Class 2025 (Spring)
Announcements
The UF teaching evaluations will be open on April 15. Here is the website for evaluations . You have login with your gator password.
Graded Assignments
HW 1 (due in February, submission via Canvas) HW1 with Solutions
Presentations in class:
44.9.6 Cauchy problem for the Klein-Gordon-Fock equation
44.9.7 Cauchy problem for the Dirac equation
44.9.8 Cauchy problem in elastodynamics
HW 2 (due in March, submission via Canvas) HW2 with solutions
HW 3 (due by the end of semester, submission via Canvas ) HW3 with solutions
Recommended Texts
[1] V.S. Vladimirov, Equations of Mathematical Physics, (main text)
[2] I. Stakgold, Green’s functions and boundary value problems, (main text)
[3] F. Riesz and B. Sz.-Nagy, Functional Analysis, (for pure math oriented students)
[4] A.M. Kolmogorov and S.V. Fomin, Elements of the theory of functions and functional analysis.
[5] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol 1-4 (for advanced reading).
Recommended readings
Banach spaces, Contraction principle and Hilbert spaces: [2] Chapter 4; [3] Chapter 5
The operator theory. Recommended reading from [1], [2], and [3] is indicated for each lecture. Several topics are taken from [5] (but simplified, so take good notes). In addition, some topics are taken from [4].
Lecture notes
S.V. Shabanov, Distributions and Operators for Theoretical Physicists
Chapter 2: Distributions
Chapter 3: Calculus with Distributions
Chapter 4: Convolution and Fourier Transform
Chapter 5: Applications to PDEs
Lecture topics
Topic 1: The Cauchy problem (13-31/2) [Chapter 5]
L1: Fundamental solution for a differential operator (Sec. 35). A general distributional solution to a linear PDE with constant coefficients and a distributional inhomogeneity. Example: Poisson equation.Uniqueness of the solution (Sec. 37). The Fourier method for finding a fundamental solution (Sec 39). Existence. Hoermander’s Theorem. Distributional regularization of the reciprocal of a polynomial as the Fourier transform of a fundamental solution. Example: 1D differential operator. Principal value regularization. Shift of a pole into a complex plane. Comparison of the two methods.
L2: Regularization of distributional solutions. Solutions in tempered distributions. Fourier transform of the convolution (Sec. 33.7). Existence. Product of distributions via the Fourier transform (Sec. 33.8). Comparison with the product defined by the localization method. The Cauchy problem for parabolic and hyperbolic PDEs. Reduction to a Generalized Cauchy problem (Sec. 40). Solution via the causal Green’s function.
TO BE UPDATED
L3: The Cauchy problem for the wave equation. Causal Green’s function for the wave operators and its convolution with distribution supported in a half-space. Wave potentials. Distributions smooth in one variable. Analytic properties of the causal Green’s function in the time variable. The Cauchy problem with distributional initial data. Distributional inhomogeneity that is smooth in the variable time. The limit properties of the wave potentials.
L4: Classical Cauchy problem for the wave equation. Huygens principle.
L5: Vector-valued distributions. Cauchy problem for Maxwell’s equations. Distributional constraints (Gauss law) and charge conservation equations. Generalized Cauchy problem.
L6: Solution of the generalized Cauchy problem. Distributional initial data. Distributional external charges and currents that are smooth in the time variable.
L7: Radiation created by the initial fields. Huygens principle. Decay of an electric string.
L8: Radiation by external distributional sources. Example: radiation by an electric dipole. Far fields. Electromagnetic energy flux in the asymptotic region.
L9: Well-posedness of the Cauchy problem for Maxwell’s equations. Helmholtz decomposition theorem for vector fields. Electromagnetic potentials.
Topic 2: Basic integral equations (01/29 – 02/26) [Chapter 4]
L10: Initial value problem for second-order non-linear ODE. Example: unharmonic oscillators. Reduction of the problem to a Volterra integral equation. The Cauchy problem for 4D non-linear wave equation. Reduction to a Volterra-type integral equation.
L11: Von Neumann series for the linear Volterra integral operator. Uniform convergence. Theorem about the existence and uniqueness of a solution to the linear Volterra integral equation. Solution by a von Neuman series. Solution to the Cauchy problem for linear wave equations with non-constant coefficients by the von Neumann series.
L12: Green’s function for the Helmhotz operator satisfying Sommerfeld outgoing wave condition. Wave scattering problem. The Lippmann -Schwinger integral equation in the wave scattering theory. Fredholm integral equations. Convergence of a von Neumann series for a linear Fredholm integral operator to a solution of the linear Frenholm integral equation.
