Formal Differential Operator Research Articles

Global Dimension of Differential Operator Rings. III

The aim of this paper is to derive formulas for the global homological dimension of the ring R[θ1,..,θu] of formal linear differential operators over a commutative noetherian ring R with u commuting derivations. Since the case when R has finite global dimension has been completed in [3], the present paper deals mainly with the case when R has infinite global dimension. Formulas are derived which show exactly when R[θ1,.. θu] has finite global dimension, and what the value of that dimension is. Examples are constructed of commutative noetherian domains R such that R is torsion-free as an abelian group, R has infinite global dimension, and R has a derivation such that R[θ] has finite global dimension.

Quantum statistical mechanics on stochastic phase space

Within the context of the theory of stochastic phase spaces, introduced in some earlier papers, a systematic mathematical procedure is developed for expressing quantum mechanical observables as generalized functions on a stochastic phase space. The states in such a theory are normalized, positive semidefinite, continuous functions of the phase space variables, satisfying marginality conditions appropriate to the stochastic nature of the underlying phase space. The action of a general quantum mechanical observable on the state space is then shown to lead in general to formal differential operators of finite or infinite order. Explicit computations of some typical operators are made to illustrate the theory. As a useful practical application, the theory is employed to derive a Bloch equation from which the Husimi transform of the canonical equilibrium state is then computed, after expressing it as an infinite series in powers of ℏ.

A general algorithm for determining state-space representations

A new and direct procedure is presented for determining state-space representations of given, time-invariant systems whose dynamical behavior is expressed in a more general, differential operator form. The procedure employs some preliminary polynomial matrix operations, if necessary, in order to “reduce” the given system to an equivalent differential operator form which satisfies four specific conditions. An equivalent state-space representation is then determined in a most direct manner; i.e. the algorithm presented requires only a single matrix inversion. An explicit relationship between the partial state and input of the given system and the state of the equivalent state-space system is also obtained.

The differential operator approach to linear system analysis and design

The time domain, state-space approach to control systems analysis and design has dominated the control literature since the early 1960's. More recently, however, there has been a resurgent interest in frequency domain methods. This is due in large part to the development of an alternative (to the state-space approach) time domain technique, namely the differential operator approach for the analysis and synthesis of linear multivariable systems. More specifically, a significant number of investigators have now noted and employed the fact that the (pxm) transfer matrix, T(s), of a linear, time-invariant, multivariable system can be factored as R(s) P-1R(s) [orP-1Q(s) Q(s)]. with R(s) and PR(s) [PQ(s) and Q(s)] relatively right (left) prime polynomial matrices in the Laplace operator's and PR(s) [PQ(s)] column (row) proper. Furthermore, such a factorization directly implies a minimal (i.e. both controllable and observable) time domain realization of T(s) in “differential operator form”, namely PR(D) z(t) = u(t); y(t) = R(D) z(t) [orPQ(D) y(t) = Q(D) u(t)], where PR(D) andR(D) [PQ(D) andQ(D)] are polynomial matrices in the differential operator D = d/dt, withPR(D) [PQ(D)] nonsingular. It might be noted that a differential operator description of the dynamical behavior of a physical system often follows as a direct result of applying well-known physical laws to describe the performance of the system. The primary purpose of this paper is to delineate and unify a number of the more important differential operator techniques which have recently been developed and to compare them to analogous, but perhaps more familiar state-space procedures.

Representation of linear operators in the form of differential operators of infinite order

Representation of linear operators in the form of differential operators of infinite order

Vestigial effects of singular potentials in diffusion theory and quantum mechanics

Repulsive singular potentials of the form λV (x) =λ‖x−c‖−α, λ≳0, in the Feynman−Kac integral are studied as a function of α. For α≳2 such potentials completely suppress the contribution to the integral from paths that reach the singularity, and thus, unavoidably, certain vestiges of the potential remain even after the coefficient λ↓0. For 2⩾α⩾1 careful definition by means of suitable counterterms at the point of singularity (similar in spirit to renormalization counter terms in field theory) can lead to complete elimination of effects of the potential as λ↓0. For α<1 no residual effects of the potential exist as λ↓0. In order to prove these results we rely on the theory of stochastic processes using, in particular, local time and stochastic differential equations. These results established for the Feynman−Kac integral conform with those known in the theory of differential equations. In fact, a variety of vestigial effects can arise from suitable choices of counter terms, and these correspond in a natural way to various self−adjoint extensions of the formal differential operator.

Cohomology of the algebra of formal universal differential operators

Cohomology of the algebra of formal universal differential operators

On the structural properties of composite systems

Composite systems are defined as multivariable systems which are obtained through a rather general interconnection procedure involving two or more subsystems, each of whose dynamical behavior can be expressed in differential operator form. A technique is outlined for representing the complete dynamical behavior of any composite system via a single differential operator equation, which can then be used to investigate the controllability and observability properties of the composite system. A reduction procedure is presented for reducing a composite system to an equivalent form which separates the system's modes into four mutually exclusive classes. Some observations regarding the minimal order, the zeros, and the ability to stabilize composite systems are then presented.

Composite system controllability and observability

A differential operator approach to modeling the dynamical behavior of linear, time-invariant, multivariable systems is employed in order to develop some new techniques for determining the controllability and observability properties of series, parallel, and feedback composite systems. The tests outlined involve polynomial matrix manipulations and rely on some recent results which deal with the controllability and observability of systems in differential operator form.

Note on a Limit-Point Criterion

A sufficient condition is given for the formal differential operator $\tau y(t) = (p(t)y’(t))’ + q(t)y(t)$ defined on the interval $[a,b),b \leqq \infty$, to be of limit-point type at $b$; this generalizes a criterion of Ismagilov given for the case $p(t) = 1$ and $b = \infty$.

