Onsager SOLUTION OF THE EIGENWERT PROBLEM (O-29) V = e H A e H B λ max Z 2 Onsager (O-77) (O-82) (O-83) Kramers-Wannier 1 1 Ons

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1 Onsager SOLUTION OF THE EIGENWERT PROBLEM O-29 V = e H A e H B λ max Z 2 OnsagerO-77O-82 O-83 2 Kramers-Wannier Onsager * * * * * V self-adjoint V = V /2 V V /2 = V /2 V 2 V /2 = 2 sinh 2H n/2 e 2 H A e H B e 2 H A O-84 V V VV X 0, X,, X n X 0, X,, X n Z 2 r = R ro-68[] expi αz r = cosαz r + i sinαz r = 2! αz r 2 + 4! αz r 4 6! αz r 6 + [ + i αz r 3! αz r 3 + ] 5! αz r 5 + = R r + R r 2! α2 + 4! α4 6! α6 + + i Z r α 3! α3 + 5! α5 + = R r + R r cos α + i Z r sin α O-83 O-83b O-83

2 Z r X r Y r cosαz r = R r + R r cos α 2 sinαz r = Z r sin α 3 Z r X r Y r expαx r = R r + R r cosh α + X r sinh α 4 expαy r = R r + R r cosh α + Y r sinh α 5 expαz r = R r + R r cosh α + Z r sinh α 6 [ X r cosαz r = X r R r + R r 2! α2 + 4! α4 ] 6! α6 + = X r X r + X r 2! α2 + 4! α4 6! α6 + = X r cos α 7 X r sinαz r = X r Z r α 3! α3 + 5! α5 = X r Z r sin α = i Y r sin α 8 Y r cosαz r = Y r cos α 9 Y r sinαz r = Y r Z r sin α = i X sin α 0 X r Y r Z r Onsager X r X r exp n r= iπn r Z r = 2n = n r= n r= [ πn r R r + R r cos + i Z r sin 2n [ R r sin πr 2n + i Z r cos πr ] 2n ] πn r 2n Onsager X r X r ω = rπ/n X r O-68 e iωzr/2 X r e iωzr/2 = { R r sin ω i Z r cos ω 2 2 }X r{ R r sin ω + i Z r cos ω 2 2 } = X r cos ω + Y r sin ω 2 e iωzr/2 Y r e iωzr/2 = { R r sin ω i Z r cos ω 2 2 }Y r{ R r sin ω + i Z r cos ω 2 2 } = X r sin ω Y r cos ω 3 e iωzr/2 Z r e iωzr/2 = { R r sin ω i Z r cos ω 2 2 }Z r{ R r sin ω + i Z r cos ω 2 2 } = Z r 4 2

