Mathematical Companion to Quantum Mechanics by Shlomo Sternberg

Author:Shlomo Sternberg
Language: eng
Format: epub
Publisher: Courier Publishing
Published: 2019-04-27T16:00:00+00:00

14.1 The Rayleigh-Ritz Method

Let H be a non-negative self-adjoint operator on a Hilbert space For any finite dimensional subspace L of with L ⊂ D ≔ Dom(H) define

Define

The λn are an increasing family of numbers. We shall show that they constitute that part of the discrete spectrum of H which lies below the essential spectrum. Here is the relevant theorem:

Theorem 14.1.1. Let H be a non-negative self-adjoint operator on a Hilbert space . Define the numbers λn = λn(H) by (14.1). Then one of the following three alternatives holds:

1.H has empty essential spectrum. Then the λn →∞ and coincide with the eigenvalues of H repeated according to multiplicity and listed in increasing order or else is finite dimensional and the λn coincide with the eigenvalues of H repeated according to multiplicity and listed in increasing order.

2.There exists an a < ∞ such that λn < a for all n and limn→∞ λn = a. In this case a is the smallest number in the essential spectrum of H and σ(H) ∩ [0, a) consists of the λn which are eigenvalues of H repeated according to multiplicity and listed in increasing order.

3.There exists an a < ∞and an N such that λn < a for n ≤ N and λm = a for all m > N. Then a is the smallest number in the essential spectrum of H and σ(H) ∩[0, a) consists of λ1, . . . ,λN which are eigenvalues of H repeated according to multiplicity and listed in increasing order.

Let b be the smallest point in the essential spectrum of H (so b=∞ in case 1). So H has only isolated eigenvalues of finite multiplicity in [0, b) and these constitute the entire spectrum of H in this interval. Let {fk} be an orthonormal set of eigenvectors corresponding to these eigenvalues μk listed with multiplicity in increasing order. We want to show that

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