Self-adjoint Operators in Mathematical Physics
(graduate course, SISSA, February-April 2015)

Lecturer: Alessandro Michelangeli

venue and schedule: room A-136, Monday 11-13, Tuesday 11-13
start: Monday 9 February 2015
end: Tuesday 14 April 2015


Self-adjointness of (in general) unbounded operators on a Hilbert space is an ubiquitous tool in the mathematical methods for quantum mechanics and in the theory of linear and non-linear PDEs of mathematical physics. It determines the interpretation of Hamiltonians (of atoms, molecules, many-body systems) as quantum observables and the existence and uniqueness of the corresponding dynamics.

This course will present the main features of the general theory of self-adjoint operators (unboundedness, spectrum, spectral measures and spectral integrals, the spectral theorem, perturbation theory, quadratic forms, self-adjoint extensions) and will provide a survey of the most significant applications. This includes "classical" results on the self-adjointness of physically relevant Hamiltonians, as well as topics that are currently at the centre of an intense research activity, such as
  1. rigorous models for cold-atom systems with zero-range interactions ("unitary gases", in the condensed matter physics jargon)
  2. rigorous models for particles constrained on surfaces, on metric graphs, on hedgehog manifolds, etc.
  3. energy methods for the well-posedness of non-relativistic and semi-relativistic non-linear Schrödinger equations with strong and singular external electromagnetic fields, for which standard methods based on Strichartz estimates are not applicable.

Pre-requisites: Only a basic knowledge of Hilbert spaces and bounded operators on a Hilbert space is expected. A number of relevant notions taught in last fall's graduate course "Topics in the Mathematics of Quantum Mechanics" (by G. Dell'Antonio) will be revisited. The course is also designed to intersect with a few seminar talks on the subject, scheduled within the Analysis, Math-Phys, and Quantum series.


Amrein, "Hilbert Space Methods in Quantum Mechanics", EPFL Press (2009)

Dell'Antonio, "Lectures on the Mathematics of Quantum Mechanics I", Springer (2015)
De Oliveira, "Intermediate Spectral Theory and Quantum Dynamics", Birkhäuser (2009)
Gitman, Tyutin, Voronov, "Self-adjoint Extensions in Quantum Mechanics", Birkhäuser (2010)
Schmüdgen, "Unbounded Self-adjoint Operators on Hilbert Space", Springer (2012)
Teschl, "Mathematical Methods in Quantum Mechanics", AMS (2009)


Additional materials and weekly diary of the lectures:


. Scope and concept. General abstract theory of (unbdd) self-adj ops on H-space. Along the way and in the end: role of self-adjointness in various branches of math-phys, significantly maths methods for quantum theories. The H-space will be abstract to a large extent. Various example of H-spaces.

. Motivations and applications. (1) When do Hamiltonians for quantum systems (atoms, molecules, quantum gases, ...), obtained from 1st quantisation from Classical Mech., give rise to a unique sol to the initial value problem for the
Schrödinger equation? Issue of the self-adjointness of Schrödinger operators / Dirac operators, etc. (2) Rigorous construction of self-adj. Hamiltonian for quantum particles constrained on metric graphs. Self-adjointness conditions at the vertices of the graph. (3) Rigorous construction of self-adj. Hamilt. for systems of particles with zero-range interaction (now feasible in the lab: "unitary gases"). (4) Well-posedness of Hartree equations with non-Strichartz magnetic fields.

§2. Need for unboundedness. Unbdd ops on H-spaces arise naturally (multiplic. by an unbdd function, differentiation, ...). Schrödinger operators are unbdd and this does not depend of fact that we chose to represent the Weyl C*-algebra in the Schrödinger representaton (von Neumann's theorem).  In general unbdd-ness and domain issues come up together. An example of everywhere defined unbdd op on a H-space (axiom of choice is needed, though).


The Hellinger-Toepliz thm, though, which is a straightfwd consequence of the Closed Graph Thm, forces a symmetric and everywhere defined linear op to be bdd.

§3. Closable and closed operators. A subclass of unbdd ops for which the theory is still sufficiently rich. General notions first: domain, range, kernel, restriction, extension, invertibility on the range. The domain of the sum of two unbdd ops may just collapse to {0} (examples). Graph Γ(T) of an operator T.  Γ(T) "contains the full info" on T. Graph norm. Definition and equivalent characterisation of closed ops and of closable ops. The operator closure. Not all ops are closable. Bdd vs closed: a bdd op is necessarily closable, and necessarily closed if it's everywhere defined. A closed linear op which is unbdd cannot be defined on the whole Hilbert space. Operator cores (advantage: dealing with cores is more manageable than dealing with domains). Closed times bdd = bdd, closed + bdd = bdd, but bdd times closed is not necessarily closed.


