On Eigenfunction Expansion of Solutions to the Hamilton Equations

Abstract

We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein’s spectral theory of J-selfadjoint operators in the Hilbert spaces with indefinite metric. Our main result is an application to the eigenfunction expansion for the linearized relativistic Ginzburg–Landau equation.

Appendices

Appendix A: Spectral Condition for |v|<1

Let us check the spectral condition (1.6) for operators B _v from (3.8) with any |v|<1 in the space of the odd states. First let us check the continuous spectrum.

Lemma A.1

The continuous spectrum of B _v lies in [δ,∞) with some δ>0.

Proof

By Corollary 2(c) of [45, XIII.4] and (3.7) it suffices to find the continuous spectrum of the unperturbed operator $B_{v}^{0}$ corresponding to V _v (x)=0. Consider the spectral equation

$$ \bigl(B_v^0-\lambda\bigr)\psi=0 $$

(A.1)

and find the solution of type ψ=e ^ikx ϕ with real k and $\phi\in\mathbb{C}^{2}$. Substituting to (A.1) we obtain

$$\left ( \begin{array}{c@{\quad}c} k^2+m^2-\lambda & ikv \\ -ikv & 1-\lambda \end{array} \right )\phi=0. $$

For nonzero vectors ϕ, the determinant of the matrix vanishes:

$$ k^2\bigl(1-\lambda-v^2\bigr)+ \bigl(m^2-\lambda\bigr) (1-\lambda)=0. $$

(A.2)

Then k ²=(m ²−λ)(1−λ)/(λ−1+v ²)≥0. This inequality holds if

$$\left \{ \begin{array}{l@{\quad}l} \lambda\in[1-v^2,1]\cup[m^2,\infty) & \mbox{for } 1\le m^2,\\ \lambda\in[m^2,1-v^2)\cup(1,\infty) & \mbox{for } m^2\le 1-v^2\le1,\\ \lambda\in[1-v^2,m^2]\cup[1,\infty) & \mbox{for } 1-v^2\le m^2\le1. \end{array} \right . $$

Now the lemma follows with δ=min(1−v ²,m ²). □

Now let us consider the discrete spectrum.

Lemma A.2

• (i) $\operatorname{Ker}B$ is generated by $(s_{0}'(x),-vs_{0}''(x))$, where $s_{0}'(x)$ is an even function.

• (ii) The nonzero discrete spectrum of B is positive.

Proof

• (i) Equation Bψ=0 is equivalent to the system

  $$ \left ( \begin{array}{c@{\quad}c} S_v & -v\frac{d}{dx} \\ v\frac{d}{dx} \ & 1 \end{array} \right )\left ( \begin{array}{c} \psi_1 \\ \psi_2 \end{array} \right )=0. $$

  (A.3)

  The second equation (A.7) implies $\psi_{2}=-v \psi_{1}'$. Substituting into the first equation we obtain

  $$ \tilde{S}_v \psi_1=0, $$

  (A.4)

  where $\tilde{S}_{v}$ is defined in (3.9). However, (3.9) with v=0 means that $0\in\sigma_{pp}(\tilde{S}_{v})$. Moreover, λ=0 is the minimal eigenvalue of $\tilde{S}_{v}$, since the corresponding eigenfunction $s_{v}'(x)$ does not vanish [31, (1.9)]. Hence,

  $$ \sigma(\tilde{S}_v)\subset[0,\infty). $$

  (A.5)

  Moreover,

  $$ \operatorname{Ker}\tilde{S}_v=\bigl(s_0'(x) \bigr), $$

  (A.6)

  since any second linearly independent solution of the homogeneous equation cannot belong to $L^{2}(\mathbb{R})$ by Theorem X.8 of [43].

• (ii) Consider equation B _v ψ=λψ with λ<0:

  $$ \left ( \begin{array}{c@{\quad}c} S_v-\lambda& -v\frac{d}{dx} \\ v\frac{d}{dx} \ & 1-\lambda \end{array} \right )\left ( \begin{array}{c} \psi_1 \\ \psi_2 \end{array} \right )=0. $$

  (A.7)

  The second equation (A.7) implies $\psi_{2}=v \psi_{1}'/ (\lambda-1)$. Substituting into the first equation we obtain

  $$ \biggl(\tilde{S}_v+\frac{v^2\lambda}{1-\lambda}\frac{d^2}{dx^2}- \lambda \biggr)\psi_1=0. $$

  (A.8)

  For λ<0 the operator is positive since $\tilde{S}_{v}\ge0$ by (A.5). Hence, (A.8) has no nonzero solutions ψ ₁∈L ².  □

Corollary A.3

Lemmas 3.1 and A.2 imply that condition (1.6) holds for |v|<1 in the space of the odd states.

Appendix B: Linearization of U(1)-Invariant Hamilton PDEs

Equations (1.2) with JB≠BJ arise in the linearization of nonlinear U(1)-invariant Hamilton PDEs. Namely, consider the U(1)-invariant Hamilton functional

(B.1)

with a real potential U(x,r) and $\psi(x)\in\mathbb{C}=\mathbb{R}^{2}$. The corresponding Hamilton equation reads as the nonlinear Schrödinger equation

(B.2)

where i can be regarded as a real 2×2 matrix J of type (3.8). The linearization at a stationary state s ₀(x) is obtained by substitution ψ=s ₀+φ and expansion |ψ|²=|s ₀|²+2s ₀⋅φ+|φ|², where s ₀⋅φ is the scalar product of the real vectors from $\mathbb{R}^{2}$. Neglecting the terms of higher order, we obtain the linearized equation

$$ i\dot{\varphi}(x,t)=-\Delta\varphi(x,t)+ U_r \bigl(x,\bigl|s_0(x)\bigr|^2\bigr)\varphi+ 2U_{rr} \bigl(x,\bigl|s_0(x)\bigr|^2\bigr) (s_0\cdot\varphi) s_0, $$

(B.3)

which can be represented in the form (1.2) with J=−i and $X(t)=(\operatorname{Re}\varphi(t),\operatorname{Im}\varphi(t))$. The last term of (B.3) is not complex linear operator of φ. In other words, it does not commute with the multiplication of φ by i. So JB≠BJ if $U_{rr}(x,|s_{0}(x)|^{2})\not\equiv0$. Let us assume that U(x,r) is a real-analytic function of r>0, and $s_{0}(x)\not\equiv0$. Then the last term of (B.3) vanishes exactly for the linear Schrödinger equation when U(x,r)=V(x)r.