L13: Linear spaces. Examples. Normed spaces. Examples: Lebesgue spaces, spaces of power-summable sequences, Sobolev spaces, space of bounded functions.
L14: Complete normed spaces. Banach spaces. Space of bounded functions with the supremum norm as a Banach space. Space of continuous functions on compact sets as closed subsets in the Banach space of bounded functions. Other examples of Banach spaces. Spaces of power-summable sequences. Lebesgue spaces. The Riesz-Fisher theorem.
L15: Transformation on sets in a Banach space. Continuous transformations. Lipschitz continuous transformations. Contractions. Fixed point of a transformation. Closed sets in a Banach space. Example: subset of continuous functions taking values in a bounded interval in the Banach space of bounded functions.
L16: The contraction mapping theorem. An extension of the contraction mapping theorem.
L17: Applications of the contraction mapping theorem to non-linear Volterra and Fredholm integral equations. The existence and uniqueness of a solution.
L18: The Cauchy problem for non-linear wave equations with certain type of non-linearity.
L19: Green’s function for the Sturm-Liouville operator in an interval. Solution to the linear Strum-Liouville boundary value problem.
L20: Non-linear Sturm-Liouville boundary value problem in an interval. Reduction to a non-linear Fredholm integral equation by the Green’s function method. A solution by means of the contraction principle (by a perturbation theory). An example of a non-linear boundary value problem.
L21: Higher dimensional boundary value problems for Sturm-Liouville operators. Green’s function. Example: Green’s function for the Laplace operator in a ball with Dirichlet’s boundary conditions. Kelvin’s transformation.
L22: Resolvent of an integral operator with a continuous kernel. Reduction of a Fredholm equation to an integral equation with integral operator of finite rank. Fredholm alternative for integral equations with continuous kernels.
Topic 3: Hilbert spaces (02/28 – 03/08) [Chapter 5]
L23: Inner product vector space. Schwartz-Cauchy-Bunyakowski inequality. Natural norm on an inner product vector space. Continuity of the inner product. Hilbert space.Orthogonal sets in an inner product space. Bases and orthogonal bases in an inner product space. Additional reading: [2], Chapter 4, Section 5; [3] Chapter 5; [4] Chapter 3
L24: The Gramm-Schmidt process. Example: Legendre polynomials. Trigonometric orthogonal set in an interval. Linear independent sets as bases in an infinite dimensional inner product space. Orthogonal bases.
L25: Separable metric, normed, and inner product spaces. Orthogonal sets in separable inner product spaces. Examples of separable spaces. The space of square summable sequences, its separability and completeness. Space of Lebesgue square integrable functions as a separable Hilbert space. Examples of non-separable inner product spaces. The existence of a countable orthogonal basis in a separable inner product space.
L26: The best approximation of a vector from a Hilbert space by a vector in a linear manifold. Solution in a finite dimensional Hilbert space. Linear manifolds in an infinite dimensional Hilbert space. Orthogonal complement of a linear manifold. Closure of a linear manifold. The projection theorem.
The best approximation problem in a Banach space. Normed versus inner product spaces. Conditions under which a Banach space can be turned into a Hilbert space. The significance of the parallelogram law.
L27: Fourier series in a separable Hilbert space. Bessel inequality. Parseval-Steklov equality. Riesz-Fisher theorem about convergence of a Fourier series a Hilbert space. The theorem about the existence and convergence of the Fourier series over an orthonormal set satisfying Parseval-Steklov equality in a separable Hilbert space. Convergence in the mean versus point-wise and uniform convergences of Fourier series in a Hilbert space of square integrable functions. Hermite polynomials. An orthogonal basis in the space of Lebesgue square integrable functions on a line. Resolution of unity in a separable Hilbert space.
L28: Operators on linear manifolds. Linear operators. Bounded operators. Norm of an operator. Continuity of an operator. Continuous and bounded operators. Bounded and unbounded operators. Linear operators in a separable Hilbert space. Matrix elements of an operator. Integral operators with square integrable kernels (Hilbert-Schmidt operators).
Spring Break 03/11-15. No Lectures
Topic 4: Operators in Hilbert spaces (03/18- 04/08) [Chapter 6]
L29: Basic operator algebra. The sum and product of operators. Properties of the sum and product of (linear, or bounded, or continuous) operators. A set of operators as a linear space. Space of operators as a normed space. The Banach space of operators with range being a Banach space. Convergence of a series of operators. Convergence of a operator geometric series. The von Neumann series for a solution of a Fredholm integral equation for a Hilbert-Schmidt operator.
