Homework and announcements

Lecture schedule

Reading for Lectures 32-34 : I. Stackgold, Green’s functions and boundary value problems, Chapter 7, Spectral theory of second-order differential operators;
V. Vladimirov, Chapter V, Boundary value problems for elliptic equations, Sections 21, 22, 28, 29.
Reading for Lectures 19- :  I. Stackgold, Green’s functions and boundary value problems,  Chapter 5, Operator theory;
A.N. Kolmogorov and S.V. Fomin, Elements of the theory of functions and functional analysis, Chapter IV, Linear functionals and linear operators.
Reading for Lectures 9- 18  : I. Stackgold, Green’s functions and boundary value problems,  Chapter 4, Hilbert and Banach spaces;
F. Riesz and B. Sz.-Nagy, Functional Analysis, Chapter V, Hilbert and Banach Spaces;
A.N. Kolmogorov and S.V. Fomin, Elements of the theory of functions and functional analysis, Chapter IV, Linear functionals and linear operators.
Reading for Lectures 5-8: V. Vladimirov, Equations of mathemtical physics, Chapter III, Fundamental solution and the Cauchy problem, Sections 12-14
Reading for Lectures 1-4: V. Vladimirov, Chapter III, Fundamental solution and the Cauchy problem, Section 11.

L39 (M,04.16): Square root of a positive bounded operator. Polar decomposition of a bounded operator. The absolute value of a compact operator. Singular values of a compact operator. Finite rank operators. Compact operator as the series with terms being operators of finite rank. Singular value decomposition of bounded operators of finite rank. Singular value decomposition of compact operators.

L38 (F, 04.13): Boundary value problems for differential and partial differential equation and eigen-value problem to the Sturm-Liouville operator. Reduction to the eigenvalue problem for a symmetric compact operator via the Green’s function formalism. Complete orthogonal sets in a space of square integrable functions.

L37(W, 04.11): Symmetric compact operator. Self-adjoint extension. The spectrum of a compact self-adjoint operator. The spectral theorem for compact self-adjoint operator. Complete orthogonal sets as the set of eigenvectors of a compact self-adjoint operator.

L36 (M, 04.09): Spectrum of a compact operator. The point spectrum. The continuous spectrum. The residual spectrum. The accumulation point of the spectrum.

L35 (F, 04.06):  Compact or absolutely continuous operators. Properties of compact operators. Every compact operator is bounded, but not every bounded operator is compact. The inverse of a compact operator in an infinite-dimensional Hilbert space is not bounded. A compact operator maps an infinite orthonormal set to a null sequence. The limit of a sequence of compact operators is a compact operator. Operators with finite-dimensional range.

L34 (W, 04.04):  Relation between the spectra of differential operators arising in boundary value problems in partial differential equations and the spectra of integral operators. The Sturm-Liouville problem in a Euclidean space. Sturm-Liouville operators. Green’s function for the Sturm-Liouville operators. Example: Green’s function for the Sturm-Liouville operator in an interval. The reduction of the eigenvalue problem for the Sturm-Liouville operator to the eigenvalue problem for an integral equation with a square integrable kernel.

L33 (M, 04.02):  The operator of second derivative in an interval (the Hamiltonian of a quantum particle in a well). The domain. The closure. The adjoint. The existence of a self-adjoint extension. The spectrum of the self-adjoint extension. The point spectrum. The continuum spectrum (via the criterion for the approximate spectrum). The resolvent via Green’s function for a Sturm-Liouville operator. The resolvent set.  Analytical properties of the resolvent. Pole singularities and the discrete spectrum. Coalescence of the pole singularities into the branch singularity in the limit of an infinite interval. The differentiation operator on an interval with various boundary conditions.  Examples with the empty spectrum and only the residual spectrum (the compression of the range).

L32 (F, 03.30):  The second-derivative operator on a half-line (the Hamlitonian of a quantum particle on a half-line). The domain. The closure. The adjoint. The existence of a self-adjoint extension. The spectrum of the self-adjoint extension. The point spectrum. The continuum spectrum (via the criterion for the approximate spectrum). The resolvent via the Fourier transform in a space of tempered distributions. The resolvent set. Analytical properties of the resolvent. Branch singularity of the resolvent and the continuum spectrum.

