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The description of dissipative processes in quantum mechanics here presented is based on the existence of a time operator canonically conjugate to the Liouville operator. The entropy can be defined as a functional of this time operator and a new dissipative semi-group of time evolution can be constructed. This formalism seems well adapted to the description of the decay phenomena and the measurement process.

In this paper, we summarize recent results on dissipative gradient systems and give applications to some hyperbolic equations.

Some questions about abstract methods for initial value problems lead us to a study of the equation (*) u′(t)=(A+B)u(t), where A is m-dissipative and B is compact. Does a solution to (*) necessarily exist? Earlier studies of this question, reviewed and then continued here, depend on an analysis of the related quasiautonomous equation (**) u′(t)=Au(t)+f(t). We say A has Gihman's property if the mapping f ↦ u is continuous from L w 1 ([0,T],K) into C([0,T];X) for every compact K⊂X; this condition is closely related to the Lie-Trotter-Kato product formula. If A has this property, then (*) is known to have a solution. In this paper, we consider linear, m-dissipative operators A which lack Gihman's property. We obtain partial results regarding the existence of solutions of (*); but in general, the existence question remains open. Our method applies the variation of parameters formula to (**), but this requires a weakened topology when Range (f) \(\nsubseteq \overline {D(A)}\). Two examples are studied: one in l∞, the other in the space of bounded continuous functions.

Let X be a Banaeh space with norm II If, A: D(A) ÷ X a closed linear map with dense domain which generates a C O semi-group on X and let R+ be the interval [0, ~). Let B(t) be a linear map defined at least on the domain of A with B(t)x a bounded uniformly continuous X valued function for each x E D(A). Also, we assume B(-)x(s) is continuous in BU when x(s) and Ax(s) are continuous. Let BU denote {f E C(R +, X): f is bounded and uniformly continuous} and make BUa Banach space by endowing it with the norm IIfIl = sup{IIf(t)ii: t ~ 0}. The space Z = X × BU will be considered with norm given by II(x, f)ll 2 = iIxlI 2 + llfIl 2. On z we will be particularly interested in the operators 01 E = , D s-~l

A given (but otherwise random) environmental time series impinging on the input of a certain biological processor passes through with over-whelming probability undetected. A very small percentage of environmental stimuli, however, are “captured” by the processor’s non-linear dissipative operator, as initial conditions, that is as solutions of the processor’s dissipative dynamics. The processor in such cases is instrumental in compressing or abstracting those stimuli, thereby making the external world collapse from a previous regime of a “pure state” of suspended animation on to a set of stable eigenfunctions or “categories” — chaotic strange attractors. The charateristics of this cognitive set depend on the operator involved and the hierarchical level where the abstraction takes place. In this paper we model the physics of such a cognitive process and the role that the thalamocortical pacemaker of the (human) brain plays in both stimulating the individual attractors and permutating them on a time division multiplexing basis. A synthesis of the Markovian processes taking place within each individual attractor-memory and the Markovian or Semi-Markovian process involving the intermittent jumping among the different attractors-memories is discussed.

An information-theoretic approach is used to give a straightforward procedure that allows one to determine whether dissipative behavior can be attained for a given Hamiltonian. This technique stresses the importance of an adequate set of initial conditions in order to obtain dissipative evolution. It is shown that, although dissipative solutions may be compatible with the set of dynamical equations, they may not be realized just because it is impossible to construct the adequate set of initial conditions that would result in dissipative behavior. This is illustrated by reference to the well-known Bateman generalized problem of two interacting harmonic oscillators. We demonstrate the impossibility of obtaining dissipative evolution in linear, time-independent Hamiltonian systems generated with any combination of bilinear products of creation and annihilation operators.

Let H be a Hilbert space. The contractions T, S in H with closed, nondense domains D (T), D (S) are said to form a dual pair (denoted by {T,S}), if the relation $$(Tx,y)=(x,Sy):\ (x\varepsilon \mathit{D}(T)),:\ y\varepsilon \mathit{D}(S))$$ holds. It is well-known that for each dual pair {T,S} of contractions in H there exists a contraction T~ in H, D (T~)=H, such that $$T\subset \check{T},:\ S\subset \check{T}^{*}$$ (1) .

It is well-known that the semigroup S on D(A) generated by a nonlinear (possible multivalued) operator A (via (2.8) or (2.13) may not be differentiable (see Crandall and Liggett [4]). However, in continuous case this semigroup is differentiable everywhere on [0,+~) (Theorem 2.3 here, of R.H. Martin Jr.). The most general condition under which an operator of dissipative type generates a semigroup is the "tangential condition" (2.9) introduced by Kobayashi [8] and extended by the author (see (2.19)) to non-autonomous case. This condition is strictly more general than the standard "range condition" (2.10). In applications to PDE the range condition is most useful. However, the tangential condition is still important in PDE, since e.g. in continuous case it is equivalent to Nagumo condition (2.14) (which can be applied in PDE). Moreover, this condition (2.9) has a unifying effect in the theory of non-linear semigroups. Indeed, first of all from (2.9) (Theorem 2.1) we can easily derive the exponential formula of Crandall and Liggett ((2.13) in Theorem 2.2 her). This is a known fact. Another important fact is that Theorem 2.3 (i.e. the differentiability of S in continuous case) can also be easily derived from Theorem 2.1. This is important since the original proof of Theorem 2.3 is more difficult than the proof of Theorem 2.1 itself. Such aspects are presented in Section 2. Finally we note that condition (2.9) can also be used in the theory of m-dissipativity of the sum of two mdissipative operators [8, 13].

