 Thank you very much. Thanks for organizing this great program, for, well, having this stimulating summer school in this nice environment. I'm glad to be here. I have to say right away that my talk is somewhat different from the talks that we've heard before. In particular, there will be no time and there will be no waves. Now that is, I have to say, not because I don't like time, but simply because I can't do it in a time-dependent way. And I think it would be great if I could somehow motivate people here to work on this problem and to do some time-dependent version of what we do here. And if you think somehow historically, there has been about 10 years ago a lot of work. This was to a great deal, motivated or initiated by Erdersch, Schlein and Jao about how some microscopic quantum dynamics give rise to an effective macroscopic dynamics. And also for their work, I think it's fair to say that the understanding of the stationary theory is basic. It was the first step and once this is understood, then one can do the time-dependent step. So therefore, I'm trying to make, I mean, it's kind of analogous to what they did except we do things at positive temperature. It's a different regime, but the setup is similar. And so I try to explain this stationary model. But keep in mind that there are time-dependent Ginsburg-Landau equations and would be a big problem to understand those and derive those from these microscopic equations. So the title, as Funk already mentioned, that's Microscopic Derivation of Ginsburg-Landau theory. Most of the results that I will present were obtained in joint work with Christian Heinzel, Robert Saringer and Jan-Philippe Soloway. And especially today's lecture is also based on joint work with Marius Lem from Caltech. And if you want to read something about this background, so we've been working on this for at least five years. Now there's a review of Heinzel and Saringer. So this is a review in Journal of Mathematical Physics 2016 where you can get some background information and some, I mean, references to related results. Okay, so there are two buzzwords that I want to mention that are really important for what's going on. The first word is that of a phase transition. So phase transition, that's a word from statistical mechanics. This happens that if you vary some parameter in the system continuously, then some property of the system changes abruptly. Changes abruptly means that the quantity or derivatives of it are discontinuous. Okay, now that's a rather surprising phenomenon when you think about it because usually these things, they are finitely many particles and usually things behave analytically in this parameter. And then somehow when we pass to this infinite volume limit, we lose this analyticity and we get these jumps. There's one phase transition that everybody knows that's that of water, right? We know water in its solid phase, its liquid phase and its gas phase. And you see the phase transition, that's not a random phenomenon. It's rather the contrary, I mean our thermometers, right? When we talk about Celsius, right? That's zero Celsius, that's where the phase transition between solid and liquid happens. And when we talk about 100 degrees centigrade, that's where we go from liquid to the gas phase, right? So it's really, we've made our thermometer based on these phase transitions of water. Now, I will also be speaking about the phase transition and its so-called superconducting phase transition. That's a transition that happens at much, much lower temperatures between either 1 and 20 Kelvin. And it's a phenomenon that certain materials lose, completely lose their electrical resistance when you pass a certain critical temperature, okay? It's a phenomenon that was discovered, I think, in 1911 by Kammerling Ons. And it's a quantum mechanical phenomenon. It's a phenomenon that cannot be explained by classical physics, classical mechanics. And it's a macroscopic phenomenon. By this, I mean you don't need a microscope or anything to look at this, but you can really watch it with your bare eyes. And it's one of the few quantum mechanical phenomenons that we have like this, okay? And that brings me, this scale thing, that brings me to the second big word that we have here. The second thing is that of effective equations or effective theories. I should draw the same. By this, I mean when you drink water, when you do something with water, you don't want to think about H2O molecules, right? You just want to have your water and you think of it and you have a piece of water, a size, and that's how you want to think of it. You want to forget all the atomic structure that's below it. And the same thing happens about superconductivity. When you want to make an experiment or use it for something, then you want to forget the atomic level and you want to think of this really on the scale of the thing that you look at. And that put differently when you say some are in this superconductivity phenomenon, they are somehow 10 to the 23 particles involved. You don't want to work with a function of 10 to the 23 variables, right? That's what quantum mechanics would tell you to do. But that's just an insane thing. You cannot do this, right? But you rather want to describe this thing simply by a single function of three variables. And so what an effective theory does is somehow it takes this huge wave function of 10 to the 23 variables and somehow cooks up an effective