Is the Spin-Boson model renormalisable? (Mathematical aspects of quantum fields and related topics)

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(1)126. Is the Spin-Boson Spin‐Boson model renormalisable? Thomas Norman Dam Aarhus University tnd@math.au.dk. January 11, 2019. 1. Introduction. The purpose of these notes is to give an explanation of the results obtained in [2]. [2]. In that paper, the authors consider the Spin-Boson Spin‐Boson model, which is a very popular model for a qubit coupled to a radiation field. field. It is proven, that ultraviolet renormalisation in the Spin-Boson Spin‐Boson model cannot be done in the same way as Edward Nelson renormalized the Nelson model in the paper [8]. [8]. More precisely, it is proven, that if Edward Nelsons method worked then the limiting operator is independent of the qubit so it would be be physically physically uninteresting. uninteresting. It It should should be be noted, noted, that that the the proof proof does does not not exclude exclude other other (more (more exotic) exotic) renormalisation methods. It might be possible to take the coupling constant to 0 as the ultraviolet cut-off cut‐off is removed and then end up at a physically useful model.. 2. Notation and definitions. \mathcal{H} will always denote The following introduction is taken almost directly from [2]. [2]. Throughout this paper, H n ‐fold tensor product of H \mathcal{H⊗n }^{\otimes n} for the n-fold \mathcal{H} and let H \mathcal⊗ {H}^{\sotimnes_{s}n}\⊂ subset \matH hcal{H}^⊗n {\otimes n} be the a separable Hilbert space. Write H subspace of symmetric tensors. The bosonic (or symmetric) Fock space is defined as symmetric) defined ∞ . \mathcal{F}_{b}(\mathcal{H})=\bigoplus_{n=0}^{\infty}\mathcal{H}^{\otimes_{ } n}. Fb (H) =. n=0. H ⊗s n .. 2 ⊗ n 2 n ⊗n (M, F, μ⊗n If H \mathcal= {H}=L^{L 2}(\mat hcal{M}, \matF, hcal{F},μ) \mu) where (M, (\mathcal{M}, \mat hcal{Fμ) }, \mu) is a σ-finite a‐finite measure space then H \mathcal{Hs}^{\otimes_{=s}n}=L_{Lsym}sym ^{2}(\mathcal(M {M}^{n}, \mathcal, {F F}^{\otimes n} ,, \mu^{ \otimes n}).) . (n) = (ψ ) and define We will write an element ψ \psi\in ∈ \mathcalF {F}_{bb}(\(H) mathcal{H}) in terms of its coordinates as ψ\psi=(\psi^{(n)}) define the vacuum † Ω\Omega=(1,0,0, = (1, 0, 0,\ldots) . . . ).. For gg\in \mathcal{H} ∈ H one defines (g) defines the annihilation operator a(g) a(g) and the creation operator aa\dagger(g) † on symmetric tensors in F = 0, aa\dagger(g)\Omega=g (g)Ω = g and \mathcbal{F}_(H) {b}(\mathcal{H}) by a(g)Ω a(g)\Omega=0, n. 1 a(g)(f1 ⊗s · · · ⊗s fn ) = √ g, fi f1 ⊗s · · · ⊗s fi ⊗s · · · ⊗s fn n i=1 √ † aa^{\dagger}(g)(f_{1}\otimes_{s}\cdots\otimes_{s}f_{n})=\sqrt{n+1}g\otimes_{s}f_ {1}\otimes_{s}\cdots\otimes_{s}f_{n} (g)(f1 ⊗s · · · ⊗s fn ) = n + 1g ⊗s f1 ⊗s · · · ⊗s fn. a(g)(f_{1}\otimes_{s}\cdot\cdot\cdot\otimes_{s}f_{n})=\frac{1}{\sqrt{n} \sum_{\dot{i}=1}^{n}\{g,f_{i}\ f_{1}\otimes_{s}\cdot\cdot\cdot\otimes_{s} \hat{f_{i} \otimes_{s}\cdot\cdot\cdot\otimes_{s}f_{n}. where where f\hat{f_i{i} means means that that ff_{\doit{i} is is omitted omitted from from the the tensor tensor product. product. These These operators operators extend extend to to closed closed operators operators ∗ in F = a† (g).. Furthermore, we have the canonical commutation relations: \mathcbal{F}_(H) {b}(\mathcal{H}) and (a(g)) (a(g))^{*}=a\dagger(g) [a(f ), a(g)] = 0 = [a† (f ),a\dagger(g)]} a† (g)] and [a(f ), aa†\dagger(g)]}=\langle (g)] = f,f, g. \overline{[a(f), a(g)]}=0=\overline{[a\dagger(f), \overline{[a(f), g\rangle.. We also define define the field field operators. ϕ(g) = a(g) + a† (g). \varphi(g)=\overline{a(g)+a\dagger(g)}.. 1.