§4. Adjoint of a densely defined operator.  T* has the domain of the y's for which  ∃z (uniquely determined by density) such that z,x〉=⟨y,Tx〉,  and T*y :=z. Equivalent definitions via Riesz. Determining T*, given T, is not immediate. Two main examples worked out explicitly: adjoint of a multiplication operator, adjoint of the operator of differentiation. Useful spaces introduced at this point: absolutely continuous functions AC[a,b] and Sobolev spaces H^k(a,b). Further examples: the operator of (weak) differentiation on a half-axis and on the real line; maximal and minimal Laplacians; the Laplacian on smooth functions compactly supported away from one point (this last example is the beginning of the set-up for delta interactions). Notice: the domain of T* need not be dense (examples). Basic properties of the operator adjoint (list). In particular: T* is always closed. D(T*) dense ⇔ T is closable. For closable (and densely defined) T, the adjoint of T and the adjoint of its operator closure coincide; also, the operator closure is nothing but the double adjoint T**. In particular, T is closed ⇔ T=T**. Adjoint of the inverse. Closure of the inverse.


If T is densely defined and closable, ker(T*-z) and Ran(T-z*) are closed subspaces that decompose the Hilbert space.

§5. Spectrum of (possibly unbdd) closed operators.  Definition of resolvent set (the z's such that (T-z) is invertible and everywhere defined on H, or simply (by Closed Graph) the z's such (T-z) is a bijection from the domain of T onto H. Resolvent operator. Spectrum σ(T). Point spectrum. The resolvent is an operator-valued analytic function of the resolvent set. Resolvent identities. It can happen that σ(T) is the whole complex plane (examples) or the empty set (example), the latter can only occur for unbdd T's for, recall, bounded T's have always a non-empty spectrum. Actually any closed subset of complex numbers is the spectrum of a closed operator (proof). Further examples of operators and spectra. Spectrum and resolvent of the adjoint. The spectrum of the multiplication operator by a function φ is the essential range of φ. Examples.

§6. Basics on (unbdd) symmetric and self-adjoint operators.  Definition of symmetric (not necessarily densely defined!): y,Tx〉=⟨Ty,x〉 for all x,y in the domain of T. Equivalently (by polarisation), x,Tx〉 is real for all x. Which justifies the def. of lower/upper semi-bounded symmetric operators. For a densely defined symmetric operator, one has T⊆T*, moreover T is closable, its closure is still symmetric, thus T⊆T**T*. For a closable, symmetric T it makes sense to define two deficiency indices, dimension of the orthogonal complement of Ran(T-z), for these numbers are constant on the whole upper/lower open complex half plane. (Proof in class only for closed an symmetric T.) For a densely defined and symmetric T (hence closable) the deficiency indices are simply dim ker(T*±i).


Same as above if T is symmetric, closable, and bounded below: then dim ker(T*-a) is constant also on all a's below the lower bound of T. Symmetric operators have real eigenvalues, and distinct EV's give rise to orthogonal eigenspaces. Definiton of self-adjoint: T=T*. Essential self-adjoint (there is a unique canonical self-adjoint extension). Note that a self-adj op is always closed. The problem to find conditions for a densely defined, symmetric T (or also: a closed symmetric T) to be self-adjoint. Von Neumann's formula and the structure of the domain of T*. Basic criterion of (essential) self-adjointness: the role of the deficiency indices. Characterisation of the spectrum of a self-adjoint T by means of a sequence of almostr eigenvalues. Examples for Schroedinger ops.

§7. Spectral measures and spectral integrals. Preparatory for the spectral theorem. Spectral measure ME(M) (aka projection-valued measure). Mx,E(M)x gives the corresponding scalar (real) measure, by polarisation x,E(M)y is the corresponding scalar complex measure.


Spectral measures vs resolutions of the identity: every pvm on the Borel sets of ℝ gives rise in a natural way to a resolution of the identity, and vice versa. Getting familiar with symbols such as dE, dE(λ), d⟨x,E(λ)y〉, ... Once we have (operator-valued) measures, we can build up operator-valued integration. Spectral integral of simple functions and of bounded measurable functions (a density argument to go from the former to the latter). Spectral integrals of (possibly unbounded) measurable functions, dE-a.e. finite. Basic properties of the spectral integrals.