object, which just depends on three variables. And everything, or at least the important things, should be describable by this new thing of three variables, okay? And so I told you that this phenomenon was discovered in 1911 by Kamaling Ons. It took quite some time until 1950 until Ginsberg and Landau came up with an effective theory. And let me write down this functional. It's not so important yet. There will be some constants and we'll talk about those later. It looks very much like a Schrodinger energy functional. A is a magnetic field. Psi is this function, which depends on three variables. It lives on some open set omega. That's where your superconducting sample. Let me put some w, psi squared, perhaps another parameter. I'll speak a lot more about these things, what it means. I just want to have it on the board here. So I want to have a minus and then plus lambda 3 psi 4, okay? So essentially this looks like this ordinary Schrodinger energy functional, except that you've added a nonlinearity. And it's the simplest nonlinearity that you can think of. It's psi to the power 4. And then what Ginsberg-Landau said was, well, you have here four parameters, lambda. The lambdas, you go measure those. And then once you have this, then you can use this functional and make predictions. And somehow it works. That's kind of a miracle because there was not much understanding why you should have those things. What these terms mean, it just is a surprising thing that using this functional, you can describe superconductivity to very high precision. It's a thing that's also numerically very convenient. And also in the mathematics literature, this has been studied a lot because it's somehow prototypical for many interesting phenomena. And then, so this was 1950, in 1957, BCS, 1957, Bardin, Cooper and Schriefer proposed a microscopic model. Now really a model for superconductivity that really works on the atomic level. And then, of course, so now you have two different models and somehow how are these two things related. And that was clarified, at least in the physics literature, in 1959 by Gorkov. And there's an alternative argument later, later Duchen. And these two, they showed how BCS theory leads in a certain limit to Ginsberg-Landau theory. And so what I'm interested in is making this statement mathematically rigorous. I mean, these papers are kind of mathematical, but not to the extent that a math that can be made rigorous. I mean, this is not hand-waving, is what I want to say. But it's not like we quantify error terms in these approaches, but we have to do something completely different. And this is actually a rather important aspect of our work. We have to really come up first of all with a model where we put length scales in and really get some well-defined limit that we can analyze mathematically. Anyway, if you want to give a name to the field that this is about, or this talk is about, you can call it non-commutative, qualitative calculus of variations. Calculus of variations, therefore, what do people do in calculus of variations? They want to minimize something. And that's actually a fundamental idea in our work as compared to these physics papers. We do not want to derive the Ginsberg-Landau equations from the BCS equations, but rather we want to derive the ground state, the minimal energy state in Ginsberg-Landau theory from the minimal energy state in BCS theory. It's one motivation why we want to do this is because we say, well, perhaps BCS is not the whole truth either, so I should backtrack a little. We do not only do it for the ground state here, but for approximate ground states. And so we're saying that our work is therefore more general because even if BCS theory does not describe the whole truth and if there's a more complicated model which has more effects, then what we would hope is that these more complicated ground states are approximate ground states for the BCS theory. So there's a stability statement involved. And these almost ground states of BCS are almost ground states of Ginsberg-Landau. That's what we prove. It's yet another question to really do something on the level of equations. And by this I mean solutions to the equation which are not ground state solutions. For instance, such vortex solutions that you might have heard about these are precursor of lattices. They are probably not ground state solutions, but they are higher up in the spectrum. And so there would be some correspondence between solutions that we do not have. That was the calculus of variations part of this name. And if you look at this functional, what I will talk about today is that to a logic's extent this part here, that's what's often called the Mexican head. You see if you look at this, this is a psi to the power 4. This goes up and then it goes down. There's a certain minimum and at zero it goes like that. You think of psi as a complex valued function. This part, this can be explained by a classical calculus of variations theory. That's what I will do today. Now the other part here, in particular that one, that's a lot harder. I hope I can tell you a little bit about this in the two following lectures. This is done using semi-classical analysis. This thing really comes from the fact that if you have a multiplication operator and a differential operator they do not commute. And that gives you commutators and these commutators they're really relevant for these terms. And that's