L31 (W, 03.28):  Properties of the spectrum of an operator. Approximate spectrum as the union of discrete and continuous spectra. The criterion to find the approximate spectrum . Compression spectrum of an operator as the union of the residual and discrete spectra. The criterion to find the compression spectrum.  Spectrum of a symmetric (hermitian) operator. Orthogonality of eigenvectors of a symmetric operator. Deficiency of the spectral parameter in the compression spectrum of an operator and its relation to the multiplicity of the spectral parameter in the discrete spectrum of the adjoint operator. The spectrum of a self-adjoint operator.

L30 (M, 03.26):  The resolvent of an operator. The spectrum of an operator. Discrete, continuum, and residual spectra of an operator. The left shift operator in the Hilbert space of square summable sequences as an example of an operator with a non-empty residual spectrum. Other examples in the space of square integrable functions.

L29 (F, 03.23):  Solvability of a general linear equation Au=f in a separable Hilbert space. The null space of the adjoint operator. The orthogonal complement to the null space of the adjoint operator and the range of the operator. The difference with a finite dimensional case. The solvability condition. Well-posed linear problem and the closure of an operator (revisited).

L28 (W, 03.21):  Further examples of self-adjoint extensions of differential operators.

L27 (M, 03.19):  Properties of the adjoint operator. The double adjoint operator. The adjoint of a closable operator. The closure of an operator as the double adjoint operator. The adjoint of the operator and its closure. Symmetric versus self-adjoint operators. Basic criterion for self-adjointness. Essentially self-adjoint operators. basic criterion for essential self-adjointness. Example: the differentiation operator on an interval. General conditions for hermiticity. Construction of the adjoint and the closure. Self-adjoint extensions. Relation to spin and statistics in physics (bosons, fermions, and anyons).

L26 (F, 03.16): The adjoint of a bounded operator. An explicit construction by means of the Riesz representation theorem for linear functionals in a Hilber space. The adjoint of an integral operator with a square integrable kernel. The adjoint of an unbounded operator with a domain dense in the Hilbert space. An example of an operator with a dense domain for which the adjoint does not have a dense domain in the Hilbert space. Symmetric or hermitian operators. Self-adjoint operators.

L25 (W, 03.14): The closure of differential operators. The existence of the inverse of the closure. The range of the closure. Examples. Well-posed linear problem Au=f in a Hilbert space. A crude classification of operators by the properties of the inverse (bounded, unbounded inverse, non-existing inverse) and by the properties of the range (closed, not closed, the closure of the range coincides with the whole Hilbert space or is a proper subset in it) and the well-posed linear problems.

L24 (M, 03.12): Closed operators. Properties of closed operators. The closure of a closed operator is closed. The inverse of a closed operator with a closed domain is bounded. The inverse of a closed operator is bounded if and only if the range of the closed operator is closed. Absolutely continuous functions. Cantor ladder.

L23 (F, 03.02): An extension of a linear operator. Theorem about an extension of a bounded operator to the whole Hilbert (or Banach) space. Null sequences. Images of null sequences under the action of an unbounded operator (three options). Examples. Closable operators. The differentiation as a closable operator. Extension of an unbounded operator. The closure of an operator.

L22 (W, 02.28): Theorem about invertibility of perturbations of an invertible operator. Convergence of a geometric series for a bounded operator. Operators bounded away from zero. Theorem about the equivalence of the boundedness away from zero and invertibility of an operator. Basic concept of a perturbation theory for a linear equation in Hilbert or Banach space.

L21 (M, 02.26): Operator algebra. Sum and product of operators. The norm of the sum and product of two operators. The set of operators as a normed space. Banach space of operators with the range being a Banach space. Invertible operators. The inverse of an operator. Examples: the second derivative in an interval and in a line. Boundedness of the inverse operator. The Banach theorem about the existence of a bounded inverse operator.