§8. Spectral Theorem. A structural theorem for self-adjoint operators. First case: the Spectral Theorem for bounded, self-adjoint operators. A=A* ⇔ A=∫λdE(λ) for a unique pvm dE(λ) on the Borel sets of the spectrum σ(A) of A. Proof of existence of dE(λ). First use Riesz-Markov to represent the evaluation px,p(A)y〉 on polynomials p as a measure dμ on C(σ(A)). (to be continued...)


(continuation) Then, by means of the mapping M μ(M) on the Borel sets of the spectrum, use the fact that (x,y)μ(M) is a bounded sesqulinear form, hence there is a bounded operator E(M) that realises it, (x,y)x,(E(M)y. Check that E(M) is an orthogonal projection and that E(σ(A)) is the identity operator. Thus we've got a genuine pvm and hence spectral integrals built up with it. Then one cheks that λdE(λ) is precisely A. Second case: the Spectral Theorem for general (possibly unbdd) self-adjoint ops. Now the pvm is on ℝ. Idea of the proof: map A into the bounded operator A(1+A*A)^{-1/2} and use the bounded spectral theorem of the latter. Sketch.

§9. Functional calculus. The way to give an unambigous meaning to functions f(A) of a self-adjoint operator A as spectral integrals in terms of the pvm dE(λ) associated with A. Definition of f(A) and of its domain. f is taken in the class of the Borel-measurable functions on which are dE-a.e. finite. Various examples in detail. Properties of the functional calculus: expectation of f(A), norm of f(A)x, f(A) is bounded ⇔ f is bdd and the norm of f(A) is the ess-sup-norm of f, linearity, adjoint of f(A), composition fg(A), invertibility of f(A), positivity of f(A), etc. Among the most relevant functions of A are the exp(itA), (A-z)^{-1}, fractional powers of A. Functional calculus and the spectrum. λ∈σ(A) E(J)≠0 for any open interval J around λ. λ is an eigenvalue iff E({λ})0. E(σ(A))=1. Each isolated point of the spectrum of a self-adjoint operator is an eigenvalue.


Proof of the previously stated facts. Stone's formula: from
dE(λ) to (A-λ)^{-1} and vice versa.

§10. One-parameter group of isometries. Strongly continuous, one-parameter, unitary group on a Hilbert space. If A=A*, then U(t):=exp(itA) is a strongly cont., one-param., unitary group; A is uniquely determined by U(t). The proof relies on functional calculus methods. In fact, each strongly cont., one-param., unitary group on a HIlbert space has the form exp(itA) for some self-adjoint A: this is the content of Stone's theorem. Such an A is the infinitesimal generator of U(t). The group of translations (infinitesimal generator: the momentum). Group of dilations. Trotter product formulas.

§11. Differential equations on Hilbert spaces. The abstract Schrödinger equation. The abstract heat equation. The abstract wave equation. The associated Cauchy (initial-value) problems. The IVP for the Schrödinger equation u'=-iAu has a unique solution u(t)=exp(-iAt)φ with initial datum φ in the domain of A. Unitary evolution. Norm and expectation of A (the "energy") are preserved.


The IVP for the heat
equation u'=-Au, t>0, where A=A* is positive, has a unique solution u(t)=exp(-tA)u_ The abstract heat equation. The abstract wave equation. The associated Cauchy (initial-value) problems. The IVP for the Schrödinger equation u'=-iAu has a unique solution u(t)=exp(-iAt)φ with initial datum φ in the Hilbert space. The proof is again via the functional calculus for A. The IVP for the wave equation u''=-Au, with u(0) in the domain of A, u'(0) in the domain of A^{1/2}, with A=A* positive. Existence and uniqueness, explicit formula for the solution. The free Schrödinger equation on ℝ^d with the self-adjoint Laplacian defined on H^2. The free propagator as multiplication operator in the Fourier space. Integral kernel of the free propagator. (In turn, this allows for the proof of dispersive estimates and Strichartz estimates.) Higher regularity for the free Schrödinger equation on ℝ^d, with H^s (Sobolev) regularity. Lower regularity for the free Schrödinger equation on ℝ^d (H^{-2} and H{-1}).