somehow this non-commutative part that will play a big role. But in order to just explain you what we want to do and you know what the difficulties are I just want to start today with a commutative version, just ordinary calculus of variation problem. It's something that I guess could have been solved like 20 years ago or something. Nevertheless, I think it's good if we go through this slowly and get used to this mechanism. So that's now really the first part here. And that's translation invariant ECS theory. And this follows to a great extent this paper with a lemma. Okay, and so the goal for today is that we will see mathematically rigorous a phase transition arise and we will see how this Mexican head is related to this phase transition. And so we have a certain class of admissible states. There will actually be pairs of functions, gamma and alpha. Okay, and this alpha, this is what first is called the Cooper pair function or Cooper pair wave function. And that's the object that describes superconductivity. I should, by the way, just a small remark. I will use the word superconductivity. There's a similar phenomenon which is called superfluidity. I will use them interchangeably and always mean superconductivity. Okay, even though this might be not completely physically correct. Anyway, so we have a functional F which depends on T. T is the temperature and we'll minimize this over all such admissible gamma and alpha. And the question is simply if we minimize this, does the minimizer have alpha equal to zero or not? Having alpha equal to zero means we're in the normal state. Having an alpha which is different from zero means we're in the superconducting state. And what we hope is that if we change the temperature, then there's a transition. Namely for low temperatures, we're in a superconducting state. So this minimization problem should have a minimizer alpha different from zero. And if T goes above something, then we want to have an alpha which is non-zero. Okay, that's the basic question that we will ask. All right, so let me do some assumptions. I think we have to start on this. Okay, thank you. So first of all, well, there's an underlying dimension and we'll always assume that this is one, two, three. This is something to do with solar left inequalities and the power of psi to the four. But that's certainly fine for all physical applications. Now, the admissible states, these are functions, gamma and alpha. They are both functions which are defined on Rd and take values in the complex numbers. And such that there are two conditions that we want to have and they will be nice functions. So I can Fourier transform them. And so the Fourier transform of alpha is point wise. I mean, it's square is point wise bounded by the Fourier transform of gamma hat times one minus gamma hat. Okay. And the second assumption is that's a regularity assumption. That's that one plus psi squared gamma hat of psi. So this in particular means that gamma hat has values between zero and one. And here we have that this integral d psi is less than infinity. Okay. This is kind of H1 assumption, right, except that somehow the gamma hat is kind of the square of something. So it enters here linearly. This implies you see that alpha hat squared is less than gamma hat. Right. And so therefore this implies in particular that alpha is an H1 function. Okay. Now what we will often do is we will write this first condition. So we'll write, we will often write capital gamma. This is a two by two matrix, which has the little gamma as its one one entry. The alpha hat as its one two entry. Then we make it self-adjoint by putting the complex conjugate down here. And there we put a one minus gamma hat. Okay. Now it's a little computation, right? This is a two by two matrix. We can all compute its eigenvalues. And we can say that this assumption that the diagonal entry squared is less than the product of the diagonal entries. That means exactly that this matrix in the sense of matrices lies between zero and one. And observe. All right. Good. And then one more thing. This I just say this for completeness. One could also assume that this function alpha is an even function. That's what we will need in the second and the third lecture. Therefore I'm writing it on the board, but the theory goes through exactly without this assumption. So don't worry about this. All right. So these are the admissible states. Okay. So the first one is gamma and the alphabet that sits in the off diagonal. Now I'm going towards defining this energy functional. And I will have the, so that's often called the free energy. And it will depend on a couple of parameters. And the first one is, well, the obvious one is this parameter T, which is the temperature. The parameter mu, which is a real number, which is called the chemical potential. The interesting case is mostly when mu is positive, but you can just treat this as a parameter. And then there is an interaction potential V, which belongs to some LP on our D. It's a real valued function. And so this is called the potential. And in order to have a minimization problem that is bounded from below, we have certain, we have to assume certain regularity conditions. So that's P greater than one in dimension one P greater than one in dimension two and P greater than three over two in dimension three. Don't worry too much about these things. This is really just if you want to push it to rough