L 30: The inverse operator. The inverse of a linear operator. Banach theorem about the inverse for linear bounded operators. Examples: The second derivative operators in a space of square integrable functions over bounded and unbounded intervals. The range of a differential operator in the space of square integrable functions. Perturbation theory for the inverse operator. Boundedness of the inverse operator. Operators bounded away from zero. Necessary and sufficient conditions for an invertible operator to have a bounded inverse. Examples: differentiation and multiplication operators on bounded and unbounded intervals.
L 31: An extension of an operator. An extension of a bounded operator. An extension of an unbounded operator. Closable operators. Examples of closable operators. Differential operators are closable. The closure of a closable operator. Closed operators. Properties of closed operators.
L32: Extensions of differential operators. Absolutely continuous functions. Closures of differential operators. Well-posed linear problems Au=f in a Hilbert space. Classification of operators by properties of the inverse and properties of the range.
L33: Boundary conditions in differential equations and the closure of differential operators. Classification of operators and well-posed linear problems Au=f in a Hilbert space. The closure of an operator and well-posedness of a linear problem Au=f. Examples. Linear functionals on a Hilbert space. The Riesz representation theorem for linear bounded functionals.
L34: The adjoint of a bounded operator. A construction via the Riesz representation theorem. Examples. Linear operators in Euclidean spaces. Matrix representation. Adjoint of a matrix. Integral operators with square integrable kernels.
L35: The adjoint of an unbounded operator with domain dense in a Hilbert space. The domain of the adjoint. Example of a non-closable operator. Additional reading: [2], Chapter 5, Section 4; [3] Chapter 8
L36: Properties of the adjoint. The adjoint of an extension of an operator. The adjoint is closed. The adjoint of the closure of an operator is the adjoint of the operator. The double adjoint of an operator is the closure of the operator. Example: the second derivative operator with the domains being the space of test functions on an interval and the space of twice continuously differentiable functions on an interval. Construction of the adjoint, the closure, the adjoint of the closure, and the double adjoint.
L37: Hermitian or symmetric operators. Self-adjoint operators. Hermitian operators vs self-adjoint operators. Comparison of domains of hermitian, double adjoint, and self-adjoint operators. Symmetric densely defined operators with range being a Hilbert space. Criterion for a symmetric operator to be sefl-adjoint. Essentially self-adjoint operators. Criterion for an hermitian operator to be essentially self-adjoint. Self-adjoint extensions of differential operators. Procedure to find a self-adjoint extension. The differentiation operator on the space of test functions with bounded support. Symmetric extensions. The adjoint operator. Verifying the criterion for the essential self-adjointness. The closure of the differentiation operator in an interval. Additional reading: [2], Chapter 5, Section 2; [3] Chapter 8
Topic 5: Spectrum of an operator in a Hilbert space [Chapter 6]
L38: The resolvent of an operator in a Hilbert space. Examples. The derivative operator in an interval. The second derivative operator in an interval. Energy operator for a quantum harmonics oscillator. Theorem about the isometry of the Fourier transform. Example: the derivative operator on the real line. The resolvent set of an operator. The spectrum of an operator. Classification of the spectra. The point, continuum, and residual spectra. Examples.
L39: The compression and approximate spectra.The spectrum of a symmetric operator. Spectral values. Deficiency of a spectral value. Theorem about the deficiency of a spectral value of an operator. The spectrum of a self-adjoint operator. A procedure to find the spectrum of an operator. Example.
L40-42: Compact operators. Properties of compact operators (Sec 15.1 you may skip proof of properties) Continuity of compact operators. Hilbert-Schmidt operators as compact operators. Spectral properties of compact operator (Sec. 15.2). Fredholm alternative for compact operators (Sec. 15.3). Compact self-adjoint operators and Green’s functions for a Sturm-Liouville operator in a bounded region. Spectrum of a compact self-adjoint operator. Spectral theorem for compact self-adjoint operators (Sec.15.4). Foundations of the Fourier method for partial differential equations.
L43-45: The spectral theorem for self-adjoint operators Sec. 16. Stieltjes integral. Examples: center of mass of an extended object, probability theory. The spectral family of operators. Examples. The spectral theorem in the form of operator-valued projection measures. A reformulation of the spectral theorem for compact self-adjoint operators via the projection measures. Example of the projection measures for self-adjoint operators with continuous spectrum. The wave packet decomposition in quantum mechanics.