3 O-86 X r X r X r Onsager P.34 R r + X r Z r X r Y r { R r + X r } X r cos 2ω + Y r sin 2ω{ R r + X r } = X r cos 2ω + Y r sin 2ω 5 { R r + X r }X r sin 2ω Y r cos 2ω{ R r + X r } = X r sin 2ω + Y r cos 2ω 6 { R r + X r }Z r { R r + X r } = Z r 7 Z r Z r V e i αzr X r cos β + Y r sin βe i αzr = X r cos2α + β + Y r sin2α + β O-85 [cosαz r + i sinαz r ]X r cos β + Y r sin β[cosαz r i sinαz r ] = [ R r + R r cos α + i Z r sin α] X r cos β[ R r + R r cos α i Z r sin α] + [ R r + R r cos α + i Z r sin α]y r sin β[ R r + R r cos α i Z r sin α] 8 R r R r 0 = [X r cos 2 α cos β + Z r X r Z r sin 2 α cos β + i Z r X r sin α cos β cos α i X r Z r cos α cos β sin α] + [Y r cos 2 α sin β + Z r Y r Z r sin 2 α sin β + i Z r Y r sin α sin β cos α i Y r Z r cos α sin β sin α] = [X r cos 2 α cos β X r sin 2 α cos β+, Y r sin α cos β cos α + Y r cos α cos β sin α] + [Y r cos 2 α sin β Y r sin 2 α sin β i X r sin α sin β cos α X r cos α sin β sin α] = X r cos 2 α sin 2 α cos β + 2Y r sin α cos β cos α + Y r cos 2 α sin 2 α sin β + 2X r sin α sin β cos α = X r cos 2α cos β sin 2α sin β + Y r sin 2α cos β + cos 2α sin β = X r cos2α cos β + Y r sin2α + β 9 O-85 2α + β = 0 2α + β = π/2 β = 0 β = π/2 X r Y r X r Y r Z r X r = X r cos rπ n + Y r sin rπ n Y r = X r sin rπ n + Y r cos rπ n O-86- O-86-2 Z r = Z r O-86-3 X 0 = X 0, X n = X n, Y 0 = Y n = 0 20 O-85-2 sinrπ/n O-85- cosrπ/n X r = X r cos rπ n + Y r sin rπ n Y r = X r sin rπ n + Y r cos rπ n 2 22 O-73- [2] n A = X0 2 Xr Xn 23 r= 3

4 O-79-[2] 2 22 n B = X 0 2 X r X n 24 r= n = X0 + 2 r= X r cos rπ n Y r sin rπ n + X n O-84 X r X sx 0 = X 0 X n = X n V = 2 sinh 2H n/2 e 2 H X 0 e H P X r e 2 H X n e H X 0 e 2H P X r e H X n e 2 H X 0 e H P X r e 2 H X n = 2 sinh 2H n/2 e 2 H X0 e 2 H X0 e H X 0 e P H Xr e 2H P X r e P H Xr e H X n e 2 H Xn e 2 H Xn [ n ] = 2 sinh 2H n/2 e H X 0 +H X 0 = 2 sinh 2H n/2 e H H X 0 = 2 sinh 2H n/2 e H H X 0 [ n r= [ n r= r= e H X r n r= n e 2H X r e H X r e 2H X r e H X r r= ] e H X r e H +H X n e H X n H X n U r ] e H +H X n O-87 U r = e H X r e 2H X r e H X r 26 U r U r X r X r U r 4 rπ/n = ω r e H X r = Rr + R r cosh H X r sinh H = R r + R r cosh H X r sinh H 27 e 2H X r = R r + R r cosh 2H X r sinh 2H U r = [ R r + R r cosh H X r sinh H ] = R r + R r [cosh 2H + X r cos ω r sinh 2H Y r sin ω r sinh 2H ] 28 { R r + R r [cosh 2H + X r cos ω r sinh 2H Y r sin ω r sinh 2H ]} [ R r + R r cosh H X r sinh H ] = [ R r + R r cosh H X r sinh H ] 2 R r + [ R r + R r cosh H X r sinh H ] 2 R r cosh 2H + [ R r + R r cosh H Xr sinh H ] 2 R r Xr cos ω sinh 2H [ R r + R r cosh H X r sinh H ]Y r sin ω r sinh 2H [ R r + R r cosh H X r sinh H ] 29 R r 0 4