§12. Strong- and norm-resolvent convergence of self-adjoint operators. One motivation: a unitary dynamics has (trivially) a continuous dependence on the initial data, what if instead the infinitesimal generator is "slightly" changed? Want to give meaning to A≈B for A=A*, B=B*, possibly both unbounded. (Application: a Schrödinger operator where the potential V is not fully known. Random Schrödinger operators.) Definition of norm-resolvent convergence and strong-resolvent convergence.


Notice that we are developing the theory in which both the An's and the A are self-adjoint. Norm (resp. strong) resolvent convergence of An to A implies norm (resp. strong) operator convergence of f(An) to f(A) for continuous functions of
vanishing at infinity (respectively, continuous bounded functions of ℝ). Proof again based on the functional calculus. In particular the convergence of resolvents implies the convergence of the unitary groups. Applications to resolvent convergence of Schrödinger operators: need to compare resolvents, crucial role of resolvent identities. Quick detour on the Konno-Kuroda resolvent identity. One can somehow go backwards: from the so-called "dynamical convergence" (i.e., strong convergence of the unitary groups) one deduces strong resolvent convergence; this is Trotter's theorem. Resolvent convergence and spectrum, both in the strong case and in the norm case (non-contraction, non-expansion of the spectrum in the limit). A remark: for linear self-adjoint operators, strong resolvent convergence is the same as Gamma-convergence of the corresponding energy functionals.

. Relatively bounded perturbations of self-adjoint operators. Definition of a relatively A-bounded operator B. Opertor bound. Infinitesimally bounded operators (B<<A). Clearly a bounded operator B is always infinitesimally bounded w.r.t. another operator A. A fundamentla perturbation result: the celebrated Kato-Rellich theorem.


The conclusion of the Kato-Rellich theorem is false when the operator bound is equal to 1. This case is covered, with a slightly weaker conclusion, by Wüst's perturbation theorem. Proof of Kato-Rellich.

§14. Self-adjointness of Schrödinger operators via perturbation methods. This would be a major chapter in a separate course on Schrödinger operators... Rellich proved his theorem in the 1939, then Kato came up with his seminal paper "Fundamental properties of Hamiltonian operators of Schrödinger type" in 1951, which was a turning point because the proof of self-adjointness bacame a necessary preliminary to spectral analysis and scattering and the attention was put on concrete examples rather than foundational properties. Nowadays it is all classical material (see Reed-Simon, Chapter X). Proof of the continuous embedding of H^2 into L^q for generic d dimensions. Kato-Rellich potentials. This leads, via Kato-Rellich, to the self-adjointness of -Δ+V on H^2 (proof). Wide applicability. Examples. The local singularity |x|^{-λ} gives self-adjointness for  all λ's up to (but not equal to) 3/2. Self-adjointness of magnetic Schrödinger operators (-i∇+A)^2+V via Kato-Rellich. Proof under the condition that the magnetic potential is L^4+L^\infty. This condition can be substantially weakened: Leinfelder-Simander theorems (1981).


Further examples of applicability of Kato-Rellich. The one-particle Dirac operator with interaction: a Hardy-like inequality guarantees that a Coulomb interaction with sufficiently small coupling is a Kato-Rellich perturbation of the free Dirac operator. Essential self-adjointness on smooth functions compactly supported away from the origin is lost, though, if the coupling is sufficiently large (Weidmann 1971). A distinguished self-adjoint extension for large coupling is given by Arrizabalaga-Vegsa 2013. Another example: Kato-Rellich applied to the semi-relativistic
Schrödinger operator: the Kato inequality.

§15. Quadratic forms and self-adjoint operators. Motivation from the expectation values x,Ax〉. Sesquilinear and quadratic forms on a Hilbert space. Symmetric and lower semi-bounded forms. Closed and closable forms. Examples. The form A[x,y] associated with a self-adjoint operator A: form domain, action.


