The Deterministic Schrödinger Operator

Abstract

Let L be the linear space of complex sequences ψ = (ψ_n) where n runs through the set of integers ZZ. The operator H is associated to two given real sequences a and b with b_n ≠ 0 ∀n ε ZZ, and acts on L by the formula:

$$ {\left( {{\text{H}}\psi } \right)_{\text{n}}} = {\text{ }}{{\text{b}}^{ - {\text{1}}}}_{\text{n}}[{\psi _{{\text{n}} - {\text{1}}}},{\text{ }} - {\psi _{{\text{n}} + {\text{1}}}} + {\text{ }}{{\text{a}}_{\text{n}}}{\psi _{\text{n}}} $$

. For a complex number X every solution of the difference equation Hψ = λψ lies in a two dimensional subspace of ob spanned by the solutions p(λ) and q(λ) constructed from the initial values p (X) = q_-1(λ) = 1, p_-1 (λ) = q_o (λ) = 0, such that:

$$ {\psi _{ n}}^{\left( \lambda \right)} = {p_n}^{\left( \lambda \right)}{\psi _o}\left( \lambda \right) + {q_n}\left( \lambda \right){\psi _{ - 1}}\left( \lambda \right) $$

. From now in order to avoid too complicated notations we don’t write the variable X in the solutions of the difference equation. A solution ψ of the difference equation is constructed from initial values ψ_o and ψ_-1. by a product of “transfer matrices” Y_n defined by:

$$ {Y_n}\left[ {\mathop {{a_n} - \lambda {b_n}}\limits_1 \mathop { - 1}\limits_{\;\;\;\;\;\;0} } \right] $$

$$ {S_n} - {Y_n}{Y_{n - 1}} \cdot \cdot \cdot {Y_o} if n \geqslant 0 $$

$$ {S_n} - Y_n^{ - 1}Y_{n + 1}^{ - 1} \cdot \cdot \cdot Y_{ - 1}^{ - 1} if n \leqslant - 1 $$

$$ Thus\left[ {\frac{{{\psi _{n + 1}}}}{{{\psi _n}}}} \right] = {S_n}\left[ \begin{gathered} {\psi _o} \hfill \\ {\psi _{ - 1}} \hfill \\ \end{gathered} \right] if n \geqslant 0,\left[ \begin{gathered} {\psi _n} \hfill \\ {\psi _{n - 1}} \hfill \\\end{gathered} \right] = {S_n}\left[ \begin{gathered} {\psi _o} \hfill \\{\psi _{ - 1}} \hfill \\ \end{gathered} \right] if n \leqslant - 1 $$

. The transfer matrices Y_n and therefore the products S_n belong to the group SL(2,c) of two by two matrices with complex entries and of determinant one. If A is real then Y and S belong to the subgroup SL(2,ℝ) with real entries. The construction of the solutions of the difference equation by such products of matrices is the essential link between the two parts of this book.