right or the eigenstates of our Hamiltonian are essentially products Of one particle states. What does it mean? It means that the particles live Along each other without interacting at all, which is not what happens in the real world, right? So In the real world particles interact For example electrons interact through kulo interactions Actually, if you add the random potential You should change the kulo interactions Into something different because there are famous screening effects, right? So they actually the decay at infinity is not because of the random potential is not anymore One over the distance, right? It's going to be exponentially Decaying at infinity. Okay, but you need to act interactions So therefore what we're going to do is we're going to add interactions. So pick you Be a function from rd and I'm going to assume that the particles Well, what I have in mind is electrons so that they should repel one another So I'm going to take an interaction which is non negative And the other thing I'm going to assume is that the particles will only be interacting pairwise Right, it means that if there is an interaction, it's between two particles Groups of three particles or more particles only interact Through the pairs you can form within such a group, right? There is no interaction attached to the fact that you have three particles Rather than to have three times two particles Okay, so that means the following it means that I'm looking now so define H omega lambda and you put the u here No, this is an n here and the lambda is here H omega Lambda u. Well, this is just our free operator Plus the sum for all pairs of particles. What does it mean? It means I take i less than j Of u xi minus xj Right, I just add this potential This interaction potential To my operator Okay, so this again gives me a Schrodinger operator Modulo the symmetries And what I'll be looking at so I can let's this to let's let's to make it simple. Let it be continuous Unbounded Okay, so obviously if this is a continuous bounded function Uh This doesn't correspond to Coulomb, right because you have the singularity when particles get close to each other Okay, but nevertheless Start somewhere just to make things simple. I don't want to have to deal with domain problems and stuff like that Well, then this potential is obviously bounded From below, right note that the bound here from below Is rather nasty because you have n square term in here So if you just do brutally a bound, right just lower bound this sum by n and minus one over two The bound for this one you get something here, which is much larger potentially much larger negatively than this operator right, so Tic-tac-tac and what do I want to say I have this And so this defines the h omega n U lambda is But the fact that this is bounded Okay, we'll tell you that you can add it and the Friedrich's extension of this will have the same domain as without the potential u Okay, so this will be given again by This will again be an well is self-adjoint on The domain The operator domain of h omega n zero lambda right domain of the Friedrich's Friedrich's extension And it's lower semi bounded Okay, so that's just for the setting and so what are the questions. What do I want to do with this? well, so the most simple question you can ask For such models is to understand what the ground state is right Of course, if you speak with physicists what they would really like to know is well, imagine that you have this operator right What happens to it? as usual with Schrodinger operators or when you let when you look at The Schrodinger group associated to this operator at large times Okay, what's going on? Of course in this case if you fix n and lambda Not much interesting is going to happen in some way because you have an operator Which is going to have discrete spectrum for the same reasons h zero had Right and you're going to have a bunch of oscillating eigenvalues Of course, they can recombine doing something subtle But the really interesting thing comes when you actually think of the fact that the sample in real life Even if it's just one cubic centimeter size Because of the number of ever gradle inside it you have 10 to the 23 Particles right of order 10 to the 23 particles Which means that lambda actually effectively in any real life sample is going to be sent to infinity Okay, so what we are interested in is the thermodynamic limit So what is called the thermodynamic limit? Which means the limit when the volume Of your space goes to infinity At the same time the number of particle goes to infinity and The ratio Which is the density of particles Goes to a limit which is positive Right, this is what I call thermodynamic limit for short. I call this Oh such that Such that and I call this t arrow Let it be t arrow right when I say in thermodynamic limit row It means that I have the following thing and what so you can do many things now You'd like to understand what's happening to this operator when you perform the thermodynamic limit. Okay, of course the problem Obviously come from here is that well in the limit There is no real limiting operator, right? You're dealing with infinitely many particles in this infinite volume So that's something that needs to be dealt with. There is no limiting operator See many quantities just will blow up Okay, or at least it's going to be some work to show that they don't blow up And so as it's a rather complicated setting, we're going to ask simple questions. Okay So what are the questions I'll be interested or I'll be dealing with later on? It's the following. So let me define e u where it has a number of indices and parameters n lambda Be the ground state So it's the lowest possible Eigen energy inside your system of h omega lambda Sorry h omega n Okay, and let phi Omega n lambda u be the ground state. Sorry. That's the ground state energy forgot the energy And that's the ground state associated or be a ground state Because there is actually nothing that guarantees in general that the ground state is going to be unique Right, it's not something because it's mainly coming from the fact that I have these symmetry conditions right things just of this one right Let me think of this one so take this right and well To take the ground state is easy. What you need to do is to take this as small as possible So you should take the n first Eigen values And take the corresponding states, but imagine that the nth eigenvalue Comes up with multiplicity two Right, then you will get two possible ground states Okay, so it's perfectly possible even if this is nice and has Well, not if the