Characterisation of D(A) as a subspace of D[A] (when A=A*). In partic. A[x,y]=
x,Ay〉 for x in D[A] and y in D(A). A is bdd below A[.,.] is bdd below, in which case the bound is the same. Explicit expression for A[x,y] when A is bdd below, or in partic. positive. Conversely, introduce the operator A_q associated with a densely defined quandratic form q. Main result: 1:1 correspondence between lower semi-bdd self-adj. ops on one side, and densely defined, lower semi-bdd, closed forms on the other side. Proof. Often physical arguments indicate the formal expression of the (energy) quadratic form: typically its domain is dense but the form is not closed. Then issue of whether the energy form is closable + issue of whether it is bounded below. The latter is attacked by vsarious powerful analytic estimates (e.g., Kato, Hardy). An example worked out in full detail: the quadratic form of the Hydrogen atom. Given the energy form, and proved that it is densely defined, bounded below, and closed, one can then exploit analytical/variational techniques to study the minimisation problem: it gives info on the EV-problem of the corresponding self-adjoint operator. In particular if the inf of the energy functional is reached, then one finds the ground state of the operator.


§16. Order relations for self-adjoint operators. The Friedrichs extension. Order realtion A≤B for A=A*, B=B*. Stated in terms of the associated forms. Special case when A,B are bounded below, or positive. AB0 and kerB={0} implies that kerA={0} and 1/B 1/A. If A and B are bounded below and z is below their lower bounds, then AB is equivalent to (B-z)^{-1}A-z)^{-1}. Heinz inequality: if AB0, then A^s B^s for s in [0,1] (prove by integral identities). Wrong for s>1. Construction of a distinguished, canonical self-adjoint extension F ("the Friedrichs extension") of a densely defined, bounded below, symmetric operator T. Start with the form x,Ty〉 defined on the same domain as T, and prove (crucial fact) that this form is closable. By closing it, one obtains a closed, symmetric, bounded below form, whose associated self-adjoint operator is F. F is indeed a self-adj extension of T and any other self-adj ext is smaller than T in the ordering introduced before. Moreover, T is the only self-adj ext of T with domain contained in the form closure of T. Example: negative Laplacian on [a,b] originally defined on smooth funtions compactly supported away from the boundary. T, F, T*. F is the Dirichlet Laplacian. Other extensions, such as the Neumann Laplacian, are smaller than F.


Further example: the Dirichlet vs the Neumann Laplacian on a bounded open domain
in d dimensions. The former is the Friedrichs extension of the Laplacian on smooth functions compactly supported inside Ω. The latter is another self-adjoint extension, "smaller" than the Dirichlet Laplacian in the sense of the operator ordering. In fact, the Neumann Laplacian is a Friedrichs extension itself, of the Laplacian with the boundary condition of zero normal derivatives at Ω.

§17. Perturbation of forms and form sums. Definition of a relatively bounded form with respect to a reference form which is lower semi-bdd, symmetric. Definition of a self-adjoint operator B that is relatively form bounded with respect to a self-adjoint, bdd-below operator A. The KLMN Theorem (Kato, Lax, Lions, Milgram, Nelson): gives meaning to the sum of a bdd-below, self-adj operator A and a quadratic form that is A-form bounded with relative bound <1. Along the same line: operator form sum A∔B vs the operator sum A+B. Applications to Schrödinger operators. V  for positive L^1-loc potentials. Examples where the operator sum is only trivially defined on {0}, whereas the opertor form sum is densely defined and self-adjoint. The form sum of ∔δ in one dimension. In three dimensions, a large class of real-valued potentials for which V exists (and is self-adj) via KLMN is the Rollnik class. Definition and properties. L^{3/2} is included in the Rollnik class (it follows from the Hardy-Littlewood-Sobolev inequality). Thus a local singularity |x|^{-a} can be taken in a Schrödinger operator with expontent a up to 2 (Kato-Rellich only could cover up to 3/2).


