Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs
Zhiwu Lin, Chongchun Zeng
Abstract
Consider a general linear Hamiltonian system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u equals upper J upper L u"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _{t}u=JLu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We assume that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L colon upper X right-arrow upper X Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mtext> </mml:mtext> <mml:mi>L</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\ L:X\rightarrow X^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induces a bounded and symmetric bi-linear form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper L dot comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>L</mml:mi> <mml:mo> ⋅ </mml:mo> <mml:mo>,</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \langle L\cdot ,\cdot \right \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which has only finitely many negative dimensions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n Superscript minus Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n^{-}(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . There is no restriction on the anti-self-dual operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J colon upper X Superscript asterisk Baseline superset-of upper D left-parenthesis upper J right-parenthesis right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> <mml:mo> ⊃ </mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">J:X^{\ast }\supset D(J)\rightarrow X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We first obtain a structural decomposition of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into the direct sum of several closed subspaces so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is blockwise diagonalized and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J upper L"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">JL</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript t upper J upper L"> <mml:semantics> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi>J</mml:mi> <mml:mi>L</mml:mi>