The course ends on L45 (Wednesday, April 24)
L1, 1/5/22 W: The classical and generalized Cauchy problems for the wave equation. A distributional solution to the generalized Cauchy problem. Retarded causal Green’s function for the wave operator. Reading [1] Ch III, Sec 13
Lecture 1
Supplementary material from Part 1: Initial value problems for linear ordinary differential equations. Reducing the problem to an equation in the convolution algebra of distributions. Uniqueness of the solution. Causal Green’s function. Example: Harmonic oscillator.
Video Lecture
L2, 1/7/22 F: Explicit forms of the retarded causal Green’s function for the 2D, 3D, and 4D wave operators. Uniqueness of the solution of the distributional problem associated with the Cauchy problem for the wave equation. Wave potentials. Reading [1] Ch III, Sec 12.
Lecture 2
L3, 1/10/22 M: Smoothness of a multi-variable distribution relative to a particular variable. Smoothness of the retarded causal Green’s function for the wave operator relative to the time variable. Reading [1] Ch III, Sec 12
Lecture 4
L4, 1/12/22 W: Properties of the surface wave potentials. Surface wave potential for densities being regular distributions (locally integrable functions). Smoothness of the surface wave potentials. Explicit form of the surface wave potentials in 2, 3, and 4 dimensional spacetime. Reading [1] Ch III, Sec 12 .4.
Lecture 5
L5, 1/15/22 F: Integral representation of the surface wave potentials for regular initial data. Integral representation of the wave potential for a regular inhomogeneity. Smoothness of the wave potential with a smooth density (a proof for the 2D and 4D cases). The classical solution to the Cauchy problem for the wave equation. Support of the surface wave potentials. Wave propagation in space, Huygence principle for wave fronts (the beginning of recorder Lecture 7 below). Reading [1] Ch III, Sec 12.3
Lecture 6
L6 1/19/22 W: The Cauchy problem for Maxwell’s equations. Vector-valued distributions. Helmholtz theorem. Vector and scalar potentials. Distributional solution for the generalized Cauchy problem. Electromagnetic wave propagation and polarization states.
Lecture 7
Chapter 5: Basic integral equations
L9, 2/23/22 W: Scattering problem for waves. Sommerfeld outgoing-wave conditions and the corresponding Green’s function.
Lectures 9,10; Correction to Lecture 9
L10, 2/25/22 F: The Lippmann-Schwinger formalism in the wave scattering theory. The Fredholm integral equation for the scattered wave. Its solution in the case of point-like scatterers.
L11, 1/31/22 M: Linear spaces. Linear manifolds. Examples. Normed and metric spaces. Convergence of a sequence or series in a normed linear space. Cauchy sequences. Convergence of Cauchy sequences. Example: The space C^0 of continuous functions on a closed bounded set in a Euclidean space as a normed linear space. Uniform convergence and the convergence in the space C^0. Example: The space C^0_2 of continuous square-integrable functions on a bounded set. Examples of Cauchy sequences that do not converge in C^0_2. Banach spaces.
Lecture 11
L12, 2/02/22 W: Space of bounded functions with the supremum norm as a Banach space. Space of continuous functions on compact sets as closed subsets in the Banach space of bounded functions. Other examples of Banach spaces. Spaces of power-summable sequences. Spaces of power-integrable functions. The Riesz-Fisher theorem.
Lecture 12. Comment for Lecture 12.
L13, 2/04/22 F: Transformation on sets in a Banach space. Continuous transformations. Lipschitz continuous transformations. Contractions. Fixed point of a transformation. The contraction mapping theorem. An extension of the contraction mapping theorem.
Lecture 13
L14, 2/07/22 M: Linear and non-linear Volterra integral equation. The existence and uniqueness of a solution to the non-linear Volterra integral equation. Example: The initial value problem for an oscillator with frequency that depends on time and amplitude.
Lecture 14
L15, 2/09/22 W: Green’s function for the Sturm-Liouville operator. Solution to the linear Strum-Liouville boundary value problem (Notes. Section 50).