spectrum is simple, right? But if the spectrum has multiplicity here, it's perfectly possible that the ground state for this operator At least for the free one is going to be multiple. It need not be Simple, right? Nevertheless, you can look at what's happening to one ground state, right? The one thing you noticed here in this example is that well, even if the ground state is multiple Right two ground states should not defer too much Because they will defer by a small number each of these states here is going to have finite multiplicity Right and so they should defer by actually a small number of states Okay, that's one thing you can hope for you're so actually the ground states if there are many of them should not be too different from one another Okay, that's something we are going to see later on in our model Okay, so And uh, well, what's the question now? Well the basic question we want to understand Or we want to answer rather say rather said Is the following Well, what happens to e u omega n lambda and psi Phi sorry, I call it phi u n This is a common lambda In the thermodynamic limit Right, how do these things behave in the thermodynamic limit? Of course, you could also be interested in more evolved statements or more evolved questions Is if you look at for example, you look at your system You let it involve so you take it on a Cube, right You let it involve for a finite time And you're interested in the results. So you start with some state Right Well defined. So for example, I don't know you take one of the typical questions that physicists would look at is the following You take the ground state And you take one particle, which is not in the ground state So you take the ground state and remove one particle put in a larger state Okay Okay in the high energy state not larger but high energy state. Okay, and now look at the evolution of this Okay of such a state After taking the thermodynamic limit At large time Does this large this particle that you remove from the ground state Does it decay and go back into the Ground state for the infinite size or for the infinite volume system or doesn't it right? That's one of the typical questions But that's As far as I understand much too hard to solve up to now, right As you'll see there are not many results on these systems. The systems are rather complicated and the study only started recently Okay, so there is one case nevertheless as we already know everything about the Non-interacting case we can start with looking at what happens When the system is non-interacting right just to get an idea of what can happen At least we have one system that we can completely solve so we can start with what happens when Well, actually before doing that just want to make a Comment Well as I'm talking to a probabilistic audience. It's not a problem So you see him for the moment randomness didn't come in right? I mean, it's just I have a general potential I didn't use randomness at all But it's going to come up. So Yeah, use the language that the particles are moving. Yes. So what does that mean? What does it mean? Well, it means that You can for example think of these functions in the case of a random system, right? If you take the free system these functions if you part if you are If your system is imagine that it's one-dimensional We are going to come back to one dimension eventually imagine that it's one-dimensional. So the spectrum is localized Right at any energy. So all of these functions are localized Right and so you can try to understand so they have a center of localization So you can try to understand If they are these centers of localization, right if they are moving When you let time evolve But time is not in here, but time it comes up in the usual way, right to solve the Ah, okay. Okay. That's fine. So you can you can use the Schrodinger equation, right? What I'm solving is I'm not interested in the heat equation I'm really interested in the actually, but I won't look even at the Schrodinger equation. That's too complicated I don't know how to do that, but Times come in. What is this? This is lambda. So this is uh, you right? Yes Okay, and yeah, sure sure psi is coming as well and psi at some time t equals zero is some state psi zero Which of course depends also on n lambda right and of course the limits If you play now and you take the thermodynamic limit here There is no reason to expect that it's going to commute Right with taking With moving time So this for me means Brownian motion Sorry, this for me is Brownian motion in the box Uh, what do you mean Brownian motion in the box? Yeah, there's a one of eye. This is not the heat equation. This is Schrodinger equation Yeah, this is one of our eye in front of you, right? Yeah, this is what you might with brown mentioned Yeah, okay. No, no, that's not the heat equation. This is what I underlined, right? I'm really interested in Schrodinger equation evolution So it won't be relaxing to the ground state Right if you let evolve in general Uh, so yeah, the one thing I want to do. Yes, please Do you have an equivalent of the green function in classical? Sure, you have an equivalent for this one. This is just this this big thing here is just a standard train operator Though it has a green function in the usual sense But it's a useful object or? Untractable. I don't know how to I don't know how to estimate it in any useful way Right, it's too it contains too much information. I don't know how to To extract the information from this object. It's too complicated Right, so I'm what I'm really going to do. I'll show you what I'm really going to do is I'm really using I really study the ground state and states near the ground state Right states that have low energy But dealing with whole green function in this setting I don't know how to do that Right So but before I just before doing that. Yeah, maybe at this point I've said the problem So what I said at first is that there are not many works on this, right? There's uh, well Few papers around Actually, uh on this problem depending so of course there are many parameters that I let free in here I didn't tell you which what the omega is Right, so but I need to mention so this is the part of what I'm going to talk about which is Less general than that is a joint work with a former student of mine. Nicolae Veniaminov But there is some other work, right by uh, vieri master opietro Where he considers well in spirit The same object Right, but for a different v omega he will he will replace r d by Well z d actually z if I remember well, right I'll be replacing r d by r Very soon. Okay to