(2) 127 A be a selfadjoint operator on H A \mathcal{H} with domain D(A). Let A define the second quantisation of A D(A) . Then we define to be the selfadjoint operator n ∞ dΓ(A) = 0 ⊕ (1⊗)k−1 A(⊗1)n−k . (2.1) (2.1). d \Gam a(A)=0\oplus\bigoplus_{n=1}^{\infty}\sum_{k=1}^{n}(1\otimes)^{k-1} A(\otimes 1)^{n-k}|_{\mathcal{H}^{\otimes} ^{n}. H ⊗s n. n=1 k=1. \mathcal{K} is another Hilbert space and U U :H \mathcal{H→ }arrow \matK hcal{K} is a bounded The number operator is defined = dΓ(1).. If K defined as N N=d\Gamma(1) operator with U. ≤ eq11 then we define define \Vert U\Vert\l ∞ . \Gam a(U)=1\oplus\bigoplus_{n=1}^{\infty}U^{\otimesn}|_{\mathcal{H}^{\otimes_ {S}n }. Γ(U ) = 1 ⊕. n=1. U ⊗n |H⊗s n . .. (n) (n) \omega is a ⊗s n and Γ ) = (A) = dΓ(A) We will write dΓ d\Gamma^{(n)}(A)=d\Gamma(A)| _{\mathcal{H}^{\oti|Hmes_{S}n}} \Gamma^{ (n(U )}(U)=\Gamma( U)|_{\matΓ(U hcal{H}^{\ot)imes_{|\H math⊗cal{sS} nn} throughout the text. If ω (n) multiplication operator then dΓ (ω) is the multiplication operator defined , knk_{n})= )= defined by the map ω n (k1 , . . .\ldots, d\Gamma^{(n)}(\omega) \omega_{n}(k_{1}, ω(k 1 ) + · · · + ω(kn ).. The \omega(k_{1})+\cdots+\omega(k_{n}) The following following three three results results are are essential essential standard standard results results (see (see [1], [1], [2], [2], [6]). [6]).. \mathcal{H} and Lemma 2.1. Let ω ≥ 00 be = inf(σ(ω)).. \omega\geq be aa selfadjoint operator operator defined defined on on the Hilbert space H and let m m= \inf(\sigma(\omega)) For nn\geq ≥ 1 we have (n) + 1·}+\cdot · ·s+\l+ambda_{ λnn} ||\lambda_{ λi i∈ σ(dΓ \sigma(d\Gamma^{ ((ω)) n)}(\omega))== \overl{λ ine{\{\1lambda_{ }\in\siσ(ω)}, gma(\omega)\} ,. (n) inf(σ(dΓ (ω))) = nm. \inf(\sigma(d\Gamma^{(n)}(\omega)))=nm.. \omega has compact Furthermore, dΓ(ω) compact resolvents if if and and only only if if ω compact resolvents. d\Gamma(\omega) will have compact. A be U:\mathcal{H}arrow \mathcal{K} Lemma 2.2. Let U :H→ K be ∈ H. \mathcal{H}, V be unitary, A be aa selfadjoint operator operator on on H, {H} . Then Then V\in \mat∈hcalU(H) {U}(\mathcal{H}) and ff\in \mathcal Γ(U and \Gamma(U)) is unitary and. Γ(U )dΓ(A)Γ(U )∗ = dΓ(U AU ∗ ). \Gamma(U)d\Gamma(A)\Gamma(U)^{*}=d\Gamma(UAU^{*}) Γ(U )W (f, V )Γ(U )∗ = W (U f, UUVU^{*}) V U ∗ ). \Gamma(U)W(f, V)\Gamma(U)^{*}=W(Uf, ∗ Γ(U )ϕ(f )Γ(U ) = ϕ(U f ). \Gamma(U)\varphi(f)\Gamma(U)^{*}=\varphi(Uf) .. .. .. )Ω = Ω. Furthermore, Γ(U )(f1 ⊗s · · · ⊗s fn ) = U f1 ⊗s · · · ⊗s U fn and and Γ(U \Gamma(U)(f_{1}\otimes_{s}\cdots\otimes_{s}f_{n})=Uf_{1}\otimes_{S} \cdots\otimes_{S}Uf_{n} \Gamma(U)\Omega=\Omega.. Lemma 2.3. Let ω ≥ 00 be ∈D(\omega^{-1/2}) D(ω −1/2 ) then ϕ(g) \omega\geq be selfadjoint and and injective. If If gg\in bounded. In \varphi(g) is dΓ(ω) d\Gamma(\omega)1/2 ^{1/2} bounded. 1/2 N^{1/2} bounded. particular, ϕ(g) bounded. We We have the following bound bound \varphi(g) is N −1/2 1)g. (dΓ(ω) + 1)1/2 ψ. ϕ(g)ψ ≤ \Vert\varphi(g)\psi\Vert\l2 (ω eq 2\Vert(\omega^{1+ /2}+1)g\Vert\Vert(d\Gamma(\omega) +1)^{1/2}\psi \Vert. 1/2 ).. In particular, ϕ(g) which holds on on D(dΓ(ω) bounded. D(d\Gamma(\omega)^{1/2}) \varphi(g) is infinitesimally dΓ(ω) d\Gamma(\omega) bounded.. We now introduce the Weyl representation. For any gg\in \mathcal{H} ∈ H we define define the corresponding exponential vector vector ∞ g ⊗n √ . ǫ(g) = (2.2) (2.2) n! n=0. \epsilon(g)=\sum_{n=0}^{\infty}\frac{g^{\otimesn}{\sqrt{n!}. D\subset⊂ \mathcal{H} One may prove that if D H is a dense subspace then {ǫ(f ) | f ∈ D} \{\epsilon(f)|f\in D\} is a linearly independent and total \mathcal{H} into H. \mathcal{H} . Let U \mathcal{H}. H. subset of F \mathcbal{F}_(H). {b}(\mathcal{H}) . Write U \mathca(H) l{U}(\mathcal{H}) for the set of unitary maps from H U\in \mat∈hcal{U(H) U}(\mathcal{H}) and hh\in ∈ Then there is a unique unitary map W (h, U W(h, U)) such that 2. W )ǫ(g)\Vert=h\Vert^{2}/2e−h\la/2−h,U ǫ(h + U \foral g) l g\i∀g ∈ H. W(h,(h, U)\epsiUlon(g)=e^{ngle h,Ug\ranglge}\epsi lon(h+Ug) n \mathcal {H}. One may easily check that (h, U ) → W W(h, (h, UU)) is strongly continuous and that (h, U)\mapsto. 1 ,U1 h2 ) , U1 )WU_{2})=e^{-i{\rm (h2 , U2 ) Im}(\langle = e−iIm(h W ((h1 , U1U_{1})(h_{2}, )(h2 , UU_{2})) W (h1U_{1})W(h_{2}, 2 )), W(h_{1}, h_{{\imath}}, U_{1}h_{2} \rangle)}W((h_{1}, ,. A is a selfadjoint operator on H \mathcal{H} and ff\in \mathcal{H} , U1 U2 ).. If A ∈ H we have where (h 1, U 1 )(h2 , UU_{2})=(h_{1}+U_{1}h_{2}, 2 ) = (h1 + U1 h2U_{1}U_{2}) (h_{1}, U_{1})(h_{2}, itdΓ(A) itA = Γ(eitA ) = W (0, ee^{itA}) ) ee^{itd\Gamma(A)}=\Gamma(e^{itA})=W(0, itϕ(if ) = W (tf, 1). ee^{it\varphi(if)}=W(tf, 1) .. We have the following lemma (see [6]): [6]): 2.

