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  "body": "\n## Statement\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]].\nLet $A$ be a bounded self-adjoint operator on $H$, that is, $A = A^*$ with $A^*$ the [[adjoint|6HSJBJ8TTQ]] of $A$.\nLet $K$ be a [[compact operator|M2NEP1K5AE]] on $H$ with $K = K^*$.\nThen\n$$\\sigma_{\\mathrm{ess}}(A + K) = \\sigma_{\\mathrm{ess}}(A),$$\nwhere $\\sigma_{\\mathrm{ess}}$ denotes the [[essential spectrum|FRZG0E15SK]].\n\n## Remarks\n\nFor a self-adjoint operator, $\\lambda$ lies in the essential spectrum if and only if there is a Weyl sequence: an orthonormal sequence $(x_n)$ in $H$ with $\\norm{(A - \\lambda) x_n} \\to 0$.\nAn orthonormal sequence tends to $0$ weakly, and a compact operator maps weakly null sequences to norm null sequences, so $\\norm{K x_n} \\to 0$.\nHence a Weyl sequence for $A$ at $\\lambda$ is also one for $A + K$, and the symmetric argument gives the reverse inclusion.\nThe theorem also holds for unbounded self-adjoint $A$, with $A + K$ defined on the domain of $A$.\nIn other words, a compact perturbation of a self-adjoint operator can only change its [[discrete spectrum|PJWY3JJQQ7]], the isolated eigenvalues of finite multiplicity.\nWith the definition of the essential spectrum through Fredholm operators, the equality holds for every bounded operator $A$ on a Banach space and every compact $K$.\nIndeed, $A + K - \\lambda I$ is Fredholm exactly when $A - \\lambda I$ is.\nExample: on $\\ell^2$, the zero operator has essential spectrum $\\set{0}$, and so does every compact self-adjoint operator, such as the diagonal operator with entries $1/n$.\nWithout compactness the conclusion fails: a self-adjoint projection $P$ of infinite rank and infinite corank has $\\sigma_{\\mathrm{ess}}(0 + P) = \\set{0, 1}$, while $\\sigma_{\\mathrm{ess}}(0) = \\set{0}$.\n\n## Explanations\n\n### L1 human\n\nThe essential spectrum is the robust part of the spectrum of an operator, the part that does not come from finitely many isolated eigenvalues.\nWeyl's theorem says that adding a compact operator, a small perturbation in a precise sense, never changes this robust part.\nOnly the isolated eigenvalues of finite multiplicity can move, appear or disappear.\n\n### L2 human\n\nThe theorem is a basic tool of mathematical physics.\nFor a Schrödinger operator $-\\Delta + V$ with a potential $V$ that decays at infinity, one shows that $V$ is a relatively compact perturbation of $-\\Delta$.\nA version of Weyl's theorem for relatively compact perturbations then gives $\\sigma_{\\mathrm{ess}}(-\\Delta + V) = [0, \\infty)$, so the negative spectrum consists of isolated eigenvalues, the bound states.\n",
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