get state results So he looks at discrete operators But more importantly what he is looking at so here what I'm looking at is what physicists call zero temperature model, right? Meaning that I don't take into account that particles should be distributed in a special way according to temperature He looks at positive temperature model And I post the temperature models And the second thing is what is important for him is that the v omega Is not a random model. Well, it's a random model, but it's more precise than that. It's a quasi-periodic potential and So typically right what he has is that v omega of x is something like so it's in one dimension of Of n to specify this is going to be two lambda cosine Of alpha n plus omega right where alpha is What did I do in two Actually has some assumptions right on diapentine conditions Right, you need some diapentine assumptions on alpha and stuff like that. I don't want to go into the details But typically the potentially be looking at will be this And but more importantly is that from the point of view of quantum statistical mechanics The way he looks at the problem is a bit different What he does is instead of looking at just this operator He looks at the scale of operators In which you incorporate all possible particle numbers, right? So he looks at the graded state the the graded scale of My h n where n goes From let's say or zero even that's just a complex space two plus infinity. I don't remember whether he looks at fermions or bosons or actually the non-symmetric with no symmetric On so this is a big Hilbert space right to take orthogonal This orthogonal sum and for each scale he looks at this operator, right? And what he studies is actually So one So what do I want beta h omega? So you look at the sum of this He looks at the operator Right minus mu where mu is some positive constant Which is called well, I don't remember what it's called Uh Yeah, no, that's the that's the model that's the ground canonical ensemble, but mu has a special name. It's called I don't remember the name the physical Physicist name for this constant, right? But roughly what it tells you is that setting mu you change the you So this is h omega n u Lambda right so he has so that's a again an operator Right, which this one is bounded right this one is not bounded actually in his case. It is bounded. Okay, it is bounded and Now what's this constant called now? Well by setting roughly by setting this constant you set the density of particles Right, this is the parameter that measures the density of particles meaning my number row here Right and what he is studying is actually the ground state. So you consider an operator Let me call it a l which depends on many things Okay on all the parameters and what he looks at is what is the ground state of this operator On this space Okay, he looks at the ground state of this operator on this space and he wants to describe this ground state Okay Yes, sure. Oh, that's the chemical potential. Thanks very much. Yeah. Yeah. Sure. Sure. You're right. Definitely Thanks You may like to I put it in the wrong. Let me think Why in the no, it's it's fine. I think if you let's if you send Oh, yeah, you're Let me think if you take the limit beta to zero you should recover this one, right? So Oh, yeah, yeah, you're right. It's one of a temperature. You're right beta is one of a temperature That's true. You're right. I mean, I'm thinking of one of a T actually T is a temperature Yeah, I could t beta the temperature This is what I had in mind Okay, so if I take t to zero a t to zero, right? I get actually my model, right? And of course everything else is going to be sent off to infinity and I get just My model at density Mu Well, that's a paradigm of quantum statistical mechanics that is telling you that you have three formalism, right? I don't want to go into Statistical mechanics, but you have what I called three formalism For a statistical system. You have this one, which is called the micro this is micro canonical, right? But this one does not incorporate temperature Right, meaning that if I know if I would know, right all of these systems for any n Okay, here I'm this fix for any n. Of course, I know this operator. That's the only thing that's coming up in here Right, I know all of this but What so that's roughly if you want a definition in physics There is another way to look at it is rather study this operator, right? That's a given if you want Okay, and to study this operator Should tell you what is happening at a temperature t Okay, and there's a third formula actually a formulation, which is the canonical, which is But I don't remember how it comes Yeah, that's grand canonical, but is it canonical Can include it's just the temperature which it keeps in the number. Ah, yeah, okay, okay So you keep this you keep n fixed and you keep just this one Right, that's the canonical But I don't I mean mathematically. I'm not sure that it was proven that these are equivalent Right, or in which sense they are equivalent, right for physicists. There are three ways to look at the same objects Okay, but I'm not sure that there's a mathematical proof of this fact And that's sort of sensitive to mechanics, okay Oh, yeah, so what uh, so what uh, maestro Pietro does he looks at this operator Studies the ground state for this operator and gives a description of the ground state Actually of something I'm going to define right now, which scored the I think yeah the two particle density functions for the ground state of this operator, right And that's the next thing I'm going to talk about before actually doing the chem computation in the Free case So one of the problems that you have here Is that you let n go to infinity, right the number of particles And so if your eigen functions you let the number of variables go to infinity Right and it's going to be a well If not impossible at least cumbersome To study the limit of such function depending on a larger and larger number of variables So the way you do it is the standard way in probability theory as you can study the global function What you're going to study is its marginals, right? So this is called in physics. It's called the k particles Okay particle density Functionals But these are just The marginals of the projector on the ground state. So what you do actually on any state. So let pick Psi n just keep the track of n a state in lambda and assume that this one is normalized Right pick a normalized state Of course, it depends on n right and Well, you can look at this state as an operator just by taking the spec the the orthogonal projector