(3) 128 and U Then \mathcal{H} Lemma 2.4. Let f, H and f, hh\in ∈ U\in \mat∈hcalU(H). {U}(\mathcal{H}) . Then. W U )ϕ(g)WU)^{*}=\varphi(Ug)-2{\rm (h, U )∗ = ϕ(U g)Re}(\langle − 2Re(U g, h) W(h,(h, U)\varphi(g)W(h, Ug, h\rangle) WW(h, (h,U)a(g)W(h, U )a(g)WU)^{*}=a(Ug)-\langle (h, U )∗ = a(U g)Ug, − h\rangle U g, h. W U )a† (g)W (h, U )∗ = a† (U g) − h,h, U g. W(h,(h, U)a^{\dagger}(g)W(h, U)^{*}=a^{\dagger}(Ug)-\langle Ug\rangle.. ∗ \omega is selfadjoint, non negative and \mathcal{H} and Furthermore, if D(ωUU^{*}) ) then if ω and injective on on H and hh\in∈D(\omega ∗ ∗ ∗ W U )dΓ(ω)W (h, U )∗ = dΓ(U ωU ) − ϕ(U ωUU^{*} h)+\{h, h) + h, U ωUU^{*}h\} h W(h,(h, U)d\Gamma(\omega)W(h, U)^{*}=d\Gamma(U\omega U^{*})-\varphi(U\omega U\omega ∗ . on the domain domain D(dΓ(U on ωUU^{*})) )). D(d\Gamma(U\omega. The last essential ingredient is the lemmas In what follows we consider two fixed \mathcal{H1}_{1} fixed Hilbert spaces H and H \mathcal{H2}_{2} . We will need the following two lemmas (see [9]). [9]). Lemma 2.5. There (ǫ(f ⊕ g)) g))= = There is aa unique isomorphism UU : F(H \mathcal{F}(\mathcal1{H}⊕ _{1}\oplH us \math2cal{)H}_{2}→ )ar ow \mathF(H cal{F}(\mathcal{H}1 _{1)})\ot⊗ imes \matF(H hcal{F}(\mathcal{2H}_{)2}) such that UU(\epsilon(f\oplus ǫ(f ǫ(g). \omega_{ii} is selfadjoint on V_{i}i is unitary on \mathcal{H}i_{i} and \mathcal{iH}_,{i}, V If ω on H on H and ff_{ii}\in ∈ \mathcalH {H}_{ii} then \epsilon(f))\oti⊗mes\epsi lon(g) . If UUW(f_{1}\oplus W (f1 ⊕ f_{2}, f2 , V_{1}\oplus V1 ⊕ V2V_{2})U^{*}=W(f_{1}, )U ∗ = W (fV_{1})\otimes (f2 V_{2}) , V2 ) 1 , V1 ) ⊗ W W(f_{2}, ∗ ⊕ ω )U = dΓ(ω ) ⊗ 1 + 1 ⊗ dΓ(ω U dΓ(ω 1 2 1 2) Ud\Gamma(\omega_{1}\opl us\omega_{2})U^{* }=d\Gamma(\omega_{1})\otimes 1+1\otimes d\Gamma(\omega_{2}) ∗ + 1 ⊗ ϕ(f2 ) UU\varphi(f_{1}, ϕ(f1 , ff_{2})U^{*}=\varphi(f_{1})\otimes 2 )U = ϕ(f1 ) ⊗ 11+1\otimes\varphi(f_{2}) f2 )U ∗ = a(f1 ) ⊗ 11+1\otimes + 1 ⊗ a(f UUa(f_{1}, a(f1 ,f_{2})U^{*}=a(f_{1})\otimes 2) a(f_{2}) ∗ UUa^{\dagger}(f_{1}, a† (f1 , ff_{2})U^{* = a† (f1 ) ⊗ 1+1\otimes 1 + 1 a^{\dagger}(f_{2}) ⊗ a† (f2 ). 2 )U }=a^{\dagger}(f_{1})\otimes .. Lemma 2.6. There There is aa unique isomorphism ∞ . U:\mathcal{F}(\mathcal{H}_{1})\otimes\mathcal{F}(\mathcal{H}_{2})ar ow \mathcal{F}(\mathcal{H}_{1})\oplus\bigoplus_{n=1}^{\infty}\mathcal{F} (\mathcal{H}_{1})\otimesS_{n}(\mathcal{H}_{2}^{\otimesn}). U : F(H1 ) ⊗ F(H2 ) → F(H1 ) ⊕. n=1. such that. F(H1 ) ⊗ Sn (H2⊗n ). ∞ . U(w \otimes\{\psi_{2}^{(n)}\}_{n=0}^{\infty})=\psi^{(0)}w\oplus\bigoplus_{n=1}^ {\infty}w\otimes\psi_{2}^{(n)}. (n). (0) w⊕ U (w ⊗ {ψ2 }∞ n=0 ) = ψ. n=1. (n). w ⊗ ψ2 .. A be B be is reduced by Let A be aa selfadjoint operator operator on on F(H and B be selfadjoint on on F(H by all all of of \mathcal{F}(\mathcal1{H}_{)1}) and \mathcal{F}(\mathcal2{H}_{)2}) such that BBiS ⊗n (n) the subspaces SS_{n}n(\mat(H Write B Then hcal{H}_{2}2^{\otimes n}).) . Write B^{(n)}=B|_{S= _{n}(\matBhcal|{HS}_n{2}(H ^{\otim⊗n es n}))} .. Then 2. ∞ . U(A \otimes 1+1\otimes B)U^{*}=A+B^{(0)}\oplus\bigoplus_{n=1}^{\infty}(A\otimes 1+1\otimes B^{(n)}) UA \otimes BU^{*}=A\otimes B=B^{(0)}A\oplus\bigoplus_{n=1}^{\infty}A\otimes B^{ (n)}.. U (A ⊗ 1 + 1 ⊗ B)U ∗ = A + B (0) ⊕. n=1. (A ⊗ 1 + 1 ⊗ B (n) ). U A ⊗ BU ∗ = A ⊗ B = B (0) A ⊕. 3. ∞ n=1. A ⊗ B (n) .. The Spin-Boson Spin‐Boson model. Let σa_{x}, \sigma_{zz} denote the Pauli matrices x, σ y, σ \sigma_{y}, 1 0 0 −i 0 1 σz = σy = σx = 0 −1 i 0 1 0. \sigma_{x}=(\begin{ar ay}{l } 0 1 1 0 \end{ar ay}) \sigma_{y}=(\begin{ar ay}{l } 0 -i i 0 \end{ar ay}) \sigma_{z}=(\begin{ar ay}{l } 1 0 0 -1 \end{ar ay}). and define define ee_{1}=(1,0) Spin‐Boson Hamiltonian is given by 1 = (1, 0) and ee_{-1}=(0,1) −1 = (0, 1).. The Spin-Boson H ω) :=\eta\sigma_{z}\otimes := ησz ⊗ 11+1\otimes + 1 d\Gamma(\omega)+ \sigma_{x}\otimes\varphi(v) ⊗ dΓ(ω) + σx ⊗ ϕ(v), η (v,\omega) H_{\eta}(v, ,. \omega selfadjoint on H. \mathcal{H} . We will also need the fiber which is here parametrised by vv\in∈ H, η\eta\i∈ \mathcal{H}, fiber operators: n \mathC bb{C} and ω. FF_{\eta}(v, = ηΓ(−1) + dΓ(ω) + ϕ(v). η (v, ω) \omega)=\eta\Gamma(-1)+d\Gamma(\omega)+\varphi(v) .. 3.

