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In this paper we consider second order self-adjoint systems of differential equations. A fairly straight-forward and self-contained proof of the equivalence of three known conditions for disconjugacy is established.

This paper describes some families of unconstrained variational principles for finding eigenvalues and eigenfunctions of symmetric closed linear operators on a Hilbert space. The functionals involved are smooth, with well-defined second derivatives and Morse-type indices associated with nondegenerate critical points. This leads to an analog of the Courant-Fischer-Weyl minimax theory, where an analysis of the second derivative at a critical point leads to the determination of which eigenvalue of the operator is associated with this eigenfunction. An extension to weighted eigenproblems is also described. The case of linear second order elliptic operators on a nice bounded set and subject to zero Dirichlet conditions is treated in some detail.

The local residue has been presented since the early days of several complex variables and plays a fundamental role in complex analysis and geometry. Nevertheless, in many cases, there are very few practical tools in the computational point of view.

After a lengthy hiatus, research on reduced density matrix (RDM) theory has re-emerged, and with results an order of magnitude better than before. There is little doubt that the theory has turned the corner and that we will see a succession of rapid advances in the near future.

We introduce the inverse of the hermitian operator (ââ†) and express the Boson inverse operators â−1 and a†−1 in terms of the operators â, at and (âât)−1. We show that these Boson inverse operators may be realized by Susskind—Glogower’s phase operators. We find a family of three two-photon annihilation operators (ââ†) -1 â 2 , â(âât)−1â and â2(ââ†)−1 and show that the eigenstates of these operators have interesting nonclassical properties including squeezing and antibunching of photons.

Our intention in this chapter is to show that the theorems on the exact conditions of uniform convergence and localization of spectral decompositions that have been established by us in Chapter 2 for an arbitrary self-adjoint nonnegative extension of the Laplace operator remain valid also for arbitrary self-adjoint nonnegative extensions of a general elliptic operator of second order L.

In this chapter we establish exact conditions for the convergence of the spectral decompositions corresponding to an arbitrary self-adjoint nonnegative extension of the Laplace operator in the domain G (not necessarily a bounded one) of the space EN.

In classical mechanics a particle in a state (x0, p0) has associated with it welldefined values for any dynamical variable ω(x, p)namely, ω(x0, p0). In quantum theory, given a state |ψ〉, one can only give the probabilities P(ω) for the possible outcomes of a measurement of Ω. The probability distribution will be characterized by a mean or expectation value

I shall at first restate the principles of quantum mechanics. We take an abstract Hilbert space H. According to a definite rule, we associate with every physical quantity a linear operator over H, bounded or not bounded. We choose an element ψ of H such that ||ψ|| = 1. We call the mean value of A in the state ψ the scalar product fy(1)|1-1 if A is hermitian, Q(A) is real.

Quantum optics has progressed a long way in the last sixty years and much is now understood about coherence. Nevertheless, the description of the quantum nature of optical phase has been a longstanding problem. Classically, phase is a useful and easily-understood concept, and it might be expected that the classical phase observable should, according to the usual quantization procedures, correspond to an Hermitian phase operator ϕ. Indeed, the existence of this operator was indeed originally postulated by Dirac,1 but problems associated with finding such an operator have led to the present wide-spread belief that no such operator exists.2 Attempts to construct a phase operator have involved the use of an infinite dimensional state space and a polar decomposition of the annihilation operator. This procedure does not provide a unique operator. Specifically, the action of exp (iϕ) on the vacuum is not determined. The usual ad hoc assumption is to set exp (iϕ) ∣ 0 > = 0. This removes the indeterminacy but destroys the unitarity of exp (iϕ), along with the possibility of extracting an Hermitian operator from the exponential.

With the last two problems we have reached the limits of our — so far — purely algebraic reasoning where we could reduce everything to the elementary theory of solutions of n linear equations in n unknowns. In fact, the limit process n → ∞ marks the emergence of functional analysis which went beyond the methods of linear algebra while being motivated by its questions and results. This occurred mainly in the study of integral equations.

Consider the motion of a particle with mass m in the spherically symmetric confinement potential V(r) = grP, g > 0, p > 0. The corresponding radial Schrodinger operator Hl (with angular momentum l) has only a discrete spectrum, 0 (p+2)/2p) $$ M_\ell ^{\left( N \right)} \equiv \sum\limits_{m = 0}^\infty {\frac{1}{{E_{n\ell }^N}}} $$ (1) (“Zeta function” of the hermitian operator Hl evaluated at S = N).

Position, in quantum theory, is described by a linear Hermitian operator Q whose action on the wave function is equivalent to a multiplication by the position parameter x Therefore the eigenvalue equation $$Qu\left( {q;x} \right) = qu\left( {q;x} \right)$$ is solved by an arbitrary real value of q and by the corresponding eigenfunction (normalized on the unit length in wave number space) $$u\left( {q;{\kern 1pt} x} \right) = \delta \left( {x - q} \right)$$ (1) which is Dirac’s δ-function. The wave function (1) predicts a fixed position q. As one can write $$u\left( {q;x} \right) = \frac{1}{h}\int {dp'exp\left[ {ip'\left( {x - q} \right)/\hbar } \right]}$$ (2) it follows that all possible values of momentum have equal probability.

It is well known that the spectrum of the one-dimensional Schrodinger operator with a periodic potential is purely continuous and consists of a countable sequence of segments of the real axis separated by lacunae and going out to + ∞. When this operator is perturbed by a real potential q(x) which vanishes at infinity, it is possible that eigenvalues will appear in the lacunae of the continuous spectrum and also to the left of the lower bound of the operator. The following facts are known.

Let K0, K denote two separable Hilbert spaces. The set of bounded linear operators mapping K° into K we denote by R(K0, K). The sets R(K, K0), R(K0), R(K) are defined similarly. When no confusion is possible, we will write simply R. We denote similarly by S1 and Soo the sets of nuclear and completely continuous operators. I0, I denote the identity operators in K0 and K respectively. The symbol s-lim denotes the strong operator limit. If A is a densely defined operator, then D(A) denotes its domain and A* the adjoint operator. We denote by Z the real (spectral) axis.