Homomorphisms and extensions of modules over certain differential polynomial rings

There is a well-known simple hereditary noethcrian domain L3; this is the @-algebra (@ a field of characteristic zero) with two generators x, y subject to the relation xy -yx = 1. A is isomorphic to the ring of formal differential operators with polynomial coefficients and so may be regarded as @]~y][.x], where @[y] is the ring of polynomials over Cp in an indeterminate y, and @[y][x] is the ring of noncommutative polynomials over @[y] in an indeterminatc .T with

Note on a limit-point criterion

A sufficient condition is given for the formal differential operator $\tau y(t) = (p(t)y’(t))’ + q(t)y(t)$ defined on the interval $[a,b),b \leqq \infty$, to be of limit-point type at $b$; this generalizes a criterion of Ismagilov given for the case $p(t) = 1$ and $b = \infty$.

The determination of state-space representations for linear multivariable systems

A new, relatively simple, constructive method is presented for obtaining state-space or normal form representations for linear, time invariant systems whose dynamics are expressed in a more general matrix differential operator form. The development employed provides new insight into various structural properties of linear systems. Equivalence is defined for a rather large class of linear systems, and an algorithm is given for reducing any member of this class to normal form. In order to outline the algorithm, a number of “well known” results involving polynomial matrices are employed for the first time, along with the “structure theorem”. An example is used to illustrate the algorithm and two appendices are employed to clarify the development.

The method of least squares for boundary value problems

The method of least squares is used to construct approximate solutions to the boundary value problem if=go, B,(f) = 0 for i=1, ...,k, on the interval [a, b], where r is an nth order formal differential operator, go(t) is a given function in L2[a, b], and B1, ..., Bk are linearly independent boundary values. Letting Hn[a, b] denote the space of all functions f(t) in C -'[a, b] with f(n -1) absolutely continuous on [a, b] andftn) in L2[a, b], a sequence of functions e1(t) (i= 1, 2, .. .) in Hn[a, b] is constructed satisfying the boundary conditions and a completeness condition. Assuming the boundary value problem has a solution, the approximate solutions 1 a'e(t) (i = 1, 2, ...) are constructed; the coefficients a' are determined uniquely from the system of equations 2 (Te, rel)a' = (go, e), I = 1.1, where (f, g) denotes the inner product in L2[a, b]. The approximate solutions are shown to converge to a solution of the boundary value problem, and error estimates are established.

Differential-operator form of the quantum kinetic equation

The kinetic equations for quantum distribution functions are usually in integral form. For example, the quantum distribution function introduced by Blokhintsev [1,2], l 1 x p F N (x, p, t) = ( 2 ~ h ~ 2 ~x~ "e h , (1) where N is the number of particles'in the system; x{x~,. . . ,XN}, P{P~' ' 9 ] ,PN} are the sets of particle coordinates and momenta; ~ is the Planck's constant; t is time; and p is the density matrix in the x, p representation, is consistent with the kinetic equation

The Transient Response of a Viscoelastic Torsional Pendulum

The pendulum consists of a circular viscoelastic rod fixed at one end and attached to a disk at the other end. The disk, initially at rest, is subjected to a step function of torque. The subsequent motion is analyzed in terms of normal modes and normal coordinates. The normal modes are found by separation of variables. The characteristic numbers associated with these modes depend only on the moments of inertia of the disk and rod. The normal coordinates satisfy integro-differential equations of the Volterra type. The mechanical properties of polyisobutylene at 25°C were represented in differential operator form. The Volterra integral equations were then solved by means of the Laplace transforms. An analysis of the numerical results shows that only the oscillatory part of the fundamental mode should persist after a short time. This result is in qualitative agreement with observations. This method of analysis may be applied to other dynamic problems in linear viscoelasticity provided the characteristic equation can be reduced to a form that does not involve the mechanical properties of the material.

The Spectral Matrix and Green's Function for Singular Self-Adjoint Boundary Value Problems

Let L denote the formal ordinary differential operator,where we assume the pk are complex-valued functions with n-k continuous derivatives on an open real interval a < x < b (a = — ∞, b = + ∞, or both may occur), p0(x) ≠ 0 on a < x < b, and L coincides with its Lagrange adjoint L+ given by.

On the existence of solutions of the helium wave equation

Introduction. The helium wave equation has been studied by several authors(') with a view to obtaining the exact solution for the ground state, or at least to establishing its existence. The subject is of importance as a matter of principle, since it implies the decision whether the present formulation of nonrelativistic many-body problems is correct(2). Also, the validity of the variational method(3) depends upon it, for this method would be meaningless if there existed no stationary state at all. Unfortunately the attempts of these authors have not been successful; their method of series expansion proved to be powerless to control manyparticle problems. In the present paper, the writer wishes to settle this problem by showing rigorously that the wave equation for the two-electron problem, in particular the helium wave equation, has a very large number-even infinity if the nucleus is assumed to be infinitely heavy-of solutions corresponding to stationary states of the system. In particular the existence of the ground state solution is established. Our method is different from those adopted by the authors cited above. We shall not attempt, for the present, to find explicit expressions for the solutions and shall resort to a method based on the abstract theory of operators in Hilbert space. The essential part of our theory may be regarded as already completed in the previous paper(4), where it was shown that the Hamiltonian operator of every atom, molecule, or ion is essentially selfadjoint(5). This means that the closed Hamiltonianfl of such a system, which is uniquely determined by the given Hamiltonian as a formal differential operator, is self-adjoint in the strict sense, that is, that it has a complete set (discrete or continuous) of eigenfunctions. Further, it was shown that eigen-