(4) 129 acting in F define \mathcbal{F}(H). _{b}(\mathcal{H}) . If the spectra are real we define E :=:=\inf(\sigma(H_{\eta}(v, inf(σ(Hη (v,\omega))) ω))) η (v, ω) E_{\eta}(v, \omega) E\mathcal{E}_{\eta}(v, inf(σ(F (v, ω))). η (v, ω) := η \omega) :=\inf(\sigma(F_{\eta}(v, \omega))) .. \omega selfadjoint on H \mathcal{H} we define For ω define. m(ω) = inf{σ(ω)} and m = inf{σess (ω)}. ess (ω) m( \omega)=\inf\{\sigma(\omega)\} m_{ess}( \omega)=\inf\{\sigma_{ess}(\omega)\}. Standard perturbation theory and Lemma 2.3 yields: −1/2 Proposition 3.1. Let ω ≥ 00 be D(ω \omega\geq be selfadjoint and and injective, vv\in∈ Then the operators operators n \mathC. bb{C} . Then \mathcal {D}(\omega^{1/2})) and η\eta\i∈ F (v, ω) and H (v, ω) are closed on the respective domains and are closed on domains η \omega) η \omega) F_{\eta}(v, H_{\eta}(v,. D(F = D(dΓ(ω)) η (v, ω)) \mathcal{D}(F_{\eta}(v, \omega))=\mathcal {D}(d\Gamma(\omega)) D(H = D(1 dΓ(ω)). η (v, ω)) \mathcal{D}(H_{\eta}(v, \omega))=\mathcal {D}(1\otim⊗ es d\Gamma(\omega)) .. \mathcal{D} of \omega the linear span of \mathcal{S}ets Given Given any any core core D of ω of the following sets. ∞ . \mathcal{J}(\mathcal{D}):=\{ Omega\} cup\bigcup_{n=1}^{\infty}\{f_{i} \otimes_{s}\cdot\cdot\cdot\otimes_{s}f_{n}|f_{j}\in\mathcal{D}\. J (D) := {Ω} ∪. n=1. {f1 ⊗s · · · ⊗s fn | fj ∈ D}. (D) J {fhcal1 {D⊗ \overl ine{\math:= cal{J} (\mat }):=\{ff_{12}\ot|imesff1_{2}|∈ f_{1}\{e in\{e_{11},, e_{e- −1 1}\},}, f_{2}\fin 2\mat∈hcal{JJ}(\mat(D)} hcal{D})\}. is aa core ω) and H ω) respectively. Furthermore, both core for F both operators operators are are selfadjoint and and semisemi‐ η (v,\omega) η (v, F_{\eta}(v, H_{\eta}(v, \omega) re\mathcal{S} olvents if \omega has compact bounded bounded if if η\eta\i∈ and they have compact compact resolvents if ω compact resolvents. n \mathR bb{R} and From From the the paper paper [2] [2] we we find find the the following following theorem: theorem: Theorem 3.2. 3.2. Let φ be an an element element in in F = (φ1 ,\phi_{-1})=e_{1}\otimes\phi_{1}+e_{-1}\otimes\phi_{-1} φ−1 ) = e1 ⊗ φ1 + e−1 ⊗ φ−1 be \phi=(\phi_{1}, \mathbcal{(H) F}_{b}(\mathcal2{H})^{= 2}=\mathF cal{F}_b{b}((H) \mathcal{H})\opl⊕ us \mathF cal{F} b_{(H) b}(\mathcal{H})\appr≈ox (k) (k) 2. ∈ {−1, 1}.. Let ii\in\{-1,1\} ∈ {−1, 1}.. Define φ\overiline{\= C Write φ\phii _= \mathbb{C}^{2⊗ }\otimes \maFthcalb{F}_(H). {b}(\mathcal{H}) . Write {i}=(\phi(φ_{ii}^{(k)})) for ii\in\{-1,1\} phi}_{i}=(\(oveφrline{\iphi}_{i}^{(k))}) where. (k) k is even φi (k). φi = (k) φ−i k is odd. \overline{\phi}_{ ^(k)}=\{ begin{ar y}{l \phi_{\dot{i}^{(k)} kisev n \phi_{-}^{(k)} kisod \end{ar y}. and V Then and 1 ,inφ V(\phi(φ_{1},1\phi, _φ{-1}−1 )=(\overl)ine{=\phi}(_{1φ },\overl e{ \phi−1}_{-1). }) . Then ∗ (1) =V. (1) VV is is unitary unitary with with VV^{*}=V.. = 1 d\Gamma(\omega) ⊗ dΓ(ω).. Furthermore, (2) If ≥ 00 is 1 ⊗d\Gamma(\omega)V^{* dΓ(ω)V ∗ }=1\otimes \omega\geq n \mathR bb{R} and If ω is selfadjoint selfadjoint and and injective injective then then VV1\otimes Furthermore, if if η\eta\i∈ and −1/2 then vv\in∈\mathcal D(ω {D}(\omega^{-1/2})) then ∗ VVH_{\eta}(v, Hη (v,\omega)V^{*}=F_{-\eta}(v, ω)V = F−η\omega)\oplus (v, ω) ⊕ Fη (v, ω). F_{\eta}(v, \omega) .. −1/2 ω) = E{−|η| ω) and (3) Let ≥ 00 be D(ω \omega\geq n \mathR bb{R} and η (v,\omega)=\mathcal Let ω be selfadjoint selfadjoint and and injective, injective, η\eta\i∈ and vv\in∈ \mathcal {D}(\omega^{1/2})).. Then Then E and E_{\eta}(v, E}_{-|\eta|(v, }(v, \omega) H ω) has aa ground state if ω) has aa