We study lower bounds of decay (or of growth) order in time for solutions to the Cauchy problem for the Schrödinger equation:
i∂_tu=−Δu+f(u), (t, x)∈R×Rⁿ   (n≥1),
u(0)=φ,   x∈Rⁿ,
where f is a linear or non-linear complex-valued function.
Under some conditions on f and φ, it is shown that every nontrivial solution u has the estimate
lim inf\limits_f→±∞ | t | ^n/2−n/q || u(t) || L^q(| x | < k | t |) > 0
for sufficiently large k>0 and for any q∈[2, ∞].
In the previous work [12] of the first named author, we imposed on the assumption that u is asymptotically free. In this article, however, we shall show the assumption is, in fact, irrelevant to the results.

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We study Hermitian representations of algebras A with involution, a→a^*. A representation π of A on a vector space D is said to be Hermitian (or to be a positive-energy-representation) if D carries a positive definite inner product (·, ·) such that
(π(a)u, v)=(u, π(a^*)v),   a∈A, u, v∈D.
Such representations arise in quantum field theory and in the study of unitary representations of Lie groups. They were introduced (in this general context) by Powers. We show that main features of von Neumann's index theory (for single Hermitian operators) carry over to representations. Moreover, we get explicit index-information directly from the representation theory, and this is applied to the study of representations of semisimple Lie algebras \mathfrak{g} (with Cartan decomposition \mathfrak{g}=\mathfrak{k}+\mathfrak{p}).
For a certain class (to be specified below) of positive energy representations π, we show that the index may be computed from the restriction of π to the compact subalgebra \mathfrak{k}. Our results are then applied to the integrability problem for representations of semi-simple Lie algebras.
Several classes of examples are included.

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A convolution algebra is a normal sequence space λ that is closed under the convolution product. Let E be a nuclear Fréchet locally convex space, and let p={p_r : r∈N∪{0}} be a family of Hilbertian seminorms defining its topology. We define the vector space T(E, λ, P)={(x_n)∈Π\limits_n{≥}0\hat{⊗}ⁿE: (⊗_σⁿp_r(x_n))∈λ, r{≥}0}, equipped with the topology obtained from the seminorms Σ\limits_n{≥}0 | u_n | ⊗_σⁿp_r where r{≥}0 and u∈λ^×, the Köthe dual of λ. For certain sequence spaces, the resulting space does not depend on p and we write T(E, λ). Such is the case for sequence spaces of type h.
Certain properties of λ transfer directly to T(E, λ, p). In particular, if λ is complete (respectively Fréchet, a topological algebra), then so is T(E, λ, p). A regular tensor algebra is a space T(E, λ) for λ a perfect topological convolution algebra with jointly continuous product which is of type h. T(E, λ) is then nuclear, and reflexive if Fréchet.
We examine the topological properties of the spaces T(E, λ, p). Other than invertibility, these are the same as for E_⊗=\overline{⊕\limits_n{≥}0}⊗ⁿE. We then consider the order properties arising from a natural involution. The positive wedge K(λ) is always a proper strict-b cone, and if λ is of type h and λ⊆h, its closure is proper. Here h is the sequence space isomorphic to the space H(C) of entire functions. In particular, T(E, h) is a regular tensor algebra which is locally multiplicatively convex and whose closed cone is proper. Finally, we present three conditions which are sufficient for K(λ) to be normal.

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Let λ be a perfect sequence space closed under the Cauchy convolution product, with respect to which it is a topological algebra with jointly continuous product. Suppose, in addition, that λ is type (h) and λ⊂h. Let E[t] be a nuclear Fréchet space, with the topology t defined by the seminorms {p_r⁽¹⁾: r≥0}. The corresponding regular tensor algebra is
T(E, λ)={x∈Π_n≥0\hat{⊗}ⁿ E: [p_r⁽ⁿ⁾(x_n)]∈λ, r≥0},
equipped with the topology t(E, λ^×) determined by the seminorms
p_{u, r}(x)=Σ_n≥0 | u_n | p_r⁽ⁿ⁾(x_n),   (u∈λ^×).
The algebra product is
(xy)_n=Σ_0≤k≤n x_n−k⊗y_k.
If S_n is the symmetrizing operator on \hat{⊗}ⁿE for all n≥0, we consider the symmetric sub-algebra S[T(E, λ)], where S(x_n)=(S_nx_n).
We show that functionals on T(E, λ) and S[T(E, λ)] can be represented by complete finite complex Borel measures if they are continuous with respect to a certain locally convex topology \hat{t}(E, λ^×) on T(E, λ) and its restriction \hat{t}_S(E, λ^×) to S[T(E, λ^×)] respectively. The topology \hat{t}(E, λ^×) is coarser than t(E, λ^×) and non-Hausdorff, whereas \hat{t}_S(E, λ^×) is Hausdorff.
If the algebraic positive cone in λ is normal, then so is the positive cone on S[T(E, λ)]. Using the isomorphism λ≅T(C, λ), we show that the cone in S[T(E, λ)] is normal for all nuclear Fréchet spaces E[t] if and only if t(C, λ^×)=\hat{t}(C, λ^×).

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