L16, 2/11/22 F: Non-linear Sturm-Liouville boundary value problem. Reduction to a non-linear Fredholm integral equation by the Green’s function method. A solution by means of the contraction principle (by a perturbation theory). An example of a non-linear boundary value problem. Recommended reading: [2], Chapter 4, Section 4; [2] Chapter 6,
Lecture 15. The end of Lecture 15
Supplementary Topic: Radon transform. Tomography problem. Abel’s integral equation and its solution.
Video Lecture
Chapter 6: Hilbert spaces
L17, 2/14/22 M: Inner product vector space. Schwartz-Cauchy-Bunyakowski inequality. Natural norm on an inner product vector space. Continuity of the inner product. Hilbert space.Orthogonal sets in an inner product space. Bases and orthogonal bases in an inner product space. Recommended reading: [2], Chapter 4, Section 5; [3] Chapter 5; [4] Chapter 3
Lecture 17
L18, 2/16/22 W: The Gramm-Schmidt process. Example: Legendre polynomials. Trigonometric orthogonal set in an interval. Linear independent sets as bases in an infinite dimensional inner product space. Orthogonal bases.
Lecture 18
L19, 2/18/22 F: Separable metric, normed, and inner product spaces. Orthogonal sets in separable inner product spaces. Examples of separable spaces. The space of square summable sequences, its separability and completeness. Space of Lebesgue square integrable functions as a separable Hilbert space
Lecture 19
L20, 2/21/22 M: Examples of non-separable inner product spaces. The existence of a countable orthogonal basis in a separable inner product space.
Lecture 20; Correction for Lecture 20
L 21, 2/23/22 W: The best approximation of a vector from a Hilbert space by a vector in a linear manifold. Solution in a finite dimensional Hilbert space. Linear manifolds in an infinite dimensional Hilbert space. Orthogonal complement of a linear manifold. Closure of a linear manifold. The projection theorem.
The best approximation problem in a Banach space. Normed versus inner product spaces. Conditions under which a Banach space can be turned into a Hilbert space. The significance of the parallelogram law.
Lecture 21
L22, 2/25/22 F: Fourier series in a separable Hilbert space. Bessel inequality. Parseval-Steklov equality. Riesz-Fisher theorem about convergence of a Fourier series a Hilbert space. The theorem about the existence and convergence of the Fourier series over an orthonormal set satisfying Parseval-Steklov equality in a separable Hilbert space.
Lecture 22
L23, 2/28/22 M: Convergence in the mean versus point-wise and uniform convergences of Fourier series in a Hilbert space of square integrable functions. Hermite polynomials. An orthogonal basis in the space of Lebesgue square integrable functions on a line. Resolution of unity in a separable Hilbert space.
Lecture 23
L24, 3/02/22 W: Linear functionals on Hilbert spaces. Continuous funtionals. Bounded functional. Unbounded linear functionals. Examples. Necessary and sufficient conditions for a linear functional to be unbounded. Necessary and sufficient conditions for a linear functional to be continuous. Null space of a linear continuous functional. Riesz representation theorem for a linear continuous functional. Dual basis.
Lecture 24; Comment to Lecture 24
L25, 3/4/212F: Operators in Banach spaces. Linear operators. Bounded operators. Norm of an operator. Continuity of an operator. Continuous and bounded operators. Bounded and unbounded operators. Linear operators in a separable Hilbert space. Matrix elements of an operator. Integral operators with square integrable kernels (Hilbert-Schmidt operators).
Lecture 25 Lecture 26
Spring Break 03/07-11. No Lectures
L 26, 03/14/22 M: Basic operator algebra. The sum and product of operators. A set of operators as a linear space. Space of operators as a normed space. The Banach space of operators with range being a Banach space. Convergence of a series of operators.
Lecture 27; Comment to Lecture 27
L 27, 03/16/22 W: Weak, normed, and strong convergence of a sequence of operators. Convergence of a operator geometric series. The von Neumann series for a solution of a Fredholm integral equation for a Hilbert-Schmidt operator. Functions of operators.
Lecture 28
L 28, 03/18/22 F: The inverse operator. The inverse of a linear operator. Banach theorem about the inverse for linear bounded operators. Examples: The second derivative operators in a space of square integrable functions over bounded and unbounded intervals. The range of a differential operator in the space of square integrable functions. Perturbation theory for linear equations Au=f. The theorem about the inverse of an operator that is a small variation of an invertible operator.