ground state. This if and and only only if if the operator operator F This is the η (v, H_{\eta}(v, \omega) F_{-|−|η| \eta|}(v,(v, \omega) i_{\mathcal{S} non degenerate case > 0,, and it is = 0. case if if m(ω) degenerate if if η\eta\neq 0 . Also m(\omega)>0. inf(σ ω))) = E−|η|\eta|(v, ω) + mess (ω) ess (F|η| \inf(\sigma_{ess}(F_{| \eta|(v, }(v, \omega)))=\mathcal{E}_{-| }(v, \omega)+ m_{ess}(\omega) inf(σ = Eη (v,\omega)+m_{ess}(\omega) ω) + mess (ω) ess (Hη (v, ω))) \inf(\sigma_{ess}(H_{\eta}(v, \omega)))=E_{\eta}(v,. and > Eh−|η| ω) if = 00 and = 0.0. and E\mat|η| if and and only only if if both both η\eta\neq and m(ω) m(\omega)\neq hcal{(v, E}_{|\etaω) |}(v, \omega)>\mat cal{E}_{-|\et(v, a|}(v, \omega) −1/2 \mathcal{S} (4) Let ≥ 00 be D(ω for H ω) \omega\geq n \mathR bb{R} and η (v, Let ω be selfadjoint elfadjoint and and injective, injective, η\eta\i∈ and vv\in∈\mathcal {D}(\omega^{1/2})).. If If φ\phi is is aa ground ground state state for H_{\eta}(v, \omega) then. e−sign(η) ⊗ ψ η = 0 Vφ= e−1 ⊗ ψ−1 + e1 ⊗ ψ1 η = 0. V\phi=\{ begin{ar ay}{l} e_{-sign(\eta)}\otimes\psi \eta\neq0 e_{-1}\otimes\psi_{-1}+e_{1}\otimes\psi_{1} \eta=0 \end{ar ay}. \mathcal{S}tate for F−|η| (v, ω) and where ψ ω).. \psi i_{\matis hcal{S} aa ground state either 00 or or aa ground state for F −1 are either 0 (v, F_{0}(v, \omega) F_{-|\eta|}(v, \omega) and ψ\psi1_{1},, ψ\psi_{-1}. 4.

(5) 130 At this point we should look at an example: 3 Example 3.3. The physically correct model we have H ,hcalB(R \mathcal= {H}=L^{2}L (\mat2hbb{(R R}^{3}, \mat {B}(\mathbb{3R}^{), 3}), \laλmbda_{33)}) where λ\lambda3_{3} is the Lebesgue. 3 2 +m2 and v \sigma ‐algebra. Furthermore, ω(k) = measure and B(R ) is the Borel σ-algebra. gω −1/2 \mathcal{B}(\mathb {R}^{3}) \omega(k)=\sqrt{||k| k|^{2}+m^{2}} v_{g,Λ g,\Lambda}==g\omega^{ -1/2}1_{\1{|k|{|k|≤Λ} \leq\Lambda\}} \Lambda>0 for some m ≥ 0, > 0 and Λ > 0.. In this case, m(ω) = m = mess (ω) and σ(ω) = [m,\infty ∞)) = σess (ω).. Note m\geq 0, gg>0 m(\omega)=m=m_{ess}(\omega) =\sigma_{ess}(\omega) \sigma(\omega)=[m, that all assumptions in Theorem 3.2 are fulfilled. fulfilled.. 4. Result and Interpretation. \omega will always denote an injective, non negative and selfadjoint operator on H. \mathcal{H}. Throughout this section ω Furthermore, we will write m = m(ω) and m ess = mess (ω).. The main technical result is the following m=m(\omega) m_{ess}=m_{ess}(\omega) theorem:. −1/2 \omega . For Theorem 4.1. Let {v P_{\omega} and P denote the spectral measure corresponding corresponding to ω. ω denote \{v_{gg}\}}_{gg∈(0,∞) \in(0,\infty)}\subset⊂\matD(ω hcal{D}(\omega^{-1/2})) and \overline{m}>0 \overline{m}>0 P = 1 − P = P ([0, m]).. Assume that there is m >0 = P (( m,. ∞)) and each > 0 we define P each m and . ω ω P_{\overl \overlm overl ine{P}_{\overline{m} =1-P_{\m ine{m} =P_{\omega}([0,\overline{m}]) m ine{m}}=P_{\omega}((\overline{m}, \infty)) \mathcal{S}uch that: such −1/2 −1/2 \omega^{-1/2} (1) {P D(ω . to vv\in∈\mathcal {D}(\omega^{1/2})) in in the the graph graph norm norm of of ω \{\overlm ine{Pv}_{\goverl}ig∈(0,∞) ne{7n}}v_{g}\}_{g\in(0,\converges infty)}converge\mathcal{S} to. −1 (2) ω as {gS}gtend_{\mathcal tends{S}} to \infty a\mathcal diverges to to ∞ to infinity. infinity. \Vert\omega^{-1}PP _{\overm line{mv} v_{gg}\Ver t diverges. Then Then the g-dependent g ‐dependent family of of operators operators given by by −1 ∗ η,inm F W:=W((ω ω)W (ω-1−1 +\omega^{. ω-1/2}−1/2 \overl e{F}_(v {\eta,g\overl, ω) ine{7n}}(:v_{g=}, \omega) \omega^{ -1P}P_{\movervline{gm,} 1)F }v_{g}, 1)Fη_{\(v eta}(vg_{g},, \omega) W(\omega^{ }P_{\overlPimne{mv} v_{gg},, 1)1) ^{*}+ \Vert P_{\overliPne{mm} v_{vg}\gVert ^{22} −1 = (2ω −1) + dΓ(ω)(\overl+ine{ϕ(P =\etaηW W(2\omega^{ -1}P_{\overlPine{m m} vv_{gg}, -1,)+d\Gamma(\omega)+ \varphi P}_{\overlim ne{mv} vg_{g)}). (4.1) (4.1). 