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Algebraic Embedding of Pell Sequences within Dirac Vector Spaces

Abstract

This document attempts to establish a rigorous mathematical bridge between discrete integer sequences (Pell and Lucas) and continuous Hilbert space mechanics. By defining a finite-dimensional state vector and an associated linear operator, the asymptotic expansion ratios of recurrence sequences are calculated utilizing standard Dirac inner products.

Methodology

1. State Vector Mapping

Discrete elements of the Pell sequence $P_n$ are mapped to a two-dimensional vector space. The quantum state $| \psi_n \rangle$ at index $n$ is defined as:

$$| \psi_n \rangle = \begin{pmatrix} P_n \\ P_{n-1} \end{pmatrix}$$

2. Operator Formalism

The standard Pell recurrence relation, $P_{n+1} = 2P_n + P_{n-1}$, is expressed via a time-evolution operator $\hat{T}$. This transfer matrix advances the state vector:

$$\hat{T} = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{pmatrix}$$

The state progression is linearly defined as $| \psi_{n+1} \rangle = \hat{T} | \psi_n \rangle$.

Results: Asymptotic Evaluation

The Dirac inner product is utilized to evaluate the dominant eigenvalue of the system, which represents the asymptotic expansion ratio. For large $n$, the expectation value of the operator $\hat{T}$ isolates the Silver Ratio:

$$\lim_{n \to \infty} \frac{\langle \psi_n | \hat{T} | \psi_n \rangle}{\langle \psi_n | \psi_n \rangle} = 1 + \sqrt{2}$$

Conclusion

This formulation attempts to resolve the domain mismatch between discrete number theory and quantum mechanics. The bra-ket notation functions as a valid matrix-algebraic structure for extracting deterministic asymptotic limits from recurrence equations, providing explicit mathematical utility for the inner product.

Spectral Decomposition

The foundational piece to complete your algebraic embedding is the Eigendecomposition of the transfer operator $\hat{T}$. This bridges the step-by-step vector evolution to a closed-form expression, acting as the matrix-algebraic equivalent of Binet's formula.

By extracting the eigenvalues $\lambda_1 = 1 + \sqrt{2}$ and $\lambda_2 = 1 - \sqrt{2}$, the quantum state can be expressed as a superposition of the operator's eigenkets:

$$|\psi_n\rangle = c_1 \lambda_1^n |e_1\rangle + c_2 \lambda_2^n |e_2\rangle$$

Non-Unitary System Dynamics

A secondary conceptual gap is the nature of the operator. In standard Dirac mechanics, time-evolution operators are unitary, preserving state probability. Your operator $\hat{T}$ is non-unitary and real-symmetric.

This means your framework describes an amplifying system rather than a conserved quantum state. The "wavefunction" norm diverges to infinity, which is why your asymptotic evaluation correctly requires normalizing by $\langle \psi_n | \psi_n \rangle$ to isolate the dominant eigenstate.