potentials. But everything is also interesting if you assume that V is a bounded compactly supported function. Okay, good. Now what's our energy functional? So f of T depends on these two parameters gamma and alpha. There's a first term that corresponds to a type of kinetic energy. XI squared minus mu gamma hat of XI D XI. Now there's another term, which I'll write later. And the last term is V of X alpha of X squared DX. Okay, that's like a potential energy if this would be a Schrodinger functional. Now you see these terms do not interact. So this is just involves a gamma. This just involves an alpha. So now the missing term in the middle, that's where they compete with each other. Okay, and that's an entropy term. And that's the following. So that's T. This is how the temperature enters in the game. Let me write it like this. So I've introduced for you these two by two matrix. And now we just take this matrix times its logarithm. Okay, that's still a two by two matrix. You do this for every fixed XI. And while you compute this trace, the C2 trace, and that's the coupling term. Okay, yes? Can you briefly touch on the physical interpretation of gamma? Yes, I mean these are the electrons. So to say, or the particles, the usual particles, that would... I don't know when this is your question. I dropped here certain interaction terms of the gamma. So there is an interaction of the gamma with itself and some exchange term. Those I've dropped in order to make the model simple. Okay, so therefore you don't see them. Therefore it looks very simplistic at this point. But I think this captures the difficulties. Okay, we can talk more about it. Let me just make this a little bit more explicit, right? If you don't like somehow traces and this notation. Well, we can just compute the eigenvalues of this gamma, right? For every fixed XI. And we'll see that the eigenvalues of gamma hat of XI are 1 half plus minus gamma hat of XI minus 1 half squared plus alpha hat of XI squared. Okay, these are the eigenvalues. So therefore what is this trace? Well, this trace is nothing else than 1 half plus the square root times log of 1 half plus the square root plus 1 half minus the square root log 1 half minus the square root. Okay, that's all there is. Okay, that's how the gamma and the alpha are coupled. So, and as I said, right, the goal is to, I've written it up there is to minimize this. And more precisely, we're not really so much interested in minimizing. I mean getting the number, but really determining whether we want to have alpha equal to zero or not when there's a coupling or not. And therefore I think the natural first step to do is to see what the enemy is that we are competing with, right? So let's see if alpha is equal to zero, can we compute what gamma wants to be? Okay, and that's called the normal state. So what we want to do is let's compute, let's solve this minimization. Now I just minimize over gamma. Well, and we'll turn out that we can can compute this explicitly. Here it is. Perhaps I should actually write it down if you want. Okay, so this is the integral. We have a xi square minus mu, a gamma hat of xi, and then we have a plus from over there t times. And now we have gamma hat xi log gamma hat of xi plus one minus gamma hat of xi log one minus gamma hat of xi. Yes. And you see there is no, well there might be one parenthesis missing, you see there is no global constraint on on gamma hat. So we can just minimize this thing point wise. Okay, so let's do this. I've written this down as a little lemma. So for all h in R, t positive. So we have to do this point wise. So there's a h rho plus t times rho log rho plus one minus rho log one minus rho. So I've just called gamma hat of xi. I called this rho. That's what I've done. Well, and now we can minimize this simple exercise. You see, if you abstractly, this is computing a Legendre transform. Okay. And so the answer is equal to minus one over minus t, sorry, minus t logarithm of one plus e to the one minus one over t times h. Right. And the inf is attained if and only if rho is equal to one over one plus e to the h over t. Okay. This might be familiar to some of you. That's what's called the Fermi Dirac distribution physics. Okay. So what we found here, if we go back now to this, it was the minimization problem that we solve, that we wanted to solve. We do this. So h is equal to xi square minus mu. We plug this in. So we find, so the minimizer. And it's in fact the unique minimizer of the problem with alpha identically zero is gamma zero hat xi equal to one over one plus e to the xi square minus mu divided by t. So that's a definition and that's a normal state. That's the one that we have to beat. If you want to have superconductivity, we have to get a better energy, a smaller energy, a free energy than this one. Good. So I don't know what I should draw a picture. I mean this, so this describes the occupation of momentum. This is the xi axis. Somewhere here is xi equal, I mean xi squared equal to mu. Let me draw it like this. Okay. So this function, it's very, very close to one and then it goes down and then it's almost equal to zero for large values of xi. Okay. And so the temperature is the parameter that the slope how fast it goes down. Okay. Good. So now let's talk about phase transitions. I need to introduce one more thing. It's a key player for the next talks. So one more, once you have this thing, so one over one plus e to some