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Finite Dimensional Space
Perturbation of the Discrete Spectrum
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91 is essentially analytic functions boundary conditions bounded inverse bounded operator closeable cluster points completes the proof complex number continuous first derivatives continuous functions convergent for small convergent power series corresponding eigenfunction Criterion dense domain differential equation eigen eigenelement eigenfunctions eigenvalue A(e eigenvalue equation eigenvalue problem eigenvectors essentially self-adjoint example exists a sequence finite multiplicity follows Friedrichs extension Hence Hermitian extension Hermitian form Hermitian matrix Hermitian operator Hilbert space independent inequality inner product lemma Let us denote linear operator multiplicity h norm u\\A obtain Obviously operator A(e operator defined orthonormal set permutation perturbation parameter perturbed eigenvalue piecewise continuous point eigenvalue polynomial positive number power series convergent prime factors prove real neighborhood real number regular analytic functions Rellich satisfy self-adjoint operator solution spectrum of A(e subspace 91 Suppose that A(e Theorem tinuous tion trivial extension unperturbed operator Zeeman effect zero