§18. Case study I. Magnetic Hartree equation with non-Strichartz external fields.. Recall that the self-adjoint Laplacian on R^d have dispersive properties encoded in the so-called Strichartz estimates. Admissible pairs. Strichartz estimates. The two main ingredients of the proof: self-adjointness of (which guarantees the L^2-conservation in time) and the L^\infty - L^1 dipersive estimates, which in turn follows from the explicit form of the integral kernel of the propagator exp(i). Any other self-adjoint operator with L^\infty - L^1 dipersive estimates admits Strichartz estimates. Conversely, an operator with trapping properties (e.g., eigenvalues) cannot satisfy Strichartz globally in time. Non-linear (semi-linear) Schrödinger equation. It is the effective equation for many-body quantum dynamics. Local and non-local non-linearities. Strichartz estimates are a fundamental tool to prove existence and uniqueness (locally in time) of the solution to the initial value problem of NLS. Explicit proof of uniqueness for local non-linearities: Duhamel + local Lipschitz property of the non-linearity + Strichartz. The magnetic Hartree equation. As long as Strichartz estimates are available for a self-adjoint realisation of the operator -(∇-iA)^2, then the standard scheme of existence/uniqueness apply. When magnetic Strichartz estimates do hold. Globally in time for rough A's with at most 1/x singularity (D'Ancona, Fanelli, Vega, Visciglia - 2010). Locally in time for smooth A's that increase at most linearly at infinity (Yajima 1991). Problem: well-posedness of magnetic Hartree out of the Strichartz regime. The answer goes through energy methods and the local Lipschitz property of the non-linearity in the energy space. Assumptions on the potentials (quite general, out of Strichartz regime). Definition of magnetic energy space. A distorsion of H^1 or of magnetic H^1_A. It is the form domain of a unique self-adj. operator realising the magnetic Schrödinger operator. Main theorem: global well-posedness of magnetic Hartree in the magnetic energy space. The non-linearity is locally Lipschitz in the energy space. Putting all ingredients together to prove the main theorem.

13/04 - 11am + 2 pm

The local Lipschitz property in the energy space follows from the diamagnetic inequality.

§19. Case study II. The alternating Kronig-Penney model for a 1D crystal with zero range or finite range interactions. Stability of the gap vanishing. The self-adjoint 1D Kronig-Penney Hamiltonian with alternating delta interactions (alternatingly pointing upwards and downwards). Quadratic form (it is closed and bounded below). Spectrum: structure of bands and gaps, noticeably all gaps at the centre of the Brillouin zone vanish. The proof so far relied on a delicate and lengthy analysis of the associated ODE and its monodromy matrix. Also, it leaves the questions open on whether this peculiar gap-vanishing at k=0 is only present in the idealised KP model of zero-range interactions or instead it is "physical" in that it is also present in an approximating model of periodic potentials with finite range of interactions. Recent theorem: the delta alternating KP Hamiltonian has a canonical approximation, in the norm-resolvent sense, by a Schrödinger operators with "squeezing potentials" of finite range of interaction, for which all gaps at k=0 vanish. Because of the norm-resolvent convergence, this re-proves elegantly that the limit Hamiltonian has all gaps vanishing at k=0, moreover this proves that the gap-vanishing is stable along the limit.

§20. Case study III. Rigorous construction of a point interaction in 3D (singular perturbation of the Laplacian). Aiming at defining -Δ+δ(x) in 3D one sees that, unlike 1D, the Dirac-δ is not form-bounded w.r.t. -Δ (in 3D H^1 does not control L^\infty). The idea, then: -Δ+δ(x) as a self-adjoint extension of the symmetric operator acting as -Δ on smooth functions compactly supported away from x=0. The latter is a symmetric operator with a one-parameter family {-Δα|α∈ℝ} of self-adjoint extensions, each of which is a modification of the Laplacian in the s-wave. The "delta"-interaction at x=0 has the form of a boundary condition. The parameter of the extension is conveniently expressed in terms of the scattering length -1/(4πα) of the interaction. Domain of -Δα: regular part (H^2) and singular part (the Green function of the Laplacian). -Δα acts on the functions of its domain as -Δ on the regular part. Locality of -Δα. The resolvent of -Δα is a rank-one perturbation of the resolvent of -Δ. Spectrum of -Δα : for negative α (one negative EV) and for positive α (no EV's). The extension for α=∞ is the Friedrichs extension and it is the self-adjoint -Δ. Quadratic form of -Δα: formal interpretation of -Δ+δ(x). Approximation, in norm-resolvent sense, of -Δα by Schrödinger operators with squeezing potentials scaling as ε^{-2}V(x/ε). Thus, the self-adjoint "delta"-interaction is much weaker than the non-self-adjoint Dirac delta, that scales instead as ε^{-3}V(x/ε).

*   *   *

INFORMATION FOR THE CLASS: for the final exam each candidate is free to choose any of the following
  • an open seminar where to discuss a research paper or other material related with the course, previously decided together with the instructor (intermediate discussions with the instructor are recommended before delivering the seminar)
  • a short essay (~10 pages at most) on themes previously agreed with the instructor
  • an oral examination
  • a 90' written test
  • a take-home exam (exercises to solve at home and to present to the instructor).
The examination committe consists of the instructor + Prof Dabrowski, Prof Dell'Antonio, Dr De Oliveira.