things, there will be a lot of cautious and tanges and so on appearing. Okay. That's just, if you differentiate this thing, play with this thing, you get these things. They're not particularly nice, but I try to emphasize their main features. So here's the operator that will play a crucial role. I call it KT. And this is minus Laplacian minus mu. Well, that would not be so frightening. But now I divide by Tange minus Laplacian minus mu divided by two T. Okay. So by this, as you all know, I mean a multiplier in Fourier space. And the crucial features is that for large moment of this thing, right, the Tange down there is like one. So this just behaves like the ordinary Laplacian. Okay. And of course I have to tell you why this thing arises. We will see it soon. Let me state one little lemma before that. There's a unique TC. Okay. Greater or equal than zero, but it's finite. So that, so we look now at the, it's asserting a type operator. So you take the KT plus V, you take this operator and add this V as a potential. And now you look at, it's a self-adjoint operator. We can look at the lowest point in its spectrum. Okay. And now this thing, the infimum of the spectrum is less than zero. If T is below TC and it's greater or equal than zero, if T is greater or equal than TC. That's not a difficult lemma, right? Because this symbol for each fixed Xi is a monotone function in T. This means if you think about the variational principle that the eigenvalues also are monotone functions of T. Okay. So it's a monotone function and now this TC is just the first point where you hit zero. Okay. So there's nothing deep in here. Okay. But now the important thing is what this TC stands for. And that's the first main theorem here. This one I should start with. Okay. So this is a theorem. And that says that if we're below the critical temperature, you know, the critical temperature might be equal to zero, then the first statement is an empty statement. Okay. That's included. I'm not claiming that it's a positive number. In the same sense, I mean this might be an empty statement if TC is equal to zero. Then we have superconductivity. Okay. For small temperatures, this is less than, and now I can write because we know what the solution is if alpha is equal to zero. It's just FT of gamma naught zero. Okay. Below TC, we have superconductivity and above TC, well, we do not have superconductivity. And we have this even in a strong sense, namely the free energy is strictly greater than the free energy of the normal state for every admissible state, which of course is different from the normal state. Okay. So the nice thing is really there is a fixed T temperature where it goes from superconducting to normal. And what's particularly nice is that we can characterize this number in a linear way, right? This is a linear theory. So now at this point, we can, and that's in a lot of these references that I mentioned up there, we can really do computations now, right? We can add parameters. So this TC depends, of course, on V, it depends on mu. And now we can, in certain regimes, see how it depends. And that has been done both in the math and in the physics literature. It really is a computationally very, very useful thing. Now, there is one part of this theorem, which is very simple. That's the one I'll show you. And that's the part that answers how this KT operator comes in. And this is the obvious thing that you want to do, right? You want to create superconductivity. So how do you do it? You ask yourself, is this normal state stable or not? Okay. It's a local question. So what does this mean? Well, you just compute. So compute is your normal state. So add a tiny little off diagonal entry A. T is a small parameter. And I'm going to differentiate twice with respect to this T. Why do I do it twice? You see here, this thing just depends on alpha squared. And over there in this formula, right, this is what I didn't write out. This is also alpha squared only. Okay. So I can, the first derivative is clearly zero. And what I get is, well, that's obviously a quadratic form in A. And that's exactly where this KT plus V operator rises. Okay. And therefore, obviously, if this guy has a negative eigenvalue, then I can make this thing smaller than its value at T equal to zero. Okay. So that's a simple thing. The thing that's not trivial in this theorem is that this local condition of stability actually is a global criterion. That if you cannot get to superconductivity in this by perturbing slightly around the normal state, then you cannot get to superconductivity at all. That requires a proof. It's not very difficult, but I want to skip it just to get ahead and do some other things. Okay. So let me just say this proofs, this proofs the first part for the second part. See these notes. Okay. Perhaps it's helpful if I draw a picture. We'll have some board space here. Picture that I would like to draw is the energy. As a function of the temperature. I don't know. I mean, there's no nothing special about zero. This is a convex function. There's something like that. Now over here, there's a TC. That's a critical temperature. This thing I've drawn here, that's the energy of the normal state. And here I leave the curve of the normal state. I know it's a simple exercise to see that the energy is continuous. Okay. And it's also a K function. So the question is somehow, how does this bifurcation here work? And as I said, by phase