Mathematical Principles of Theoretical Physics

3.6 Spectral Theory of Differential Operators Since the orbital electrons are bound in the interior of the atom, the following c ...

168 CHAPTER 3. MATHEMATICAL FOUNDATIONS Theorem 3.34(Spectral Theorem of Elliptic Operators).Let the matrices in (3.6.7) are Her ...

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 169 and the constraint functionalGis given by (3.6.11) G(ψ) = ∫ Ω |ψ|^2 dx. It is ...

170 CHAPTER 3. MATHEMATICAL FOUNDATIONS Proof.Assertion (1) follows directly from the following classical formula for the first ...

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 171 3.6.4 Estimates for number of negative eigenvalues For simplicity, it is phys ...

172 CHAPTER 3. MATHEMATICAL FOUNDATIONS Theorem 3.38.Under the assumptions of (3.6.22) and (3.6.23), the number N of the negativ ...

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 173 It is known thatφnis weakly differentiable, andφn∈H 01 (Ω). These functionsφn ...

174 CHAPTER 3. MATHEMATICAL FOUNDATIONS Example 3.40(Number of Atomic Energy Levels).As an application of Theorem3.38, we consid ...

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 175 whereφ= (φ 1 ,φ 2 )T:Ω→C^2 is a complex-valued function with two components, ...

176 CHAPTER 3. MATHEMATICAL FOUNDATIONS 1) The eigenvalues of (3.6.40) are real and discrete, with finite multiplicities, and sa ...

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 177 In the following, we consider the estimates of the number of negative eigenva ...

178 CHAPTER 3. MATHEMATICAL FOUNDATIONS ...

Chapter 4 Unified Field Theory of Four Fundamental Interactions Once again, the goal of this book is to derive experimentallyver ...

4.1 Principles of Unified Field Theory 2) the coupling of the four interactions is achieved through PID and PRI in the unified f ...

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 181 We know now that the four interactions are dictated respectively by the following sy ...

182 CHAPTER 4. UNIFIED FIELD THEORY which also stands for the scalar curvature of the spinor bundle:M⊗p(C^4 )^2. Here Wμ νa =∂μW ...

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 183 based on the Geometric Interaction Mechanism, which focuses directly on the four int ...

184 CHAPTER 4. UNIFIED FIELD THEORY 3) the action (4.1.11) also becomes the simple sum of these actions given in (4.1.5)- (4.1.8 ...

4.1. PRINCIPLES OF UNIFIED FIELD THEORY 185 First, the unified field model of (4.1.14) and (4.1.15) not only solves the most bas ...

186 CHAPTER 4. UNIFIED FIELD THEORY are the Riemannian metric{gμ ν}and the gauge fields{Gaμ}. In addition, the symmet- ric princ ...