Advanced Mathematical Methods for Physics
A.A. 2024/25

Perturbative Expansions

• Introduction to perturbative expansion methods.
• Regular versus singular perturbation expansions: simple examples.
• Regular versus singular perturbation expansions: Integral Representations.
• Iterative approximations.
• Nondimensionalization; Example: one dimensional damped mass-spring system.

Asymptotic Expansions

• Order relations: symbols "O", "o" e "∼".
• Asymptotic sequences and asymptotic expansions.
• Generalized asymptotic and Poincaré type expansions.
• Uniqueness of Poincaré type expansions and subdominance.
• Uniform versus non uniform asymptotic expansions.
• Convergent series versus asymptotic series.
• Example: The error function Erf(x).
• Asymptotic power series: Taylor's theorem and Borel's theorem.
• Stokes phenomenon.
• Elementary operations on Poincaré type expansions.
• Generalized Stieltjeis Series and Integrals.

Layer-Type Problems

• Internal and external asymptotic expansions and their domain of validity.
• Intermediate limit and Matched asymptotic expansions.
• Uniform asymptotic expansions.
• Example: uniform asymptotic expansion to O(1) and O(ε) of the solution of an ordinary second order differential equation.
• Distinguished e undistinguished limits, dominance balance.
• Nonlinear boundary layers.
• Boundary layer at either extrema.
• Nested boundary layers.
• Internal Layer.

WKB expansion

• Dispersive and Dissipative behaviors.
• WKB expansion.
• Geometrical optics and physical optics approximantions.
• WKB expansion for the Schrödinger equation.
• WKB expansion for Boundary Layers.
• Asymptotic behavior of Airy functions for |x| >> 1.
• Asymptotic behavior of the Parabolic Cylinder functions for |x| >> 1.
• WKB expansion with one linear turning point.
• The Liouville-Green transformation and the Langer solution to the linear turning point problem.
• WKB expansion with two turning points.
• Eigenvalue condition and semiclassical quantization.

Multiple Scales Methods

• Resonances and secular terms.
• Strained Coordinates Method: The Duffing's equation.
• Strained Coordinates Method: Limit cicle of the Van der Pol oscillator.
• Derivative Expansion Method: Duffing's equation analysis with two time scales.
• Derivative Expansion Method: Duffing's equation analysis with three time scales.
• Derivative Expansion Method: Duffing's equation analysis with three time scales.
• Derivative Expansion Method: Harmonic oscillator with non linear damping analysis with two time scales.
• Derivative Expansion Method and WKB Method: Harmonic oscillator with slow time varying frequency.
• Derivative Expansion Method for Boundary Layer problems.
• Derivative Expansion Method: Van der Pol oscillator with two time scales.
• Cole-Kevorkian Method: Duffing's equation to order ε2.
• Naive Renormalization Method: Duffing's equation to order ε2.

Renormalization Group and Asymptotic Expansions.

• Ill defined expansion and Renormalizability hypotesis.
• Regularization procedure, Renormalization prescription.
• Perturbative Renormalization.
• Renormalization group.
• Perturbative Renormalization Group and partial resummation.
• Duffing's equation: solution to O(ε).
• Callan-Symanzik's equation.
• Duffing's equation: solution to O(ε2).
• Renormalization Group approach to Boundary Layers.
• Schrödinger with one turning point.
• Van der Pol oscillator to O(ε): renormalization conditions, minimal subtraction, approach to the limit cicle.
• Van der Pol oscillator: iterative renormalization to O(ε).
• Mathieu equation: stability analysis to O(ε2) near the critical value a = 1/4.
• Stabilità soluzioni equazione di Mathieu: analisi vicino al valore critico a = 1/4 mediante RG.

Diagrammatic Theory

• Generating funtion Z[J] of the n-points functions <φ1...φn>.
• Perturbative expansion of Z[J].
• Wick and Novikov theorems.
• Pertutbative expansion of <φ1φ2> for the φ4 theory and its digrammatic representation.
• Cancellation of vacuum-fluctuations diagrams.
• Dyson-Schwinger equation.
• Perturbative calculation of the moments <φn> of the one-dimensional double well in the limit T << 1.
• Connected n-points functions <φ1...φn>c and their generating function W[J].
• Perturbative calculation of W[J] for the one-dimensional double well in the limit T << 1, evaluation of <φ2>.
• 1PR and 1PI diagrams: Self-energy Σ and Dyson equation.
• n-points proper vertices Γ⁽ⁿ⁾_1...n and their generating function Γ[φ].
• Legendre Transform and diagrams 1PI.
• Connected n-points functions and proper vertices.
• Effective Potential, spontaneous symmetry breaking.
• Γ[φ] for the one-dimensional double well in the limit T << 1, evaluation of <φ2> to order T.
• Loop expansion of Γ[φ].
• Self-consistent equation for ϕ and Dyson equation.
• 2PR and 2PI diagrams, double Legendre transform and effective Potential Γ[φ,G].
• Γ[φ] and Γ[φ,G] for the one-dimensional double well: 2-loop and 3-loop calculation.

Complements

• Integral representation of the Airy function Ai(x).
• Asymptotic behavior of Ai(x) for x >> 1: Steepest descent and Saddle point methods.

Grassmann Variables^*

• Definition and properties.
• Differentiation and integration.
• Gaussian integrals: Pfaffian and Determinant.
• Existence of a finite critical temperature of the two-dimensional Ising model (Pierls' argument).
• Computation of the partition function of the two-dimensional Ising model on a square lattice using Grassmann variables.
• Crtical temperature of the two-dimensional Ising model on a square lattice.