2 −1/2 is uniformly bounded bounded below below by by −|η|− con‐ a,\overgline{,mω)} } (v_{g}, \omega)g∈(0,∞) \}_{g\in(0,\infty)} con-| \eta|-\sup_{g\insup (0,\infty)}\g∈(0,∞) Vert\omega^{-1/2}\ ω overline{P}_{\overlinPe{7n}m} v_{vg}\gVer t^{2} . Furthermore, {\{\Fover lη,ine{Fm}_{\et(v \infty. verges to dΓ(ω) + ϕ(v) in norm resolvent sense as a\mathcalg{S}gtetends nd_{\mathcal{S}} to ∞. d\Gamma(\omega)+\varphi(v). \omega is small can lead to problems (see [3] for a The assumption in part (1) is critical. Divergence where ω [3] counter example). The following Corollary easily proved:. \omega is a Corollary 4.2. Assume H :M→ and ω a multiplication operator operator on on this space. Let vv:\mathcal{M}arrow \mathcal={H}=L^{L2}(2\mat(M, hcal{M}, \matF, hcal{F},μ) \mu) and \mathbb{R} into [0, \mathb {C} is measurable and C }g∈(0,∞) 1].. Assume gg\mapsto\chi_{g}(x) → χg (x) is and {χ be aa collection collection of of functions from R [0,1] \{\chig_{g}\}_{g\i n(0,\infty)} be A_{\mathcal{S}}sume furthermore that kk\mapsto\chi_{g}(\omega(k))v(k)\in increasing and converges ∈ R. Assume → χg (ω(k))v(k)\mathcal{D}(\omega^{-1/2}) ∈ D(ω −1/2 ) converges to 11 for all xx\in \mathbb{R}. −1/2 −1 \overline{v} := \overline{m}>0 and > 0 such that v 1 v ∈ D(ω ). If k → ω(k) v(k)1 (k) ∈ / and that there is m . If {ω≤ m} {ω>1} :=1_{\{\omega\leq\overline{m}\} v\in \mathcal{D}(\omega^{-1/2}) k\mapsto\omega(k)^{-1}v(k)1_{\{\omega>1\}}(k)\not\in \mathcalH {H} there are unitary maps {U and {V of η\eta such that: g }g∈(0,∞) g }g∈(0,∞) \{U_{g}\}_{g\in(0, \infty)} and \{V_{g}\}_{g\in(0, \infty)} independent of. ∗ 2 (1) {U , ω)U. ω-1/2}−1/2 1{ω>ine { converges in in norm norm resolvent resolvent sense sense to to the the operator operator m} m}\v \{U_{gg}F_{F\etηa}(v(v _{g}, g\omega) U_{g}^{*}g+\Vert+\omega^{ 1_{\{\omega>\overl } v_{gg} \Vert^}{2}g∈(0,∞) \}_{g\in(0,\infty)} converges dΓ(ω) + ϕ( v ) as g tends to infinity. d\Gamma(\omega)+\varphi(\overline{v})a\mathcal{S}g ∗ 2 (2). ω-1/2}−1/2 1{ω>ine { g , ω)V (2) {V is uniformly uniformly bounded bounded below below and and converges converges in in norm norm m} m}\v \{V_{gg}HH _{\etaη}(v(v _{g}, \omega) V_{g}^{*}g+\Vert+\omega^{ 1_{\{\omega>\overl } v_{gg} \Vert^}{2}g∈(0,∞) \}_{g\in(0,\infty)} is resolvent sense to the operator operator. in:= H + (ϕ( (dΓ(ω) + ϕ( \overl e{H}:=(d(dΓ(ω) \Gamma(\omega)+\varphi \overlvine{)) v}) \opl⊕us(d\Gamma( \omega)+ \varphi (\overlvine{))v}). \infty . In particular, as as gg tends to ∞.. −1/2 2 −1 −1/2 2 −1 (H , ω) + \ ω 1{ω>m} g , ω) \+ {ω>m} inv ine{mv (H_{\etηa}(v (v_{gg}, \omega)+\Vert omega^{-1/2}1_{\1{\omega>\overl e{mg}\ } v_{g+i) } \Vert^{2}+i)^{−-1}-(H_{(H 0}(v_{0g}(v , \omega)+\Vert omega^{ ω-1/2} 1_{\{\omega>\overl }\}gv_{ g}\Vert+i) ^{2}+i)^{-1} \infty. will converge g tends to ∞. converge to 0\theta in norm as a\mathcal{S}g. Example 4.3. We continue Example 3.3. Let us consider self-energy self‐energy renormalisation schemes as invented in [8]. [8]. In such schemes proves that H , ω) Eη (v ω) η (vg,Λ g,Λ ,\omega) H_{\eta}(v_{g, \Lambda}, \omega)-−E_{\eta}(v_{g, \Lambda}, Ren converges in strong or uniform resolvent sense to an operator H , ω).. Using Corollary 4.2 we see: H_{\ηeta}^{Ren}(v (v_{gg}, \omega). 5.