Lecture 29
L 30, 03/21/22 M: Continuity of a solution to a linear problem Au=f with respect to f. Boundedness of the inverse operator. Operators bounded away from zero. Necessary and sufficient conditions for an invertible operator to have a bounded inverse. Examples: differentiation and multiplication operators on bounded and unbounded intervals. Recommended practice problems: 2-4 in Section 26.5 of the notes
Lecture 30
L 31, 03/23/22 W: An extension of an operator. An extension of a bounded operator. An extension of an unbounded operator. Closable operators. Examples of closable operators. Differential operators.
Lecture 31
L 32, 03/25/22 F: The closure of a closable operator. Closed operators. Properties of closed operators. A closed operator with a closed domain is bounded. The inverse of a closed operator is closed. The inverse of a closed operator is bounded if and only if the range is closed.
Lecture 32
L33, 03/28/21 M: Extensions of differential operators. Absolutely continuous functions. Closures of differential operators. Well-posed linear problems Au=f in a Hilbert space. Classification of operators by properties of the inverse and properties of the range.
Lecture 33
L34, 04/01/22 W: Well-posed linear problems Au=f in a Hilbert space. The closure of an operator and well-posedness of a linear problem Au=f. Examples. Boundary conditions in differential equations and the closure of differential operators.
Lecture 34
L35, 04/03/22 F: The adjoint of a bounded operator. A construction via the Riesz representation theorem. Examples. Linear operators in Euclidean spaces. Matrix representation. Adjoint of a matrix. Integral operators with square integrable kernels.
Lecture 35
L36, 04/01/22 F: The adjoint of an unbounded operator with domain dense in a Hilbert space. The domain of the adjoint. Example: second derivative in a space of square integrable functions on an interval. The double adjoint and the closure of an operator. The closure of the adjoint. The necessary and sufficient condition for an operator to be closable. Example of a non-closable operator. Additional reading: [2], Chapter 5, Section 4; [3] Chapter 8
Lecture 36
L37, 04/04/22 M: Hermitian or symmetric operators. Self-adjoint operators. Hermitian operators vs self-adjoint operators. Comparison of domains of hermitian, double adjoint, and self-adjoint operators. Criterion for a hermitian operator to be sefl-adjoint. Essentially self-adjoint operators. Criterion for an hermitian operator to be essentially self-adjoint. Self-adjoint extensions of differential operators. Procedure to find a self-adjoint extension. The differentiation operator on the space of test functions with bounded support. Symmetric extensions. The adjoint operator. Verifying the criterion for the essential self-adjointness. The closure of the differentiation operator in an interval. . Recommended reading: [2], Chapter 5, Section 2; [3] Chapter 8
Lecture 37
L38, 04/06/22 W: Review: the spectrum of a square matrix. The resolvent of an operator in a Hilbert space. Examples. The derivative operator in an interval. The second derivative operator in an interval. Energy operator for a quantum harmonics oscillator. Theorem about the isometry of the Fourier transform. Example: the derivative operator on the real line. The resolvent set of an operator. The spectrum of an operator. Classification of the spectra. The point, continuum, and residual spectra. Examples.
Lecture 38
L39, 04/08/22 F, L39a, 04/11/22 M: The compression and approximate spectra.The spectrum of a symmetric operator. Spectral values. Deficiency of a spectral value. Theorem about the deficiency of a spectral value of an operator. The spectrum of a self-adjoint operator. A procedure to find the spectrum of an operator. Example.
Lecture 39
L40-42, 04/13/22 W-04/18/22 M: Compact operators. Properties of compact operators (Sec 15.1 you may skip proof of properties) Continuity of compact operators. Hilbbert-Schmidt operators as compact operators. Spectral properties of compact operator (Sec. 15.2). Fredholm alternative for compact operators (Sec. 15.3). Compact self-adjoint operators and Green’s functions for a Sturm-Liouville operator in a bounded region. Spectrum of a compact self-adjoint operator. Spectral theorem for compact self-adjoint operators (Sec.15.4). Foundations of the Fourier method for partial differential equations.
Lectures 40, 41
L43, 04/20/22 W: The spectral theorem for self-adjoint operators Sec. 16. Stieltjes integral. Examples: center of mass of an extended object, probability theory. The spectral family of operators. Examples. The spectral theorem in the form of operator-valued projection measures. A reformulation of the spectral theorem for compact self-adjoint operators via the projection measures. Example of the projection measures for self-adjoint operators with continuous spectrum. The wave packet decomposition in quantum mechanics. This lecture is not mandatory for the final assignment and is intended for physics students to fill out a possible “gap” in mathematical foundations of quantum theory in physics courses, regarding “eigenstates” of operators with continuous spectra.