L20 (F, 02.23): The norm of an operator. Methods to calculate the norm. The multiplication and differentiation operators in the space of square integrable function. Boundedness of integral operators with square integrable kernels. A method to show that an operator is not bounded.

L19 (W, 02.21): Operators on a Banach space. The domain, range, and null space of an operator. Linear operators. Continuous operators. Bounded operators. Theorem about equivalence of the continuity and boundedness of a linear operator.

L18 (M, 02.19): Dual basis in a separable Hilbert space. Existence and construction by means of the Riesz representation theorem for linear functionals. Expansion of an element in a Hilbert space over a complete linearly independent set.

L17 (F, 02.16): The projection theorem for Hilbert spaces. Orthogonal decomposition of an element in a Hilbert space. Approximation in functional Banach spaces. Non-uniqueness of the best approximation (example in the space of continuous functions). The Parallelogram law in Hilbert and Banach spaces. Conditions under which a Banach space can be turned in a Hilbert space. The Riesz representation theorem for linear functionals. Linear functionals on the space of square integrable functions.

L16 (W, 02.14): Linear functionals on linear manifolds. Continuous linear functionals. Bounded linear functionals. Examples of bounded and unbounded functionals. The theorem about equivalence of the continuity and boundedness of a linear functional. Examples.

L15 (M, 02.12): The Riesz-Fisher theorem about orthonormal sets in a Hilbert space. Isomorphism of all separable Hilbert spaces. Riesz-Fisher theorem about the completeness of the space of square integrable functions. The space of square intergrable functions as an analog of  an “infinite-dimensional” Euclidean space.

L14 (F, 02.09): Separable Hilbert space. Two examples of non-separable Hilbert spaces. Separability of the space of square integrable  functions. Complete orthogonal sets. The existence of a countable complete orthonormal set in a separable Hilbert space. Parseval-Steklov equality.

L13 (W, 02.07):  Inner product space. Inner product axioms. Schwartz inequality. The natural norm in an inner product space. Hilbert space as a complete inner product space with respect to the natural norm. Orthogonal sets in a Hilbert space. Examples. Orthogonal sets in the space square summable complex sequences. Orthogonal sets in the space of square integrable functions.  The Schmidt process.

L12(M, 02.05): Fredholm operators as contractions in the Banach space of continuous functions. Example of a nonlinear boundary value problem solved by the contraction principle. Reduction of a wave scattering problem to an integral equation.

L11 (F, 02.02): Initial value problems for ordinary differential equations and integral equations. Example: the initial value problem for  harmonic and non-harmonic oscillators with variable frequencies. The equivalence of the initial value problem to a Volterra equation. Integral operators mapping the space of Lebesgue integrable functions to the space of continuous functions. Fredholm equations of the 1st and 2nd kind.

L10 (W, 01.31): Transformations on Banach spaces. Continuous transformations. Fixed point of a transformation. Contraction principle.

L9 (M, 01.29): Linear spaces. Metric spaces. Normed spaces. Convergence in normed spaces. Linear independence and bases in normed spaces. Banach spaces. Examples.

L8 (F, 01.26): Classical and generalized Cauchy problem for Maxwell’s equations. Reduction to the case of the scalar wave equation. Decomposition of a vector field into the sum of rotational and conservative vector field. Solutions to the generalized and classical Cauchy problems. Example: A decay of an electromagnetic string (a HW problem, essential steps will be posted in the WH1).

L7 (W, 01.24): Calculation of the wave and surface wave potentials. Smoothness of the wave and surface potentials. The existence and uniqueness of the solution to the classical Cauchy problem for the wave equation. Continuity of the solution with respect the initial data and the source term.  Propagation of waves in three-dimensional space. The Huygens principle.

L6 (M, 01.22): A solution to the generalized Cauchy problem for the wave equation via wave potentials and surface wave potentials. The class of distributions for which the convolution with the fundamental solution to the wave operator exists. Uniqueness of the solution to the generalized Cauchy problem for the wave equation.

L5 (F, 01.19): Classical Cauchy problem for the wave equation. Generalized Cauchy problem for the wave equation. A relation between them.

