Semiclassical soliton-type solutions of the Hartree equation
The authors study the Hartree equation: \[ -ih\delta_t+\widehat H_k \Psi=0,~\|\Psi\|=1,~\widehat H_k=\widehat H_0+k \int_{\mathbb{R}^n}| \Psi(y,t)|^2V(x-y)\,dy,~ \widehat H_0=\widehat p^2/2m +U(x),~\widehat p=-ih\nabla. \] (I) Linear case \(k=0\). \(\Psi|_{t=0}= \Psi_0 (x,h)=N_0\exp[(i/h)(p_0,x-x_0)]\times\exp[(i/h)(x-x_0,B_0(x-x_0))]\). Hamiltonian system: \(\dot p=-\nabla_U(x)\), \(\dot x=p\). Phase trajectory of the Hamiltonian system: \(p=P(x_0,p_0,t)=p_{c1}(t)\), \(x=x_{c1}(t)\). An asymptotic solution \(\Psi(x,t,h)\) is given by using the WKB-Maslov complex method. \[ \Psi(x,t,h)=\operatorname{Nex} p[(i/h) \{(S_{c1}(t)+(p_{c1} (t),~ x-x_{c1}(t))+ (x-x_{c1}(t),~B(t)C^{-1} (t)(x-x_{c1}(t)))\}] \times(\text{det}\,C(t))^{-1/2}. \] (II) Case \(k\neq 0\). Let \[ P(t,h)=\langle \Psi,-ih\nabla_x\Psi\rangle,~X(t,h)= \langle\Psi,x \Psi\rangle,~Z(t,h)= (P(t,h),X(t,h)), \] and \(\Delta z=z-Z(t,h)\). \[ K=\{\Phi;\Phi=\varphi((x-X(t, h))/h^{1/2},t)\times\exp[(i/h)(S(t,h)+(P(t,h),x-X(t,h)))\},~\varphi (\xi,t)\in{\mathfrak S}(\mathbb{R}^n) \] w.r.t. \(\xi\). \(\Delta_\alpha (t,h)= \langle \Phi,\{\widehat\Delta Z\}^\alpha\Phi\rangle/\|\Phi\|^2\), \(\Phi\in {\mathfrak K}\). Hamilton-Ehrenfest system is given by using quantum averages: \[ \dot p=-\nabla_xU(x)-k\nabla_\tau V(0)-\nabla_x[(1/2)Sp (U_{xx}^{\prime \prime}(x)+ 2kV_{xx}^{\prime\prime}(x-y)) \sigma_{xx}]|_{y=x,} \] \[ \dot x=p,~\dot\Delta=J M_k(x)\Delta-\Delta M_k(x)J;~ U_{xx}^{\prime\prime}(x)=(\partial^2U(x)/\partial x_i \partial x_j)_{n \times n}, \] and \(\Delta\) is a \(2n\times 2n\) real block symmetric matrix. The Cauchy problem of a variational system: \[ (d/dt)^t (B,C)= JM_k(X_0(t)) \cdot^t(B,C);~X(t,h)=X_0(t)+hX_t(t)+0(h^2), \] and the form of the asymptotic solution \(\Psi_v(x,t,h)\) are shown. (III) Assuming \(U(x)=0\), the authors solve the problem for the Hartree equation with more general Cauchy data than those in (I) under the condition \(kV_{\tau \tau}^{\prime\prime}(0)>0\) etc.
scientific article; zbMATH DE number 5375909