on this state right you can take Look at pi of n Right take the orthogonal projector on this state and you can then reduce this operator. So this has a kernel I identify it with the kernel. So it has x1 xn variables y1 yn variables and you can now look at the marginals of this kernel just in the following way. So this is called what is called Gamma sub k psi of n define Uh, when it's just integrate over d x k plus 1 d xn, of course you require that n is larger than k plus 1 dy1 dy k d sorry k plus 1 Tn of pi n x1 xn y1 yn So it is Psi k psi n it depends on two k variables right And of course it satisfies It's anti-symmetric in both sets of variables because it was on this side right when you integrate this you keep the same property You're integrating an operator of rank one Okay, so of course the thing is only defined almost surely right in y1 1n by the fubini theorem, for example Okay, but it's going to be a trace class operator This one was rank one, but this one is just going to be trace class and one thing I forgot to do is Normalize it properly I take N choose k right why do I take n choose k because this gives me the normal the that's the number of So if you integrate this so the one thing is that if I take the trace of gamma k psi n This is going to give me n choose k so it's the number of k tuples you can pick among n particles Right and the second thing so this is trace class right the gamma k Psi n is trace class And the gamma k psi n Is non-negative right these are the main properties Okay, and so that's the one thing I wanted to say So what So when we want to study now Uh, so instead as I'm not able to study Psi to the ground state or one of the ground state psi omega n u Lambda We are going to study its k particle Gamma k u How does it behave or how it behaves in the thermodynamic? study We study this right to just take these marginals and study the marginals in the limit. So here we have a nice a fixed number of Particles right and we can make sense of this So the first thing I wanted to do is see what happens if I do it in The free case right without any interactions so We'll study this and the other thing is we're going to study of course EU omega n So what happens in the free case? So when u equals zero There everything is simple Because we actually know everything that's why it's simple So when u equals zero As I said e omega n zero of lambda Well, we see it here. It is just given by the sum For j going from zero to n minus one Of e j omega lambda I keep the same notations and these are just the n first eigenvalues Of my one particle system, right? Just you have one particle in a box with random potential and you look at the n first you look at the n first eigenvalues So what i'm going to see immediately is that there's no reason why this thing should converge right, but what is likely to converge is The intensive quantity meaning if you renormalize by the number of particles Right, it's not sure to expect that if you increase the size of your system The energy is going to be increased by the number of particles within your system okay, so This actually can be computed because we can rewrite this actually its limit can be computed in a in a rather general Case we can Well, we can do the following Introduce so recall that Or let Let what let n sub lambda of e be the counting function for the eigenvalues Of h omega restricted to lambda. What does it mean? It means that n lambda of e Is just the number of e j omega lambda less than e Right, I just take this counting function. So this is a well defined Okay for the counting function and there is another thing that actually well another feature of random operators that we know how to control is that If we know that the operator actually the random operator we're looking at Is ergodic like it was the case in the talks of zimona wadzah at the beginning of this summer school Right, for example the anderson model or continuous anderson models Well, then one can show under rather general condition On the random potential v omega. You can actually show that oops. I didn't hurt anybody. Okay, that's already something Uh one over the volume of lambda divided by n lambda of e Well, see this depends on omega. I forget. Yeah, let me put the omega here Thanks, christophe Well, this converges When lambda goes to infinity Almost surely to some limit which is called the integrated density of states Right, so this is known on the fairly general conditions on the operator both in discrete and the continuous case uh well so this suggests that A way to rewrite so what i'm study what i want to study is one over n e omega n zero lambda Is equal to one over n integral from minus infinity So this doesn't matter this doesn't hurt right because we are looking at we are dealing with the lower semi bounded operators to energy to energy what to energy e n minus one omega lambda right And here I have e times d n. So I just take The distributional derivative or derivative measure of this at e Right So i'm going to divide this by lambda Multiply this by lambda All right, and how is this defined? E n minus one omega lambda by definition It is the lowest point where n lambda omega e Is equal to n so if you divide this by lambda this is divided by lambda, right? It's the lowest energy Is the infimum so let me write it here It's the infimum of e such that You have this right so The one thing in your favor is that well this measure is non negative meaning that this function is non decreasing Right, so to make things simple Imagine that the limit this limit is a continuous function. This is also known to be the case in many random systems Right, not always right our random system, especially with magnetic fields where it's not the case this function can have jumps But if you don't have magnetic field in many systems with a real symbol This is known to be continuous. Well then by Dini theorem, right you have some uniformity of the convergence Okay using Dini theorem and so you can show Or expect you have to show that e n minus one omega lambda converges when in the thermodynamic limit row to e row defined by Defined by n of e row. Well, this converges to row. We said Is equal to rule Right, you can expect that actually The solution to this minimization problem converges to the solution of the limit of the minimization problem And this can be shown under suitable assumptions on the regularity of n Especially if n is continuous actually it's enough to have n continuous Uh, so that's a fair thing. So it tells you where is my integral here? So what you expect now Using the same argument is that