L4 (W, 01.17): Example: A fundamental solution to the 2D wave operator from a fundamental solution to the 3D wave operator by the dimensional reduction.

L3 (F, 01.12): The method of a dimensional reduction (or descent) to obtain a fundamental solution for a linear differential operator with constant coefficients.

L2 (W, 01.10): Linear differential operators with constant coefficients. General method for finding a fundamental solution to such an operator by means of the Fourier transform in the space of tempered distributions. The existence of a fundamental solution. A regularization of the reciprocal of a polynomial to obtain a tempered distribution. Example: a fundamental solution for the Helmholtz operator by the Fourier method.

L1 (M, 01.08): Fundamental solutions for linear differential operators. A review.

Course topics and suggested readings

The topics are not studied in the order given below and given just as a detailed description of the course. The actual order will be posted above in the lecture schedule

Topic 1: Hilbert and Banach spaces.

1. Linear spaces. Metric. Normed linear spaces. Sequences in a normed linear space. Cauchy sequences. Completion of a normed space. Complete metric spaces. Banach space. Compact sets in a metric space.

2. Continuous transformations in a metric space. Lipschitz transformations. Contractions in a metric space. Fixed point. Contraction mapping theorem. Applications to differential equations. Example: a harmonic oscillator with a time-dependent frequency, the existence and uniqueness of the solution.

3. Applications of the contraction mapping theorem to linear and nonlinear Fredholm and Volterra integral equations. Solutions of linear Fredholm and Volterra equations in the form of Neumann series.

4. Inner product spaces. Hilbert spaces.Relations between Banach and Hilbert spaces. Parallelogram law and Jordan-von Neumann theorem.

5. Projections in a Hilbert space. Linear manifolds in a Hilbert space. Orthogonal projection on a linear manifold. Orthogonal complement to a linear manifold. Projection theorem. A closest element in a linear manifold to a given element of a Hilbert or Banach space. Conditions under which the closest element is unique.

6. Linearly independent sets and bases in a Hilbert space. Separable Hilbert spaces. Gram-Schmidt orthogonalization process. Example: Space of square integrable function on an interval, the set of monomials and Legendre’s polynomials.

7. Best approximation problem in a separable Hilbert space. Riesz-Fisher theorem.

Suggested reading:

– I. Stakgold, Green’s functions and boundary value problems, Chapter 4.
– F. Riesz and B. Sz.-Nagy, Functional analysis, Chapter 5
– V.S.Vladimirov, Equations of mathematical physics, Chapters 1 and 4.

Topic 2: The operator theory

1. Linear functionals on Banach and Hilbert spaces. Continuous and bounded linear functionals. The relation between continuous and bounded linear functionals. Riesz representation theorem.

2. Best approximation problem in a separable Hilbert space (revisited). Dual basis. Representation of a linear functional in a dual basis.

3. An operator in a Banach or a Hilbert space. The domain, range, and null set of an operator. Bounded operators. The norm of a bounded operator.

4. Continuous and bounded linear operators. Representation of an operator in a basis and its dual basis. Linear operators in a finite dimensional Hilbert spaces and their matrix representation.

5. Differential operators in a space of square integrable functions as an example of unbounded operators.

6. An extension of an operator. An extension of bounded operators to the whole Hilbert space.

7. Null sequences. Classification of images of null sequences under the action of an unbounded operator. Closable operators. An extension of closable unbounded operators. The closure of an operator. Closed operators. Example: Differential operators and their extensions in the space of square integrable functions. The space of absolutely continuous functions. Properties of closed operators.

8. Solvability of the equation Au=f where A is an operator in a Hilbert space. Well posed problems. Inverse operators. Conditions under which the inverse operator exists. Conditions under which the inverse operator is bounded. Classification of operators by properties of the inverse (if it exists) and by properties of its range.

9. Adjont operators.The domain of the adjoint. The case of an operator in a finite dimensional Hilbert
space. A matrix and its adjoint. The adjoint of an integral operator (with square integrable kernel).
Theorem about properties of the adjoint of an operator with a domian dense in the Hilbert space.

