Problems and Solutions in Theoretical and Mathematical Physics

The following sections are included:

• Preface

• Contents

• Notation

Let G be a finite group. A group element h is said to be conujugate to the group element k, h ˜ k, if there exists a g ∈ G such that k = ghg⁻¹. For G the number of conjugacy classes is equal to the number of irreducible matrix representations. In a character table the rows correspond to irreducible group representations and columns to classes of group elements. If the irreducible matrix representation is given by an n × n matrix (n ≥ 2) then the trace of this square matrix is the element in the table…

Let A be an m × n matrix and let B be a p × q matrix. The Kronecker product of A and B is defined as

In Nambu mechanics the equations of motion are given by

Let W₁ be some topological vector space and f a (nonlinear) mapping f : W → W₁. We call f Gateaux differentiable in u ∈ W if there exists a mapping θ ∈ L(W,W₁) such that for all υ ∈ W

Let S be an arbitrary set. Consider a map f : S → S. Let x^* ∈ S. Then x^* is called a fixed point of f if f(x^*) = x^*. Consider the difference equation u_t+1 = f(u_t) with t = 0,1, … and given u₀ (initial value). Then u^* is a fixed point if f(u^*) = u^*…

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Let

A Hilbert space is a set, $ \mathcal{H}$ of elements, or vectors, (f, g, h, …) which satisfies the following conditions (1) to (5)…

The vector space D(ℝⁿ) denotes all infinitely-differentiable functions in ℝ with compact support. The functions ø ∈ D(ℝⁿ) are called test functions. Each linear continuous functional over the vector space of test functions D(ℝⁿ) is called a generalized function…

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If a partial differential equation admits continuous symmetries (Lie symmetries) we can reduce it to an ordinary differential equation using a similarity variable which is constructed from the Lie symmetries…

Soliton equations are special partial differential equations. They have a number of interesting properties such as a Lax representation, solvable by the inverse scattering method, infinite many conservation laws. They also admit Bäcklund transformations and Lie-Bäcklund symmetries. Most of them also pass the Painlevée test…

Let L and M be two linear differential operators. Assume that Lv = λv

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Let f, g be smooth functions. The Hirota bilinear operators D_x and D_t are defined as

A partial differential equation has the Painlevé property when the solution of the partial differential equation considered in the complex domain is single-valued about the movable, singularity manifold. One requires that the solution be a single-valued functional of the data, i.e., arbitrary functions. To prove that a partial differential equation has the Painlevé property one expands a solution about a movable singular manifold ø(z₁, …, z_n) = 0…

A Lie algebra is defined as follows: A vector space L over a field F, with an operation L × L → L denoted by (x, y) → [x, y] and called the commutator of x and y, is called a Lie algebra over F if the following axioms are satisfied:

(L1) The bracket operation is bilinear…

A Lie group is a group G which is also an analytic manifold such that the mapping (a, b) → ab⁻¹ (a, b ∈ G) of the product manifold G × G into G is analytic. Thus a Lie group is set endowed with compatible structures of a group and an analytic manifold. A subgroup H of a Lie group G is said to be a Lie subgroup if it is a submanifold of the underlying manifold of G. A group G is called a Lie transformation group of a differentiable manifold M if there is a differentiable map φ : G × M → M, φ(g; x) = gx(x ∈ M) such that (i) (g₁ · g₂)x = g₁ · (g₂x) for x ∈ M and (ii) ex = x for the identity element e of G and x ∈ M…

We define differential p-forms of class C^∞ on an open set Ω of ℝⁿ to be the expressions

Let

Let M be an n-dimensional C^∞ differentiable manifold with local coordinates x_j, j = 1, …, n. Real valued C^∞ vector fields and real valued C^∞ differential forms on M can be considered. The components of the vector field V are denoted by V_j∂/∂x_j…

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Let M be a smooth manifold and g be a metric tensor field. Then V is called a Killing vector field with respect to g if L_Vg = 0 where L_V(˙) denotes the Lie derivative. If V and W are Killing vector fields of g, then the commutator [V,W] is also a Killing vector field…

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For Fermi operators we have to take into account the Pauli principle. Let be Fermi creation operators and let c_j be Fermi annihilation operators, where j = 1, …, N, with j denoting the quantum numbers (spin, wave vector, angular momentum, lattice site, etc.)…

Consider a family of linear operators b_j, , j = 1, 2, …, m on an inner product space V, satisfying the commutation relations (Heisenberg algebra)

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The following sections are included:

• Bibliography

• Index