(6) 131 131 −1/2 2 (1) Λ → E \eta. g,Λ has aa limit limit independent independent of of η. {ω>1} \Lambda\mapst o E_{η\et(v a}(v_{g,Λ g,\Lambda}, ω) , \omega)++\Ver ω t\omega^{-1/2} 1_{\1{\omega>1\} }v_{g,\vLambda} \Ver t^{2} has −1/2 −1 −1/2 2 −1 (2) (H + ω\omega^{. 2 +i) , ω) + \ ω g,Λ, \omega)+\Vert converges to to 0\theta (H_{\etηa}(v (v_{gg,Λ ,\Lambda}, ω) , \omega)+\Vert -1/2}11_{{ω>1} \{\omega>1\}v}vg,Λ _{g, \Lambda} \Vert^{2}+i)^{−-1}-(H_{(H 0}(v_{0g,(v \Lambda} omega^{-1/2} 1_{1\{{ω>1} \omega>1\}}vv_{gg,Λ ,\Lambda} \Vert+i) ^{2}+i)^{-1} converges as Λ \Lambda tends to ∞. \infty. in norm as Ren , ω) must be From this we conclude that if a self-energy self‐energy renormalisation scheme exists then H H_{\ηeta}^{Ren}(v (v_{gg}, \omega) independent of η, which is not physically interesting. In other words, the contribution from the qubit \eta , disappears, as the ultraviolet cutoff is removed. This result is similar to the result in [4], [4], where it is shown, that the mass-shell mass‐shell in a certain model becomes "almost “almost flat" fla t^{11} as the ultraviolet cutoff is removed. Thus the contribution from the matter particle vanishes as the ultraviolet cutoff is removed.. 5. Sketch of proof of Theorem 4.1.. \omega will always In this section we give the central ideas behind the proof of Theorem 4.1. From now on, ω \mathcal{H} . As a simplifying assumption we have denote an injective, non negative and selfadjoint operator on H.. m(ω) > 0. m(\omega)>0. We will also assume {v satisfies the assumptions of Theorem 4.1. It is easy to see that if they g }g∈(0,∞) \{v_{g}\}_{g\in(0, \infty)} satisfies \ov erline{m} then it will also hold for m \overline{m}=m \overline{m}=m hold for some m = m.. Hence we will now assume that m = m.. Using Lemma 2.4 we see that −1 η,m F\overl (vg ,mω) ηW (2ω v1g}v_{g}, , −1) + dΓ(ω) ine{F}_{\eta, }(v_{g},=\omega)=\eta W(2\omega^{-1)+ d\Gamma(\omega). ⊂hcalH Hence it is enough to prove that if {h satisfies g }g∈(0,∞) \{h_{g}\}_{g\i n(0,\infty)}\subset \mat {H} satisfies. galim r ow\infty 1\d h ot{ \imathg} m\ = Vert h_{g}\Ve∞ rt=\infty. g→∞. then := ηW (hg , −1) + dΓ(ω) TT_{\eta}(h_{g}, η (hg , ω) \omega) :=\eta W(h_{g}, -1)+d\Gamma(\omega) converges converges to to dΓ(ω) d\Gamma(\omega) is is norm norm resolvent resolvent sense sense as as gg tends tends to to infinity. infinity. The The following following Lemma Lemma goes goes back back to to [5] [5] first fundamental observation. and is the first \omega has compact Lemma 5.1. Theorem Theorem 4.1 holds if if ω compact resolvents.. Proof. (hg , −1) Proof. First we see that W infinity. By [10, [10, Theorem 4.26] W(h_{g}, -1) converges to 00 weakly for gg tending to infinity. it is enough to check exponential vectors. We calculate 2. ǫ(g \langle\epsi1 ),lon(gW _{1}), W((vv_{gg}, -,1)\−1)ǫ(g epsilon(g_{2})\rangl2e) =e^{-\Ver=t v_{eg} −v \Vert^{g2}/2+\l/2+v angle v_{g},g_{g2},g\rangl2 e}\ǫ(g langle\epsi1lo),n(g_{ǫ(v 1}), \epsiglon(−v_{g}-gg_ {22)})\} 2. = =e^{-\Veret v_{−v 9}\Vert^{g2}/2+\langl/2+v e v_{g},g_{2}\ragngl,ge+\la2ngl+g e g{\imath},v1_{g},v \ragngl−g e-\langle g_{11},g,g_{2}\2rangle,},. 0 . We calculate which converges to 0.. − i)−1 − (dΓ(ω) − i)−1 =η(Tη (vg , ω) \omega)-i)^{-1}W(v_{g}, − i)−1 W (vg , −1)(dΓ(ω) − i)−1 (T η (vg , ω)\omega)-i)^{-1}-(d\Gamma(\omega)-i)^{-1}=\eta(T_{\eta}(v_{g}, (T_{\eta}(v_{g}, -1)(d\Gamma(\omega)-i)^{-1} 2 =η (Tη (vg , ω) − i)−1 W (vg , −1)(dΓ(ω) − i)−1 W (vg , −1)(dΓ(ω) − i)−1 =\eta^{2}(\overline{T}_{\eta}(v_{g}, \omega)-i)^{-1}W(v_{g}, -1) (d\Gamma(\omega)-i)^{-1}W(v_{g}, -1)(d\Gamma(\omega)-i)^{-1} + η(dΓ(ω) − i)−1 W (vg , −1)(dΓ(ω) − i)−1 . +\eta(d\Gamma(\omega)-i)^{-1}W(v_{g}, -1)(d\Gamma(\omega)-i)^{-1}. This implies. ( F η (vg , ω) − i)−1 − (dΓ(ω) − i)−1 ≤ (|η|+1)|η| (dΓ(ω) − i)−1 W (vg , −1)(dΓ(ω) − i)−1. \Vert(\overline{F}_{\eta}(v_{g}, \omega)-i)^{-1}-(d\Gamma(\omega)-i)^{-1} \Vert\leq(| \eta|+1)|\eta|\Vert(d\Gamma(\omega)-i)^{-1}W(v_{g}, -1) (d\Gamma(\omega)-i)^{-1}\Vert. By Lemma 2.1 we see (dΓ(ω) − i)−1 is compact so the result is finished. finished. (d\Gamma(\omega)-i)^{-1}. \square. The next Lemma is very technical. The full proof can be found in [2] [2] and we only make a short sketch: 6.

(7) 132 \omega is a \mathcal{H} . Let P Lemma 5.2. Assume ω P_{\omega} a selfadjoint, non negative and and injective operator operator on on H. be the spectral ω be \omega . Define the measurable function f \mathbb{→ R}arrow \matR hbb{R} measure of of ω. f_{k}k : R. ∞ . f_{k}(x)= \sum_{n=0}^{\infty}(n+1)2^{-k}1_{(n2^{-k},(n+1)2^{-k}]\cap(m,\infty)} (x). fk (x) =. (n + 1)2−k 1(n2−k ,(n+1)2−k ]∩(m,∞) (x). .. n=0.