this actually is going to converge to well this converges to so this is in the thermodynamic limit row one of a row in the graph from minus infinity to e row e d n of e Right, so you get that this converges to what i'm going to call easy row Uh Well, which depends on your row. You see that this is a non random number Right, which just depends on The initial random operator Right on rd Okay, so in a way Well, not completely half of the mission is accomplished in the case in the free case because you already got The limit we wanted to compute Of course in general, you don't know this number. You don't know this function, right? But you can say a number of things about this function Okay And what happens? What about well to make it simple the gamma k Of psi What did I call it zero omega n lambda Right, can you show something of the same kind? For the k particle reduced density matrix for the ground state Okay, what do we know? We know that this thing Well, we know what it is. It's just phi zero Omega lambda exterior phi n minus one Omega lambda So we can compute from this Right from this we can compute Gamma, let's say one just to one particle density matrix Okay, and so what is this by definition? Uh, well, it is just as we said the integral Of a day x1 the Sorry, I did something. What did I know? Uh, here I did something which is not Correct here. Why did I integrate over all of this? No, that's not what I wanted to do What I want to do is this and here I take Y k equals x sorry y j equals x j for j going from k plus one To n that's going to be better for marginal, right? I project on to all of these Right, I take the y's to be equal to the x's Right, so I project on the diagonal over the k n minus k last Terms, right what I wrote here was false Okay, so if you want I can do it right in this way. So it's x1 x k x k plus one x n comma y1 y k x k plus one x n but these have to be taken the same right I want to project down onto these Okay, so here I have dx1 dx2 dx n and I have what I have my determinant Phi j of x k Phi j of x k J running from j k one To n okay this determinant Sorry Well these are I can assume that these are real right my potential is real I didn't say that I want something self-adjoint So I don't need to put any bars and here I have Phi j x y one Or y k Right one j k n right where I'm going to write it in here where y k sorry y2 equals x2 y n Is equal to xn Right I take this product Okay, so this is this at x1 y1 Okay, and if you do the computation So this is left as an exercise Right you integrate this if you do the computation. What happens? Well, what you get is that this is just the sum For j going from zero to n minus one of Phi j x one Phi j y one Right, so what is it? It's just the orthogonal projector, right? It's the spectral projector on the energies from minus infinity to en minus one Omega lambda Of my operator on the box Right, it's just a spectral projector on all the states That have energy less than en minus one Okay And one can show That this actually in the thermodynamic limit row converges to the spectral projector For the full operator random operator On the energies below euro. Yeah, one thing I didn't say is that is euro is called the fermi energy Okay, one can show this so of course here. We are dealing with compact operators, right? So that's a limit Actually of Sorry a little bit. What did I say? Okay, that's the limit of So this is a trace class operator, right? But the trace of this thing blows up right the trace of the thing I forgot the n here Okay need to normalize this by n Okay, the trace of this thing Blows up this has incident trace Right, so we have the conversions of two operators Okay, they are bounded what you can show this that these operators are bounded, but nevertheless you need to specify a sense for this convergence, right? So depending on the model you can show for example, if the model is regular enough you can show strong convergence Right, or you can control show and this is more much more general converges Per number per particle meaning that you have convergence in the sense 1 over n The trace of a depending on n minus b n Right, sorry This goes to zero Right, so you take trace per particle Okay, so this depends on the model you have what kind of convergence you control I don't get the interpretation They'll probably seek interpretation of the gamma clip science, maybe you explain it to them But it's it's it's an operator version if you had imagined that you take right The joint distribution functions of your random variables Okay, to get the marginals you just integrate out A number of these random variables, right? So here what you have is that you don't have these are not random variables. These are kernels, right? kernels of operators But nevertheless, they have positivity properties. These operators have positive properties And what you do is you just take partial traces instead of integrating out I take a partial trace with respect to n minus k Particles and that's another way to see this right taking this integral is just taking partial traces It's the same thing if I take an orthonormal basis well chosen, right of The n minus k last variables and I take the trace over this for fixing the k first take a trace over this I get exactly this integral Right, so you trace out A fixed number of actually a growing number of variables Is there an interpretation in terms of the dynamic of the particle of the gamma k of psi n? No, because this is for a fixed state. There is no dynamics here. This is this is for if you take you take a sequence of states, right? With increasing number of particles Okay, and what you're doing and what you want to understand is Define a limit For such a sequence of states of number increasing of particles, right? So you could define in a way a weak limit by saying that all the Right finite density matrix will converge Of course, if they do there is a compatibility condition that is going to be satisfied so it must be at the potatoes as Test functions if you want. Yeah, it's it's a way of testing. It's a it's a notion of weak convergence right It's just like I imagine probably theory you do this the same way. I mean imagine that you have A family an increasing of increasing size, right of random variables You can look at the marginals and you can say that this family of increasing size convergence in the whatever sense The sign means that you look at the test again a certain function that the If you want yeah, you can you can say this yeah another way to look at it If you want to look at this operator you can apply this operator