Large Time Behavior Of Solutions To Dissipative Schr(?)dinger Equations

Schrodinger equation is a fundamental equation in quantum mechanics. It was formulated in late1925, and published in1926, by the Austrian physicist Erwin Schrodinger (see [64]). This doctoral thesis is dedicated to the researches on the large time behavior of the semigroups generated by the dissipative Schrodinger operator H=-Δ+V(x) with a complex function V(x)=RV(x)+i(?)V(x), where(?)V(x)≤0is sufficiently small. The Schrodinger operator with complex potentials may describe some non-closed physical system. The quantum scattering for non-self adjoint operators appears in many physical situations such as optical models of nuclear scattering (see [19],[20],[21]). Furthermore we assume that the potential function satisfies that V(x)=O(<x>-ρ0), j=1,2, x∈Rn, n≥3(0.0.2) where ρ0>2is some constant and (x)=(1+|x|2)1/2The Schrodinger equation can be written as follows In [58], it is indicated that there are three general mathematical problems which arise in any quantum mechanical model:(1) self-adjointness;(2) spectral analysis;(3) scat-tering theory. Since V(x) is complex, thus H is non-selfadjoint. So the main content of this thesis is to discuss the last two problems.For a real potential V(x) satisfying some additional conditions such as (0.0.2), V(x) is H0-compact perturbation and H is a selfadjoint operator on L2(Rn). The results are very rich in selfadjoint case for the short-range potential. In this case, the spectrum of H is [0,+∞) U{λ1,..., λk}. Here [0,∞) is the essential spectrum and {λ1,..., λk} is the set of negative eigenvalues of H. These results can be found in many books about functional analysis such as [5,41,58,85]. On the other hand, one also can find the results about the properties of the resolvent, especially low-energy analysis on the unitary e-itH in [1],[6],[7],[8],[9],[17],[18],[23],[25],[27],[31],[32],[33],[37],[35],[46],[49],[51],[75],[76],[81],[84] and the references therein. The main difficulty is to analyze the threshold eigenvalue and resonance.0is called an eigenvalue if there exists a L2function u such that Hu=0and called a resonance if there exists a function u satisfying Hu=0for some u∈L2((x)sdx)\L2(dx), s>1/2. For the scattering operator for the pair (Ho, H), it has been discuss in [1],[2],[3],[4],[6],[7],[8],[9],[12],[16],[28],[45],[46],[50],[53],[52],[56],[58] and the reference therein. There also exist lots of works for the non-selfadjoint Schrodinger such as [11],[13],[14],[15],[19],[26],[36],[38],[40],[43],[47],[60],[59],[66],[67],[68],[77],[78] and the references therein.Here we will consider the dissipative Schrodinger operator H(ε)=-Δ+V1(x)-iεV2(x) for ε>0sufficiently small and real functions Vj, j=1,2satisfying (0.0.2) and V2≥0. The core part of this thesis is the analysis of the spectrum and the asymptotic behavior of the resolvent near the complex eigenvalues of H. In [77], it has been proven that its discrete spectrum σdisc(H) is a perturbation of{λ1,...,λk) if0is neither an eigenvalue nor a resonance of H’s real part H1=-ΔRV(x) and{0, λ1,...,λk} if0is an eigenvalue or a resonance of H1. Thus we can apply the argument of perturbation to obtain the asymptotic expansion of the resolvent near the discrete spectrum and zero. Based on the delicate spectral analysis of H, we can prove our main results of this thesis. They are listed as follows:In Chapter2, we assume that0is a regular point of H1, which means that0is neither an eigenvalue nor a resonance of H1. Then, one can obtain a uniformly global resolvent estimate on R from the upper complex plane. In light of the self-adjoint dilation for the dissipative operator (see [22]) and Kato’s smoothness estimates(see [40]), one can prove the codimension the range of the incoming wave operator is of finite dimension and so it is a closed subspace of L2. Theorem2.1.1comes from the fact that the range of the incoming wave operator([14]) is closed is equivalent to that the scattering operator for the pair (Ho, H) is bijective.In chapter3, we mainly focus on three cases respectively:zero is only an eigenvalue but not a resonance of H1in dimension three(in Section3.3); zero is only a resonance but not an eigenvalue of H1in dimension four(in Section3.5); zero is both a resonance and an eigenvalue of Hi in dimension four(in Section3.6). We will apply the Grushin method to get the expansion of the resolvent near zero, which projects the problem into some finite dimensional space. Thus the resolvent is divided into two parts:the first one is uniformly bounded on ε;the second has singularities on ε but it is of finite rank. Based on the delicate low-energy analysis, the long-time expansion of the semigroup e-itH is obtained respectively in Sections3.4,3.5and3.6. Meanwhile, we show that there exist singularities depending on ε with respect to the expansions of the resolvent of H for low energies and the semigroup e-itH for large time. Noting that resonance can only appear in dimension smaller than, the results are complete except the three-dimensional resonance case. The reason is that the three-dimensional resonance case is different from the other cases essentially and the perturbation method is invalid in this case. Moreover in Section3.3.3, we can derive some properties for the Riesz projections with respect to the discrete spectrum of H(ε) near0and we show that the global resolvent estimate similar to the one in Chapter1holds with bound O(ε1/2) in three-dimensional eigenvalue case. But because we don’t have this estimate for H1(see [31]), the perturbation method applied in Chapter2cannot be applied in this case.