5 23 = [ R r R r R r cosh H X r sinh H + cosh H X r sinh H 2 R 2 r] R r = [ R r + cosh 2 H 2X r cosh H sinh H + X 2 r sinh2 H R r ] R r = [ R r + R r cosh 2H X r sinh 2H ] R r = R r = [ R r + R r cosh 2H X r sinh 2H ]R r cosh 2H = cosh 2H R r cosh 2H X r sinh 2H = [ R r + R r cosh 2H X r sinh 2H ]R r X r cos ω r sinh 2H = R r cosh 2H X r sinh 2H R r X r cos ω r sinh 2H = X r cos ω r sinh 2H cosh 2H R r cos ω r sinh 2H sinh 2H = [ R r + R r cosh H X r sinh H ]Y r sin ω r sinh 2H cosh H X r sinh H = Y r cosh H + X r sinh H sin ω r sinh 2H cosh H X r sinh H = Y r cosh H + X r sinh H cosh H X r sinh H sin ω r sinh 2H = Y r cosh2 H R r sinh 2 H sin ω r sinh 2H = Y r cosh 2 H Y r sinh 2 H sin ω r sinh 2H = Y r sin ω r sinh 2H U r = R r + cosh 2H R r cosh 2H X r sinh 2H + X r cos ω r sinh 2H cosh 2H R r cos ω r sinh 2H sinh 2H Y r sin ω r sinh 2H = R r + R r cosh 2H cosh 2H sinh 2H sinh 2H cos ω r X r sinh 2H cosh 2H cosh 2H sinh 2H cos ω r Y r sin ω r sinh 2H O-87a U r 26 U r U r U r U r = e H X r e 2H X r e H X r 34 T U r = T U r T 35 T = R r + i Z r 36 5

6 T = R r i Z r 37 T T = [ R r + i Z r ][ R r i Z r ] = R r + Zr 2 = R r + R r = Z r Z r = Z r X r X r Z r e H X r e 2H X r e H X r = e H X r e 2H X r e H X r Zr 39 2 R r e H X r e 2H X r e H X r = 0 40 Z r R r + Z r = R r 4 [ R r + i Z r ]e H X r e 2H X r e H X r [ Rr i Z r ] = e H X r e 2H X r e H X r 42 T O-87a O-87aR r X r Y r D E F [ R r i Z r ][ R r DR r EX r F Y r ] = R r + iz r DR r + iez r X r + if Z r Y r = R r + idz r iex r Z r if Y r Z r 43 [ R r + i Z r ] U r [ R r + idz r iex r Z r if Y r Z r ][ R r + i Z r ] = R r DZ 2 r + EX r Z2 r + F Y r Z2 r = R r DR r + EX r + F Y r 44 X r Y r U r O-87a O-89a U r + U r = 2 R r 2sinh 2H cosh 2H cosh 2H sinh 2H cos ω r = 2 R r 2R r cosh γ r 45 cosh γ r 2H 2H ω r [3, 4] OD D ω δ δ γ 2H 2H cosh γ = cosh 2H cosh 2H sinh 2H sinh 2H cos ω O-89a cosh 2H = cosh γ cosh 2H sinh γ sinh 2H cos δ 46 cosh 2H = cosh γ cosh 2H sinh γ sinh 2H cos δ 47 2 Z rx r = X rz r Z rxr = X r Zr Z re H Xr X = Z r n! H Xr n = X n! H Xr n n Z r = X n! H Xr n Z r 6

7 O : O-89a 46 sin ω sinh γ = sin δ sin δ = sinh 2H sinh 2H O-89c sinh γ cos δ = sinh 2H cosh 2H cosh 2H sinh 2H cos ω O-89b O-89c O-89b cot δ = sinh 2H coth 2H cosh 2H cos ω sin ω O-89d 45U + U R r R r = X 2 r 0R r U + U R r χ = χ χ 45 U + U χ = e γr + e γr χ 48 U U U U U e γr e γr R r 0 U + U χ = 2χ 49 U U r R r = X 2 r = X 2 r X r X r U r e γrxr e γrx r X r X r O U r O-87a O-89a O-89b O-89c U r = R r + R r cosh γ r X r sinh γ r cos δ r Y r sinh γ r sin δ r 50 = R r + R r cosh γ r X r cos δ r + Y r sin δ r sinh γ r X r cos δ r + Y r sin δ r O-86 X r+δ X r cos δ + Y r sin δ r = X r cos ω r + Y r sin ω r cos δ r + X r sin ω r + Y r cos ω r sin δ r = X r cosω r + δ r + Y r sinω r + δ r X r+δ 5 7