to two Vectors by taking a scalar product and then it's going to be testing Right plus taking a partial trace and here it's not it's it's not really testing here, right? Because you're taking a partial trace not to say the same thing. It's a different it's a different operation But this operation corresponds to taking marginals in So but really the idea is that on another way to interpret it imagine now that your state Giving rise to this is pure right in the sense that it's exactly Of this type right is just a slater determinant Then what you obtain in this case is that you obtain exactly all the k groups or the groups Or rather said all the Way you can group k particles together This is exactly what's coming out from this computation All right, okay Yes, please Yes, and the right side too actually Oh, yeah, sure, I forgot I forgot to put So you see that's also an operator This is a projector. That's just a spectral projector Of your random Hamiltonian Right So I wrote it as an operator without forgetting about the kernel variables No, no, but that's what does it mean This is a function applied to the operator So you have an author you have a self engine operator and you take a function of this operator Okay, it's just like a function of a matrix So what does it? Oh, okay, so this is If you want you can if you take I don't know in which way to explain it But if you take for example the spectral resolution, so that's that's if you want this would be the definition of this, right? That's the definition of this But you can write it in a different way if you have a self-adjourned operator h Right You know that you can actually write h as the integral of lambda And here you have some spectral valued measure of some projection valued measure Right and you integrate lambda and so what is a function of h? Is just the integral of this function of lambda d e lambda Right my function here is this Right. So what I'm doing here is I just take integral from minus infinity to e omega Oh, sorry At minus one omega lambda d e lambda up to yeah, but yeah, I wrote it. Sorry. I start with minus infinity and I write it here So it's going to be clearer And minus one omega lambda. It's the same thing So that's the spectral projection of my random Hamiltonian Okay, it's the same thing as this just a function of the operator Right the projection that you take that this is a projection. This thing is a function of the operator Okay, sure So Do you know do you know this language like the terminal processes Uh, not very much but a little bit Okay, because I think that that should somehow be related very closely to what you're doing And and you know like the GUI algorithms for example, that's uh, that's uh, their distribution is a very simple example Yeah, but it's not it's my Well, you see Okay, it's it is it's not really related to this in the sense that Of course, this comes up. You see this this story is I start with a single particle model and go up increasing the number of particles and increasing the size of space The thing you are talking about is actually something which is valid for the single particle Model, right? We are looking at just one particle model and its eigenvalues For this model follow a special Low right as random objects here. I mean in this thing I actually Need well almost nothing about this low, right? This is not what I'm studying here. I'm really doing the following thing is I'm I'm Trying to so this is still in the free case. So it's more or less. It's pretty it's obvious It's trivial. All of this is a very simple computation, right? It's just a free case, but it's more A way of an algebra. It's an algebraic procedure if you want, right? I'm doing algebra Yeah, but it's not the same kind of determinants I agree of of course they're the same kind of it's the determinant. There's only one, okay But it's not it's not determinant in the sense that you're taking determinant of the special of some special functions That comment in your random process in your in your random operator one particle random operator so I'm not sure I answered your question, right? But it really I think that the There is a difference in point of view in the sense that here what I'm looking at is I start with one particle I reproduce it I take any of them of the same kind and what I'm interested in is not really this computation because this is Well, it's finished, right? This is the case the free case nothing else to do we've got everything Okay It's what is happening is that if you put many of these particles together Okay, and add some interaction. What can you do then? Okay, and then I don't see why the terminal third person Yes and no actually the the fact that you're fermionic of course there is there is you have some interaction But it's not going to change the If you look at for example The gamma case the fact that they are fermionic will not Change this in if you take if I take the same computation, right? and look At the non-symmetric not the bosonic bosonic is going to be completely different story because then to get the lowest ground state As you can put everybody together into the same state You're going to look at everybody in the ground state. So it's going to get a completely different picture, right? But so in this way, you're right in this way. There is already some interaction due to the fermions That's correct I think if you just take that interaction, yes, and nothing else Yes, and no no no extra stuff You already get the g u e I can buy you Uh, but where do you I don't understand what you mean with with the g u e and good eigenvalues? Just clear the story that you're doing Maybe use the potential Yeah, but with g u e where where do you get the g u e? But what do you where where do you Position the particle as eigenvalue Ah But that's another story. Okay, sure. No no sure the Yeah, okay, but that's that's a completely different it's a reinterpretation of the model That's not the particles. I'm interested in Right, that's not the particles. I mean the position of the eigenvalues are not the particles. I'm interested in that's the different story Yeah, yeah, that's that's another story. Yeah, yeah, okay Okay, the particles I'm interested in are really something which is they are living in here But space and actually that's the question to come back to the question you asked before right if you take a true interacting system There is no way you're going to be able to single out a particle Right, it's going to be a mix and there's there'll never be a true particle that you're going to be Able to pull out of the whole thing