transition, we understand that there is a non continuity, at least in some derivative. And that's what the second theorem will be about. And which is really the part that gets us close to the Skinsburg-Landau theory, the Mexican hat. Okay. And what I will show you in the next theorem, that's what this picture says, is that this leaves here quadratically. Okay. So that is quadratic. And that's what's called a second order phase transition in physics. So the first derivatives are continuous. The second are not continuous. That's really the infimum of Ft, gamma alpha. Okay. And we will compute the coefficient here. And later on, to tell you what will happen in the second and the third lecture, this is where the Skinsburg-Landau arises, just in a non commutative version of this talk today. Okay. So here's the theorem. Obviously, there is only something to prove if the critical temperature is positive. So let's assume this. And so I'm looking at the free energy for T very close, but below the critical temperature. Well, we said that it's continuous. So I put here the energy of the normal state. By the way, I put here a T, right? Not a Tc. That's slightly more convenient, but you can just expand that. And then what you'll find is that this is Tc minus T over Tc to the power 2, as I told you. And there is a number E here plus and the remainder, which is power 3, as T approaches Tc from below. And I want to tell you what the E is. Well, the E is again the solution of a variational problem, some problem script E, some functional that I will tell you in a minute. And what we've seen here in this computation, really the interesting states are those that make this KT plus V operator equal to zero, right? So we're close to the critical point. So this thing is about zero. So therefore we will minimize here a certain functional of all A in the kernel of KTc plus V. And that's unfortunately a little bit lengthy. I don't know whether I will write everything down. Minus 1 over 2 Tc integral kosh. So please don't copy that. It's really going to be horrible. Divided by 2 Tc times xi squared minus mu divided by 10. I just want to show you that we can do these things explicitly that while you get formulas particularly nice. But the important thing here is destruction. So that's the first term, right? It depends on the Fourier transform of a squared and then multiplied by some positive function in Fourier space. And that comes with a minus sign. And now there's another term which comes with a plus and well g1 xi squared minus mu divided by 2 Tc divided by xi squared minus mu. And now this thing here to the power 4 a head of xi to the power 4 d xi. So this thing while I didn't tell you what g1 was it's again something related to exponential function. We will get some idea next time how these things come up. Z squared 1 plus e to the z squared. It's terrible I know. Anyway the structure of the second term here is again the Fourier transform of a now this time raised to the power 4 and again multiplied by a certain function and the important thing is that this function is positive. So we have minus a quadratic plus a quadratic term. And so the formula becomes much nicer in particular if the dimension in particular if we're in the non degenerate situation that this a kernel is one dimensional and I pick an a, should I call it a star yes, then e of right then you can write everybody in the kernel as some number psi, psi is a complex number times this guy that spans the kernel and what you get here is minus lambda 2 psi squared plus lambda 3 psi to the power 4. Now obviously we can minimize this. I mean that's not the point I mean. So e is equal I guess minus a quarter lambda 2 squared divided by lambda 3. That's not the point that we can compute the number. The point is that we see here arising the psi squared plus the psi to the 4 term. It's just right I don't have anything that breaks the translation variance so I don't have a gradient up there on the first board the first thing I in the line down there. Nothing breaks the translation variance size a constant but that's exactly how this thing comes about. And somehow the point is, well I guess I explain this later in the proof, you see here but it's a quadratic expansion around that point. Good. And these are explicit formulas for these lambda 2 and lambda 3 and such things I mean that's what physicists really care about and really I mean they plug in numerical values and they mean something even for mathematicians often they're just formulas. Alright let me tell you a little bit on how one proves something like that. And so first it's convenient to introduce a notation. I call it H. It has nothing to do with a blanks constant but it's just a usual notation for a small constant and will be even more important in the next lectures where really there are commutators involved. Okay and it turns out that the natural thing is the square root of this temperature difference. Good. And so what's the, how does one prove such a theorem? Well we want to prove an asymptotic equality by proving two different inequalities. So there's an upper bound and a lower bound. What does the upper bound say? Well the upper bound says that find somehow a pair gamma and alpha such that the energy of this pair is less than well the energy of the normal state plus now there's an H to the power 4, right? If I look at the square that's H to the power 4 E plus a capital O of H to