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Why can every symmetry operator have a self-adjoint expansion? Are you ready to understand the secret?
In functional analysis, the problem of self-adjoint expansion of symmetry operators has attracted the attention of many mathematicians. Especially in quantum mechanics, when it is necessary to specify a self-adjoint domain for the formal expression of an observation quantity, the existence of self-adjoint extension is of great significance. This article will delve into why every symmetry operator has the potential for self-adjoint expansion and explore its applications in various problems.
Basic knowledge of symmetry operators
First, we need to understand what a symmetry operator is. In a Hilbert space H, a linear operator A is said to be symmetric if its densely defined domain dom(A) > Satisfy the condition ⟨Ax,y = ⟨x,Ay for all x and y belonging to dom(A) code>. This property means that symmetry operators not only have elegant structures in mathematics, but also realize many practical applications in physics.
Every densely defined symmetry operator is closable, which means that a minimal closed extension can always be found.
Necessary conditions for self-companion expansion
For a symmetry operator A, we need to know when it has self-adjoint expansion. When an operator has a unique self-adjoint extension, it is said to be intrinsically self-adjoint. On the contrary, if the closure of the operator A* is also self-adjoint, then this operator must also be essentially self-adjoint. Here, we face a challenge: some symmetry operators may have multiple self-adjoint extensions, or even no self-adjoint extensions.
The necessary and sufficient condition for a symmetric operator to have self-adjoint expansion is that the dimensions of its insufficient-dimensional subspaces are the same.
Specific examples of self-companion expansion
Take the definition of operator A in Hilbert space L^2([0,1]) as an example. We define the operator as Af=i(d/dx)f. After partial integration, we can show that A is symmetric, but the domain of its dual A* is set without boundary conditions. Such an extension means that we need to adjust the boundary conditions to enhance the domain of dom(A).
Theoretical basis of self-adjoint expansion
The theoretical basis of self-adjoint expansion relies on the concepts of Cayley transformation and insufficient dimensional space. Among them, for each partially defined unit operator V, it can always be expanded to a unit operator. This feature ensures that each symmetry operator has self-adjoint expansion. By having a well-defined subspace of insufficient dimensions, we can build a structured parametric model for self-adjoint expansion.
Characteristics of positive symmetry operators
Positive symmetry operators are a special case. This type of operator satisfies the condition of ⟨Ax,x ≥ 0 in its domain. This means that all eigenvalues of these operators are nonnegative, and for some specific mathematical and physical problems, this property can make the problem of self-adjoint expansion relatively simple.
In summary, for every symmetry operator, regardless of its properties, the possibility of self-adjoint expansion can be found. This not only highlights the mathematical beauty of symmetry operators, but also provides a solid theoretical foundation for many applications. With the deepening of research, thinking about how to better understand these phenomena in different mathematical structures may reveal more mathematical mysteries and application potential. Are you ready to further explore the possibilities of these operators?