except you can fix a state like this And but once you've let it evolve through the operator. It's going to be all mixed right so Speaking of a single particle is something which is rather difficult Okay, so Well now so this is nice It's very general and it's very simple makes it nice also and let's come now to uh, well something where you have an interaction So of course i'm not able to give any statement Uh about general operators in this case But what we're going to do is we're going to when i'll be doing in the next Well three hours not the following ones right the ones coming tomorrow and the day after tomorrow I'll be developing two things First I speak about the special model. I'm a special model. I'm just going to introduce right now And then I speak about the model which is a true The special model will not be of Schrodinger type Right, it'll have some flavor of Schrodinger and some simplifications and then I'll explain how you can use the intuition Gained studying that model this toy model To study what actually what actually happens for a true schrodinger model Okay, so the first thing is the toy model So it's called also what actually it has two names depending it was coined in the 70s at almost the same time on both sides of the iron curtain and uh, well Because of this it bears two different names on the two sides of the iron curtain So in on this side of the iron curtain that doesn't exist anymore, but Some of us remember it's called the letting us see model and on the other side It's called simpler the pieces model They called it they found it so simple that actually they didn't bear to give the name of someone to it They just called it the pieces model. So what is it? So that's going to be a model for my one particle system Right what I'm going to define now is the one particle system. It's simple. What you do is the following thing You take on r You take a Poisson process So let's xk k being z of omega Be a your Poisson process And while normalizing things I take intensity one Okay So I look at my points And I look at the points falling between minus l over two l over two Right Let me take some control here so I look at the x k's between minus l over two l over two and at all the x k's I'm going to put directly boundary conditions Right, so right this here also in I'm going to use purple So I put directly boundary conditions What does it mean? It means that I'm going to look at the random model h omega l We put it l here l is just my interval Is the direct sum of minus laplacian On x k x k plus one so I put a tilde At omega For k going from some random number k zero of omega to some random number k or k minus omega Plus of omega right with directly boundary condition Right and of course the x tilde k Is x k if x k belongs to Minus l over two l over two Right and it's equal x tilde k is equal to minus l over two well when you guess right And it's going to be equal to l over two on the other side. Okay, just complete it You just cut it off here Okay, so the nice feature of this model Is that of course it's localized Right, it's trivially localized actually it's even worse than that It has eigenfunctions. It has only eigenfunctions of compact support almost surely Right, so of course we know what the eigenfunctions and eigenvalues are right We know that Eigenvalues Are exactly the number so let me call these pieces Let me I'll use this Delta k is equal to x tilde k x tilde k plus one right of course x tilde k and x tilde k plus one never meet Okay, almost surely And so the eigenvalues Are just the pi divided by the length of delta k Okay, so Yeah, okay, well the Laplacian will not come up anywhere I'm gonna get rid of as an analyst it may be a bold move, but I'm gonna get rid of the Laplacian Okay, so let me keep delta k otherwise. I'm gonna I won't remember Okay, so these and the associated eigenfunction Associated eigenfunction is going to be just sign of x right Well actually to be correct I need to normalize it properly right to the way I define my intervals And maybe one over a square root of two coming up somewhere here right and maybe I forgot some Yeah, I forgot some Lengths of delta k one half Something like this right there is a square root of the length If when I integrate out otherwise, it's not normalized properly, but okay, that's fine. So we know everything Okay, so this this thing it's not a Schrodinger operator right because not minus Laplacian plus potential But it is perfectly localized And the one thing it doesn't have that the Schrodinger operator has Which is a very important feature right which is actually the feature why Schrodinger operators were invented in the first place is tunneling There is no tunneling for such an operator in the sense that No particle can go from here to there Right these are hard walls. They are perfectly separating. There is tunneling within each interval It can happen because the intervals can be rather large But there is no tunneling over the whole length of the interval minus l over 2 l over 2 Because you have put hard walls right everything is separated. So this is why the thing has compactly supported. So yeah, I forgot of course one over I forgot to Say that they're compactly supported Okay, they're compactly supported eigenfunctions. So this is Well, this is one of the problems you'll have to if you want to go to Schrodinger operators That's one of the problems you'll have to understand right how you can build in tunneling again. Okay But how much time do I have four three four minutes? Okay, so this should be enough for If I find my page again, yeah, here we are Okay, so let me well in the three four minutes. I'm just going to do one thing. What happens? Maybe two Well, let me just put these is to put some theorems. So look for this we can define the h u omega l sorry n And l right I define I replace My h omega lambda by h omega l right and use the same formulas and define this is the n particle with interaction u Okay, and what do I assume over you? Assumption on you. I just need one assumption I assume as before you pin on negative, right? I didn't really use it before but here because I didn't study anything with you I assume Here I have only pair of interactions. I assume that u belongs to some lp for some p larger than one Right, I don't assume it to be compactly supported, but it's And I assume one last thing I define z of x is x cube integral from x to plus infinity of u of t dt right and I assume that this thing Goes to zero When x goes to infinity. So I assume u to decay a little bit faster than x to the minus four Right, so it has long