10. Self-adjoint operators. Relations between symmetric (hermitian) operators, symmetric and closed operators,
and self-adjoint operators. Necessary and sufficient conditions for a symmetric operator to be self-adjoint.
Essentially self-adjoint operators. Necessary and sufficient conditions for a symmetric operator to be
essentially self-adjoint. Examples of differential operators in an interval (analysis of various extensions
and self-adjoint extensions).

11. Properties of the range of an operator and solvability
of the equation Au=f. The Fredholm alternative for finite-dimensional Hilbert spaces. A reformulation using
the matrix representation.

12.  Regular values of an operator. The resolvent set. The spectrum of an operator. Point spectrum and
eigenvalues of an operator. Continuous spectrum. Residual spectrum. Approximate point and compression spectra
of an operator. The theorem about the compression spectrum (necessary and sufficient conditions for a complex
number to be in the compression spectrum). Necessary and sufficient conditions for a complex number to be
in the approximate point spectrum of an operator. The spectrum of a symmetric operator. The spectrum of a self-adjoint
operator (real and the residual spectrum empty). Spectra of some differential operators (examples).

13. Elements of the measure theory. Measure spaces. Measuarable functions. Integration theory on measure spaces.
Square integrable functions on a measure space. Examples.

14. Unitary transformations of a Hilbert space to the space of square integrable functions on a measure space.
Examples: Fourier series and Fourier transform. The spectral theorem in terms of multiplication operators.
Examples:  a finite-dimensional Hilbert space, simple (self-adjoint) differentiation operators in the space of square integrable
function on an interval and on a real line.

15. Spectral family of operators. Examples: a finite-dimensional Hilbert space, the differentiation operator in the space of square integrable
function on an interval and on a real line. The spectral theorem in terms of projection measures. Examples.
Stone’s theorem about unitary operators (mathematical foundations of quantum mechanics). Example: Laplace
operator and the energy operator (Hamiltonian) of a free particle. Unitary operator generated by it.
Projection measures and the “wave-packet” representation.

16. Compact (or completely continuous) operators. The inverse of a compact, invertible operator on an infinite-dimensional
Hilbert space. The Fredholm alternative for compact operators. Further properties of compact operators.

Suggested reading:

– I. Stakgold, Green’s functions and boundary value problems, Chapter 5.
– F. Riesz and B. Sz.-Nagy, Functional analysis, Chapters 6-10.
– V.S. Valdimirov, Equations of mathematical physics, Chapter 4.

 

Topic 3: Applications of the operator theory to PDEs

1. Integral operators with Hilbert-Schmidt kernels and polar kernels as an example of compact  operators. Spectrum of  a compact, self-adjoint operator.

2. The Sturm-Liouville problem for the operator Lu = -div (p grad u) + qu. Basic properties of L.  Eigenvalues of L.

3. Regular and singular Sturm-Liouville problems in an interval. Reduction to an integral equation  for a compact integral operator with a symmetric kernel. Examples. Bessel equation, Bessel functions, zeros of Bessel functions, and Fourier-Bessel series.

4. Dirichlet and Neumann problems for the Laplace and Poisson equations. Basic properties of harmonic function. Theorems about solvability of the Dirichlet and Neumann problems and the uniqueness of the solution.

5. Fourier method of solving the Cauchy (initial value) problem for u” = -Lu +f in a region.   Elliptic, parabolic, and hyperbolic problems in a bounded region. Example: wave equation in a ball in a three-dimensional Euclidean space; spherical harmonics, Legendre polynomials.

6. Reduction of the Dirichlet and Neumann problems to integral equations. Properties of the simple- and double-layers potentials. The Fredholm alternative for compact operators and solvability of the Dirichlet and Neumann problems.

7. Green’s function for the operator L in a region.  Example: Green’s function of the Laplace operator in a ball for Dirichlet problems.

Suggested reading:

— I. Stakgold, Green’s functions and boundary value problems, Chapter 6.
— V.S. Valdimirov, Equations of mathematical physics, Chapters 5-6