the power 6. So that means we're guessing something. I mean of course we should guess the right thing but this is somehow at our disposal we have to write down some gamma and some alpha and that proves the upper bound. Now the lower bound we have to prove something for every gamma and every alpha and we have to show that f of t of gamma alpha is greater than this. And so usually this means that well you guessed something in the first step and now you really have to ask yourself why is this guess really correct? So in some sense you want to reduce the second thing always to the first thing. By saying well you didn't really lose the generality by picking some particular alpha and gamma. That's a general strategy for such minimization problems. Now I want to write down a somewhat stronger statement which is actually what we'll prove and then I will stop for today. Let me write this down here which really says that in a certain energy region this E problem there is really the effective problem that you should care about. So here's what we really prove. In this paper I don't think I will prove all of this. In this course here in fact we show so the upper bound for every A so you give me an A in this kernel. Remember this kernel of the KTC plus V operator that's over which we have to minimize E script E. Now you give me any such A and then I can produce for you a gamma and an alpha so that the energy of this pair alpha and gamma well of course to leading order I want to get the energy of the normal state otherwise there is nothing to be said I want to get the script E energy of the A that you gave me plus perhaps some small remainder. Conversely I'll show that for every gamma and alpha there exists an A in the kernel of KTC plus V such that the H1 norm of A is bounded by 1 such that alpha is given now I have to construct an A for you which such that H times A is close to alpha it's actually an H squared what I'm saying here is alpha really is close to is of order H plus some small remainder of A order H squared and such that the energy of this thing that's given is the normal state of course plus H to the power 4 E of the A that you constructed here plus a capital of order H to the power 6 so now obviously this thing implies the lower bound because the largest this can be is just order A it's just E and for the upper bound well you just choose as A the minimizer in this optimization problem so once you have proved these two things then you get this theorem what I really want to emphasize here what's important is where you see the subtlety of the problem is that the off diagonal entry alpha that's only of order H whereas the energy difference is order H to the power 4 so there are lots of computations here that's here that's somehow doable here but that will really create problems once we have to do this all semi-classically anyway that's where I want to stop for today let's see whether there are any questions everybody's hungry I have a question concerning BCS actually Gorkov and the genes did they also provide a framework for a time-dependent problem? there's a paper by Gorkov and and then there's another paper by Schmitt German mathematician Schmitt and both of them derive a time-dependent Gainsburg-Landau equation from BCS theory there are some sort of disputes it's not somehow as universally valid as what I'm telling here so there are some additional assumptions and it's not completely clear to us what they mathematically mean but there's certainly work that's accepted in the physics community good what kind of evolution equation? it's very very interesting because it's so I don't know I'm running out of my lambda letters so perhaps there's a lambda 4 plus I lambda 5 dT psi okay so there is both a dissipative part and a dispersive part and in principle one in principle of course somehow this part should win right the the dissipative part it should go what we have so far managed is rather seeing the dispersive part so there is a paper by Heinzel well myself I guess Heinzel, Schlein and Seringer appeared I think this year where we look at an equation like this and where we get a dispersive equation but I think first of all it's rather believed in the dissipative effect so it depends maybe you have to wait more that's right I mean there are lots of questions exactly right I mean this is really an interesting question thanks other questions for Mark? this one so below the critical temperature can you say anything about whether the free energy is actually minimized and can you say anything about the minimizer is such a minimizer? yes yes it is both in this problem and in the more complicated non-commutative version one can prove existence of a minimizer and one can actually compute the Euler equation and I think I'll say a little bit about that but somehow it's kind of an energy subcritical problem so ordinary tools like Rayleigh or Rayleigh-Kondratsov give you existence of a minimizer okay thanks do you think that this can be seen for excited states this kind of transition? it would be very interesting to do something really with non-ground state solutions and somehow establish a correspondence between these microscopic solutions excited solutions and Ginsberg-Lander solutions also excited exactly I mean that would be very interesting I think physically it should be possible there's nothing that contradicts it at least in certain regimes but I don't know any books on that thank you