range. Oh, well, it doesn't have compact support, right? It's a little bit better than that So under these assumptions Whether I try to prove is the following Uh, notations are the same as before So first theorem is that Omega almost surely 1 over n e u omega n L Tends to some function e u that depends on rho When in the thermodynamic limit rho And What is the more interesting part? Is that if I compute I can give an assumption in expansion of e u rho the first term Is the free ground state zero energy plus pi What is it pi square? Gamma star divided by times rho divided by log rho modulus to the cube One plus little of one Where where what I need to define gamma star Gamma star is 1 minus e to the minus gamma 2 pi 8 pi squared sorry 8 pi squared and I need to be telling you where and Gamma is given by the following proposition consider now Let e Two particles with potential u l to be the infimum Of the spectrum Of minus d d x So this is just a laplacian in two dimensions plus u of x minus y Right on L2 of zero l tensor exterior so it's anti-symmetrics on l2 of zero l exterior l2 of zero l so Okay, so I take two particles in a cube of size l But of course they are anti-symmetric then For l large you have that e to u l Is equal 5 pi squared over l squared plus gamma over l cube One plus little o of one And this defines the constant gamma Right, that's the gamma coming in here. It comes. I don't know how to compute it, right? So it depends of course on you This is gamma of u Right, but it's defined as The second term in the asymptotic. So that's the ground state For two particles anti-symmetric particles with the laplacian alone Right, and that's the correction term coming from adding Interaction and interacting potential between the two particles Okay, I think I'll stop here Yes, sure The little o here is when l goes to infinity in this one And this one the little o is when rho goes to zero, right? That's a little o Little o of one goes to zero with rho But here it goes to zero. There's no rho here. It goes to zero when l goes to infinity So it's not a small interaction limit This is a small density limit. This is rho is small, right? For this for this So one thing I should say is that this one thing, but we'll discuss this tomorrow. This is a size Log rho to the minus two Right, so this is much larger than this, right? You have a rho here And this thing is to have full power rho here. It's only one of log rho squared Right Okay, thanks very much for your attention Any more questions today? So For the final population, yes, e to u is just the eigenvalue or what they call What this No, this one. Yeah, exactly. The infimum of the spectrum is the ground state You take the ground state of this operator. So you take this operator On this space, right? You can show that the ground state is unique in this case Right, and so that's the ground state energy. So it's the bottom lowest eigenvalue for this operator And this what you show is that this eigenvalue has an asymptotic expansion when you let the size of the square go to infinity Right, and this asymptotic expansion looks like that and it defines this constant gamma of u Right, and this constant gamma of u is what comes in here And another question that Just know you you you see something about this This gamma k per side zero Exactly. This is going to sorry You you say We are you say something about the the u equal to z is okay, right? But how about u is not not too okay Well time is over today, but it's going to come tomorrow, right? You're going to you're going to get the results for gamma Actually, I'm going to give the results for gamma one and gamma two tomorrow This is a trace class operator exactly this is a trace class operator, right But that incorporates Many eigenfunctions. Well actually not eigenfunctions in general what you have is It's kind of a projector, right It's it's not a projector in general. It's not a projector. It's something more complicated than that. Yes, right, but it's It it is a characterization of this function, but of course you cannot you cannot recover In the same way as you do with random variables You cannot recover this function from a single marginal Yeah, right. This is not enough You need to have all of them Right and to a pretty good precision meaning that you need how fast this goes there Okay, but it's going to tell you this is going to tell you roughly where your particles There the the difficulty here is the following thing you have a system with n particles, right? Yeah But the system with n particles You are not able to single out because the system is interacting There is no way you can single out the particle you can and you're not being able to tell you that Well, I picked my particle number k It doesn't make any sense, right because the whole thing in a state like this is over mixed Okay, but nevertheless What i'm going to do is I give you a result on this But I will also do something else which is really particular to this model I'll be showing you How the particles are distributed within my pieces Right in this model, you can do something which is I come to that is part of the proof Actually, and this is why this model is interesting. You can actually say that In the ground state a given piece contains a number of particles Okay, I explain how to do that tomorrow. Okay, so this will complete Actually this from this description that you get the description of this That we chose to give a description of this thing because these are the traditional objects That physicists look at When they are dealing with Many particle systems, right Okay, that's that's that's really the the the objects that physicists look at but they're really exactly the analogues of marginals for probability distribution So whether it's a margin or it's gonna be of us a lot of information. Yeah, exactly. It's it's going to give us Not all the information of course, but anyway There's one difficulty is that a single particle in such a systems You remember the total energy of your thing is going to be of order the number of particles So a single particles anyway weighs very little There is no way you're going to ever get a description which is precise enough to tell you what a single particle does You'll only be able to tell what groups Such large groups of particle are doing Right, and I'd be happy if I'd be able to do it with groups that are not of size Proportional to the total system size Right, if I even doing this is very hard Right going to little o of n taking groups of particles which are little o of n is difficult right Sure, okay. Thanks very much