1. Topological Vector Spaces 1 1.1. The Krein-Milman theorem 7 2. Banach Algebras 11 2.1. Commutative Banach algebras 14 2.2. ∗–Algebras (over complexes) 17 2.3. Problems on Banach algebras 20 3. The Spectral Theorem 21 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 26 3.2. Integration with respect to a Projection Valued Measure 27 3.3. The Functional Calculus 34 4. Unboun...

The self-adjointness of an evolution operator corresponding to the model of the flat FRW universe with massless scalar field and cosmological constant quantized in the framework of loop quantum cosmology is studied in the case > 0. It is shown that, for < c 10:3‘ 2 Pl , the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for c the operator i...

1. Topological Vector Spaces 1 1.1. The Krein-Milman theorem 7 2. Banach Algebras 11 2.1. Commutative Banach algebras 14 2.2. ∗–Algebras (over complexes) 17 2.3. Problems on Banach algebras 20 3. The Spectral Theorem 21 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 26 3.2. Integration with respect to a Projection Valued Measure 27 3.3. The Functional Calculus 34 4. Unboun...

We discuss connections between the essential self-adjointness of a symmetric operator and constancy functions which are in kernel adjoint operator. then illustrate this relationship case Laplacians on both manifolds graphs. Furthermore, we Green’s function when it gives non-constant harmonic is square integrable.

We discuss the Hamiltonian for a nonrelativistic electron with spin in the presence of a magnetic monopole and note that it is not self-adjoint in the lowest two angular momentum modes. We then use von Neumann's theory of self-adjoint extensions to construct a self-adjoint operator with the same functional form. In general, this operator will have eigenstates in which the lowest two angular mom...

1. Topological Vector Spaces 1 2. Banach Algebras 10 2.1. ∗–Algebras (over complexes) 15 2.2. Exercises 18 3. The Spectral Theorem 19 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 24 3.2. Integration with respect to a Projection Valued Measure 25 3.3. The Functional Calculus 32 4. Unbounded Operators 35 4.1. Closed, symmetric and self-adjoint operators 35 4.2. Differentia...

We study the self-adjoint Pauli operators that can be realized as the square of a self-adjoint Dirac operator and correspond to a magnetic field consisting of a finite number of Aharonov-Bohm solenoids and a regular part, and prove an Aharonov-Casher type formula for the number of zero-modes for these operators. We also see that essentially only one of the Pauli operators are spin-flip invarian...