The Trotter product formula for perturbations of semibounded operators (original) (raw)

A general Trotter–Kato formula for a class of evolution operators

Journal of Functional Analysis, 2009

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t; s) 0 s t T generated by the sum (A(t) + B(t)) of two linear, time-dependent and generally unbounded operators de…ned on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t; s) 0 s t T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by A(t) and B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a …xed set D \ t2[0;T ] D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in e¤ect, allows us to reconstruct UA(t; s) 0 s t T very simply in terms of the semigroup generated by A(t). We then illustrate our results by considering various examples of non-autonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We …nally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.

Results of a perturbation theory generating a one-parameter semigroup

Open journal of mathematical analysis, 2022

This paper consists of the results about ω-order preserving partial contraction mapping using perturbation theory to generate a one-parameter semigroup. We show that adding a bounded linear operator B to an infinitesimal generator A of a semigroup of the linear operator does not destroy A's property. Furthermore, A is the generator of a one-parameter semigroup, and B is a small perturbation so that A + B is also the generator of a one-parameter semigroup.

Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

International Journal of Partial Differential Equations, 2013

Let be a generator of an exponentially stable operator semigroup in a Banach space, and let be a linear bounded variable operator. Assuming that is sufficiently small in a certain sense for the equation , we derive exponential stability conditions. Besides, we do not require that for each , the “frozen” autonomous equation is stable. In particular, we consider evolution equations with periodic operator coefficients. These results are applied to partial differential equations.

Asymptotic stability of Schr�dinger semigroups: path integral methods

Mathematische Annalen, 1992

Let H =-89 + V be a Schr6dinger operator on R N, where A is the Laplacian and V is a suitable potential. One considers H as an operator on each /f(RN)(1 <p<oo),-H being the infinitesimal generator of the Schr6dinger semigroup, S(t) = e-tn, on each space. Much work has been done on the behaviour of the semigroups and the spectral properties of H-see [8,9,21], and the references cited therein. In addition, Hempel and Voigt [10] showed that the spectrum a(H) of H on LP(R ~) is independent ofp. Earlier, Simon [19] had shown that the type (or growth bound) of the semigroup is independent of p, or equivalently that infa(H) is independent of p. In this paper, we consider non-negative potentials V > 0, so that a(H)c= [0, oo), IlS(t)llLp< 1 (1 <p< o0), and IIS(t)fllp~O (t-~c~3, feLP(RN), 1 <p< ~). The questions which we address are: For which V does: (a) IlS(t)flll-oO (t-,~, feLI(RN)); (b) ItS(t)llL,~0 (t~)? Property (a) [strong stability on LI(RN)] is equivalent to 0 not being an eigenvalue of H on L ~176 (RN), while property (b) (exponential stability) is equivalent to 0 not being in the spectrum of H, and also to strong stability on L| Property (a) always occurs if N = 1 or 2 (unless V = 0). Otherwise, the necessary and sufficient conditions which answer the questions are: V(x) .

Semistable Operators and Singularly Perturbed Differential Equations

Journal of Mathematical Analysis and Applications, 1999

We investigate conditions for the existence of the limit exp(sA + B) as s → ∞ where A, B are bounded linear operators in a Banach space X. We extend a result for matrices due to Campbell which states that the above limit exists for any B if A is semistable. The result is applied to the study of the solutions to a parameter dependent differential equation in a Banach space and their convergence with respect to the parameter.

Short-time asymptotics for the trace of one- and multi-dimensional Schrödinger semigroups

Proceedings of the American Mathematical Society, 1989

Using Brownian motion we derive the leading asymptotic behaviour, as / 1 0 , of the (normalized) trace of e'L-e'L , where L is the operator A/2 + q(x) on Rd (with zero boundary condition at infinity), H is a hyperplane of Rd and LH is the direct sum of A/2 + q(x) acting on H+ , with Dirichlet boundary condition on H (and 0 at infinity), and the same operator acting on H~(H+ and H~ are the two half-spaces defined by H). The function q is taken bounded and continuous on Rd and, if d > 2 , we also assume that q is integrable on Rd (in fact we need a little less than that). We also show how to get higher order terms in our expansion, but in this case q is required to be smoother. In the one-dimensional case our result extends a result of Deift and Trubowitz (see the [D-T, Appendix]), since they proved a similar formula under the additional assumption that q(x)-> 0 as |.v|-> oo. The asympotic formula we give implies that q can be recovered from certain spectral properties of L and LH .