Friedrichs Extension and Min–Max Principle for Operators with a Gap

Abstract

Semibounded symmetric operators have a distinguished self-adjoint extension, the Friedrichs extension. The eigenvalues of the Friedrichs extension are given by a variational principle that involves only the domain of the symmetric operator. Although Dirac operators describing relativistic particles are not semibounded, the Dirac operator with Coulomb potential is known to have a distinguished extension. Similarly, for Dirac-type operators on manifolds with a boundary a distinguished self-adjoint extension is characterised by the Atiyah–Patodi–Singer boundary condition. In this paper, we relate these extensions to a generalisation of the Friedrichs extension to the setting of operators satisfying a gap condition. In addition, we prove, in the general setting, that the eigenvalues of this extension are also given by a variational principle that involves only the domain of the symmetric operator. We also clarify what we believe to be inaccuracies in the existing literature.

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Correspondence to Lukas Schimmer.

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The authors were supported by ERC Advanced Grant No. 321029 and VILLUM FONDEN through the QMATH Centre of Excellence (Grant No. 10059).

Communicated by Jan Derezinski.

Appendix A: Essential Self-adjointness of the Brown–Ravenhall Operator

Appendix A: Essential Self-adjointness of the Brown–Ravenhall Operator

Let \(H_0\) be the self-adjoint free Dirac operator with domain \(H^1({\mathbb {R}}^3;{\mathbb {C}}^4)\subset L^2({\mathbb {R}}^3;{\mathbb {C}}^4)\) and denote by \(\Lambda _\pm \) the projections onto the positive/negative spectral subspace of \(H_0\). For \(\gamma \in {\mathbb {R}}\), the Brown–Ravenhall operator [5] is defined as

$$\begin{aligned} B_\gamma =\Lambda _+(H_0-\gamma /|x|)\Lambda _+ =\Lambda _+(\sqrt{1-\Delta }-\gamma /|x|)\Lambda _+ \end{aligned}$$

on the Hilbert space \(\Lambda _+L^2({\mathbb {R}}^3;{\mathbb {C}}^4)\). For a comprehensive review, we refer to the textbook of Balinsky and Evans [2]. While the physically relevant case is \(\gamma >0\), we are interested in the case where \(\gamma =-\nu \in [-1,0]\). For \(\gamma <3/4\), the operator \(B_\gamma \) was proved to be self-adjoint on \(\Lambda _+H^1({\mathbb {R}}^3;{\mathbb {C}}^4)\) by Tix [32]. Since \(\Lambda _+H^1({\mathbb {R}}^3;{\mathbb {C}}^4)\subset H^1({\mathbb {R}}^3;{\mathbb {C}}^4)\), we obtain from Hardy’s inequality

$$\begin{aligned} {\left\| \Lambda _+\frac{\gamma }{|x|}\Lambda _+\psi \right\| }_{L^2({\mathbb {R}}^3;{\mathbb {C}}^4)} \le 2\gamma {\left\| \frac{1}{2|x|}\Lambda _+\psi \right\| }_{L^2({\mathbb {R}}^3;{\mathbb {C}}^4)} \le 2\gamma {\left\| \nabla \Lambda _+\psi \right\| }_{L^2({\mathbb {R}}^3;{\mathbb {C}}^4)} \end{aligned}$$

for any \(\psi \in H^1({\mathbb {R}}^3;{\mathbb {C}}^4)\). To prove that \(B_{-\nu }\) is essentially self-adjoint on \(\Lambda _+{\mathcal {C}}_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\), it thus suffices to prove the statement for \(B_{0}\). This is an immediate consequence of the fact that the free Dirac operator \(H_0\) is essentially self-adjoint on \({\mathcal {C}}_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\).

We can also conclude that the operator \(\Lambda _-(\sqrt{1-\Delta }+\nu /|x|)\Lambda _-\) is essentially self-adjoint on \(\Lambda _-{\mathcal {C}}_0^\infty ({\mathbb {R}}^3;{\mathbb {C}}^4)\) since it is unitarily equivalent to the Brown–Ravenhall operator \(B_{-\nu }\) via the transform \(U\!\!:\!L^2({\mathbb {R}}^3;{\mathbb {C}}^4)\!\rightarrow \! L^2({\mathbb {R}}^3;{\mathbb {C}}^4)\)

$$\begin{aligned} \left[ U{\psi _1\atopwithdelims ()\psi _2}\right] (x)={\psi _2(-x)\atopwithdelims ()\psi _1(-x)}. \end{aligned}$$

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Schimmer, L., Solovej, J.P. & Tokus, S. Friedrichs Extension and Min–Max Principle for Operators with a Gap. Ann. Henri Poincaré 21, 327–357 (2020). https://doi.org/10.1007/s00023-019-00855-7

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Mathematics Subject Classification

  • 49R05
  • 49S05
  • 47B25
  • 81Q10