(8) along along with ω Then the following holds k = \omega_{ k}=\int_{\matRhbb{fR} kf_{k}(λ)dP (\lambda)dP_{\omega}ω ((λ). \lambda) . Then. 1. 1. F converges to F \ov erηline{(v, F}_{\eta}(v,ω\omegak_{k)}) converges \ov erηline(v, {F}_{\eta}(v, ω) \omega) in norm resolvent sense uniformly in v.v.. \mathcal{H} . For each \mathbb{N} 2. Let {h N,, there are be a collection collection of of elements elements in H. each kk\in ∈ are Hilbert spaces H g }g∈(0,∞) \mathcal1,k {H}_{1,k,}, H \mathcal2,k {H}_{2,k,}, \{h_{g}\}_{g\in(0, \infty)} be selfadjoint operators hoverg,k operators ω collection of of elements elements {\{\ and aa collection collection 1,kk,}, ω 2,k ≥ \omega_{1, \omega_{2, k}\geq0, 0 , aa collection line{h}_}{g,kg∈(0,∞) }\}_{g\in(0,\infty)}\subset⊂\matH hcal{H1,k }_{1,k} and of }g∈(0,∞) of unitary maps {U \{u_{g,g,k k}\}_{g\in(0, \infty)} such that ∞ ⊗n Fb (H1,k ) ⊗ Sn ((H2,k ) ) , Ug,k : Fb (H) → Fb (H1,k ) ⊕. \mathcal{U}_{g,k}:\mathcal{F}_{b}(\mathcal{H})ar ow\mathcal{F}_{b} (\mathcal{H}_{1,k})\oplus(\bigoplus_{n=1}^{\infty}\mathcal{F}_{b}(\mathcal{H} _{1,k})\otimesS_{n}( \mathcal{H}_{2,k})^{\otimesn}). ,. n=1. −k ω has compact > 0 and compact resolvents, h all gg>0 and 1,k ≥ \omega_{1, k}\geq22^{-k} \Vert h_g{g}\ = Vert=\Vert\ over hline{hk,g }_{k,g}\Ver t for all ∞ . . \mathcal{U}_{g},{ _{k}\overline{F}_{\eta}(h_{g},\omega_{k})\mathcal{U}_{g,k}^ {*}=\overline{F}_{\eta}(\overline{h}_{g,k},\omega_{1,k})\oplus\bigoplus_{n=1}^{ \infty}(\overline{F}_{(-1)^{n}\eta}(\overline{h}_{g,k},\omega_{1,k})\otimes1+1 \otimesd\Gam a^{(n)}(\omega_{2,k}). ∗ Ug,k F η (hg , ωk )Ug,k =F η ( hg,k , ω1,k ) ⊕. n=1. for all all η\eta\i∈ n \matR. hbb{R}.. F (−1)n η ( hg,k , ω1,k ) ⊗ 1 + 1 ⊗ dΓ(n) (ω2,k ). Proof. Proof. Part Part (1) (1) can can easily easily be be derived derived from from the the fact fact that that 2−k. (F η (v, ω) − F η (v, ωk ))ψ = (dΓ(ω) − dΓ(ωk ))ψ ≤. dΓ(ω)ψ . m. \Vert(\overline{F}_{\eta}(v, \omega)-\overline{F}_{\eta}(v, \omega_{k}) \psi\Vert=\Vert(d\Gamma(\omega)-d\Gamma(\omega_{k}) \psi\Vert\leq\frac{2^{-k} {m}\Vert d\Gamma(\omega)\psi\Vert.. for all ψ finishes the proof. In part (2), one constructs \psi\in∈\mathD(dΓ(ω)). cal{D}(d\Gamma(\omega)) . Standard resolvent formulas then finishes Hilbert spaces H Hhcal1,k g,k : H \mathcal1,k {H}_{1,k} and H \mathcal2,k {H}_{2,k} and a unitary map UU_{g,k} \mathcal→ {H}arrow \mat {H}_{1,k}\⊕ oplus \matH hcal2,k {H}_{2,k} such that ∗ U ωk Ug,k = ω1,k ω2,kk} U_{g,g,k k}\omega_{k}U_{g, k}^{*}=\omega_{1, k}\opl⊕ us\omega_{2, −k where ω\omega_{1, has compact resolvents and h\overl ∈hcalH ∈ (0, ∞).. One now uses 1,k ≥ k,g \infty) k}\geq22^{-k} ine{h= }_{k,g}=UU_{g,k g,k}h_{hg}\gin \mat {H}1,k _{1,k} for all gg\in(0, \square Lemmas 2.2, 2.5 and 2.6 to construct U \mathg,k cal{U}_{g,k.}.. We can now prove that Theorem 4.1 is true. From Lemma 5.2 part (2) (2) and Lemma 5.1 we see that k . Lemma 5.2 part (1) the theorem holds if ω = ωk for some k. \omega=\omega_{k} (1) then finishes finishes the proof.. Acknowledgements Thomas Norman Dam was supported by the Independent Research Fund Denmark through the project Mathematics of Dressed Particles" Particles ”’ "Mathematics 1\dag er. Bibliography [1] [1]. V. V. Betz, Betz, F. F. Hiroshima Hiroshima and and J. J. Lorinczi. Lorinczi. Feynman-Kac-Type Feynman‐Kac‐Type Theorems Theorems and and Gibbs Gibbs Measures Measures on on Path Path Space, with applications to rigorous Quantum Field Theory, 2011 Walter De Gruyter GmbH and Co. KG, Berlin/Boston. Berlin/ Boston.. [2] [2]. Dam, T. N.,Møller N.,M\emptyset 1ler J. S.: Spin Boson Type Models Analysed Through Symmetries. arXiv:1803.05812 [math-ph]. [math‐ph].. [3] [3]. Dam, T. N.,Møller N.,M\emptyset 1ler J. S.: Asymptotics in Spin-Boson Spin‐Boson type models. arXiv:1808.00085 [math-ph]. [math‐ph]. 7.

(9) 133 [4] [4]. Deckert D. A., Pizzo A.: Ultraviolet Properties of the Spinless, One-Particle One‐Particle Yukawa Model. ComCom‐ mun. Math. Phys. 327, 327, 887–920 887‐920 (2014).. [5] [5]. M. M\emptyset 1ler and M. Hirokawa, Hirokawa, J. J. S. S. Møller and I. I. Sasaki. Sasaki. A A Mathematical Mathematical Analysis Analysis of of Dressed Dressed Photon Photon in in Ground Ground State of Generalized Quantum Rabi Model Using Pair Theory, Journal of Physics A: Mathematical and Theoretical 50(18): 184003, 2017.. [6] [6]. Oliver Matte and Thomas Norman Dam. SPECTRAL THEORY OF NON-RELATIVISTIC NON‐RELATIVISTIC QED with Supplements.. [7] [7]. J. S. Møller. M\emptyset 1ler . The translation invariant massive Nelson model: I. The bottom of the spectrum, Ann. Henri Poincaré 6 (2005), 1091–1135. 1091‐1135.. [8] [8]. E. NELSON. Interaction of nonrelativistic particles with a quantized scalar field, field, J. Math. Phys. 5 (1964), (1964), 1190-1197. 1190‐1197.. [9] [9]. K.R. Parthasarathy. An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics, vol. 85, Birkhauser, Basel, 1992.. [10] J. Weidmann. Linear Operators in Hilbert Spaces. Springer, New York (1980).. 8.

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