We give an exponential formula for the representation of any propagator on a Banach space, U: (s,t) ↦ U(t,s), “pregenerated” in a certain sense by a time-dependent family of dissipative operators {A(t)} with resolvents depending continuously on t. This formula is a direct generalization of the exponential Hille-Yosida representation \(P(t) = \mathop {\lim }\limits_\infty {e^{tA\lambda }}\) of a semigroup (P(t)) with generator A. In the time-dependent case we still have an analogous limit as λ → ∞ of an exponential expression, where that exponential expression is again computed by a power series.

(a) In this chapter we shall study the wave equation with dissipative term: (1.1) where, as usual, t ∈ J, ζ={ζ 1, · · ·, ζm} ∈ Ω, open, bounded, and connected set of Rm.

[1] K. Yosida, Functional Analysis, Springer-Verlag, Berlin (1965).[2] G. Lumer and R.S. Phillips, "Dissipative operators in a Banach space," Pacific J. Math,11, (1961) 679-698.[3] W. Feller, "On the generation of unbounded semi-groups of bounded Linear Operators,"Ann. of Math, 58, (1953) 166-174.

Krein spaces, the most important type of inner product spaces, can roughly be characterized as non-degenerate, decomposable, complete spaces. Sections 1–2 deal with their global properties. In Sections 4, 7, 9, 10 maximal semi-definite subspaces of Krein spaces are investigated, while Sections 3, 5, 6, 6 are essentially concerned with ortho-complemented subspaces. Theorems 1.3, 3.4, 3.5, 4.2, 5.3, 5.7, 6.3, 7.1 and 9.1 give an idea of the contents of the chapter.

Let H be a Hubert space, with real or complex scalars. A function F: H-+H is called monotonie provided, for any xu a^G-H", we have Re(#i—#2, Fxi — Fx2)^0. If ( ^ ) is replaced by ( > ) , it is strictly monotonie, and if 0 is replaced by c||xi--#2||, with c>0, it is strongly monotonie. Examples are: the gradient of a convex (resp. strictly or strongly convex) function, the negative of a linear dissipative operator, a linear operator satisfying Re(x, i

In this article we consider a second-order Sturm-Liouville operator with a spectral parameter in the boundary condition on bounded time scales. We construct a selfadjoint dilation of the dissipative Sturm-Liouville operators. Using by methods of Pavlov [40,41,42], we prove the completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales.

The physical properties of a weakly dissipative system in a steady state are approximately those of a certain macroscopic state of a conservative system.

It is shown that Hashimoto's recent construction of a nonlinear wave equation on the basis of stochastic mechanics for a dissipative system submitted to a general velocity-dependent frictional force is not valid, except for the trivial one-dimensional case.

The work of E. Kanai shows that the quantization of dissipative systems by use of a Hamiltonian formalism involving the time explicitly leads to results in disagreement with experience. We discuss the quantization of dissipative systems in a manner such that the time does not enter explicitly. It is shown that in the Heisenberg representation the only kind of dissipative forces consistent with the formalism are those which are functions of the coordinates alone. There exists no Schr\"odinger representation unless the forces are non-dissipative.

In this paper we study the existence, uniqueness and differentiability of solutions (in a certain generalized sense) for the nonlinear parabolic equation \[ u t = u x x − F ( u , u x ) ( 0 > x > 1 , t > 0 ) , {u_t} = {u_{xx}} - F(u,{u_x})(0 > x > 1,t > 0), \] under the maximal monotone boundary conditions: \[ ( − 1 ) i u x ( t , i ) ∈ β i ( u ( t , i ) ) , t > 0 , i = 0 , 1. {( - 1)^i}{u_x}(t,i) \in {\beta _i}(u(t,i)),t > 0,i = 0,1. \]

Except in one dimension, strictly incoming waves cannot be propagated by the wave equation with dissipative boundary conditions so that they disappear asymptotically in forward time. In [4] Lax and Phillips consider the acoustic equation in an exterior domain G c Rn )ut = Au in G, ( anu + ou, = 0, a > 0 in aG. They assume G contains the complement of the ball of radius p. As in [4], we define H to be the Hilbert space of all initial data d with finite energy in G. Let T(t) be the (strongly continuous) contraction semigroup formed by mapping initial data into data at time t. If G = Rn (and the second part of (1.1) is vacuous) we will denote H by Ho and T(t) by Uj(t). We note that U0(t) is a unitary group. We denote the cogenerator (see Chapter 3 of [5]) of T(t) by T and the cogenerator of U0(t) by U0. Let D+ c H be the set of all initial data vanishing on {xl lxi 01. We will prove the following THEOREM. Let n be greater than 1. (Recall that G c Rn.) If d E D and d m 0. Then limt,+oT(t)d =# 0. Before starting the proof we recall some of the material in [2], [3], and [4]. We represent the action of U0(t) on Ho as right translation on L2(R, N) (i.e., the set of all square integrable N-valued functions on R) for some auxiliary Hilbert space N so that D is mapped onto L2(R -p, N). In this representation D+ is mapped onto L2(R+ + p,N)ifn isodd and 9CL2(R+ + p, N) if n is even where Received by the editors December 5, 1975 and, in revised form, February 2, 1976. AMS (MOS) subject classifications (1970). Primary 35P25; Secondary 47A40.

Using the ideas of Livsic on closed systems (with no input), a set of state equations with special properties for a class of dissipative time-invariant systems is derived.

In this note we shall prove that a closed dissipative operator A with dense domain in a hilbert space H generates a contraction semigroup if the set A kx; k = 0, 1, 2, , x is quasi-analytic}

Necessary conditions on dissipative operators for the existence of shock‐like solutions of the Korteweg‐de Vries equation, which limit the damping tolerable at long wavelengths are derived.