8 X r+δ X 2 r+δ X r R r 50 Xr+δ = X r cos δ r + Y r sin δ r 2 = Xr 2 R r cosh γ r cos2 δr + Y r 2 sin 2 δr = R r cos 2 δ r + R r sin 2 δ r = R r 52 U r = R r + R r cosh γ r X r+δ sinh γ r 53 R r cosh γ r = X 2 r+δ r + 2! γ2 r + 4! γ2 r + = Xr+δ 2 + 2! γ2 r X 2 r+δ + 4! γ2 r X 4 r+δ r + = X 2 r+δ + X r+δ sinh γ r [ 2! γ2 r X 2 r+δ + 4! γ2 r X 4 r+δ = R r + coshγ r X r+δ 54 ] X r+δ sinh γ r = X r+δγ r + 3! γ3 r + 5! γ5 r + = γ r X r+δ + 3! γ3 r X r+δ + 5! γ5 r X r+δ + = γ r X r+δ + 3! γ3 r X 3 r+δ + 5! γ5 r X 5 r+δ + = sinhγ r X r+δ 55 U r = exp[ γ r X r+δ ] = exp[ γ r X r cos δ r + Y r sin δ r] O-88 U r X r O-88 Y r X r X r O-85 O-85 e i αzr γx r cos β + Y r sin βe i αzr = γ[x r cos2α + β + Y r sin2α + β] 56 γ n X r cos β + Y r sin β n i αzr e = {e i αzr γx r cos β + Y r sin βe i αzr } n = γ n [X r cos2α + β + Y r sin2α + β] n 57 e i αzr exp[ γx r cos β + Y r sin β]e i αzr = exp{ γ[x r cos2α + β + Y r sin2α + β]} 58 X r cos2α + β + Y r sin2α + β 2 22 X r cos2α + β + Y r sin2α + β = X r cos ω r + Y r sin ω r cos2α + β + X r sin ω r + Y r cos ω r sin2α + β = X r cosω r + 2α + β + Y r sinω r + 2α + β = X r cosπ ω r 2α β + Y r sinπ ω r 2α β 59 8

9 58 e i αzr i αzr exp[ γx r cos β +Y r sin β]e = exp{ γ[xr cosπ ω r 2α β+yr sinπ ω r 2α β]} 60 O-88 U r π ω r 2α β = δ r 6 e i αzr i αzr exp[ γx r cos β + Y r sin β]e = exp[ γxr cos δr + Yr sin δr] 62 β = 0 γx r β = π ω r γx r β = 0 α = 2 π ω r δ r e 2 π ωr δ r izr e γrxr e 2 π ωr δ r izr β = π ω r α = 2 δ r = exp[ γxr cos δ r + Y r sin δ r ] 63 e 2 δ r izr e γrx r e 2 δ r izr = exp[ γxr cos δ r + Y r sin δ r ] Y r U r U r exp[ 2 π ω r δ r iz r]u r exp[ 2 π ω r δ r iz r] = exp γ r X r exp 2 δ riz r U r exp 2 δ riz r = exp γ r X r O-90b O-90a rv O-87O-90a O-90br = n n n exp 2 π ω r δr iz r V exp 2 π ω r δr iz r r= X r = 2 sinh 2H n/2 e H H X 0 = 2 sinh 2H n/2 e H H X 0 [ n r= [ n r= r= e 2 π ωr δ r izr U r e 2 π ωr δ r izr ] e H +H X n e γrxr ] e H +H X n n = 2 sinh 2H n/2 exp H H X 0 γ r X r H + H X n r= O-9b exp n r= 2 δ riz r n V exp r= = 2 sinh 2H n/2 e H H X 0 = 2 sinh 2H n/2 e H H X 0 2 δ riz r [ n r= [ n r= e 2 δ r izr U r e 2 δ r izr ] e H +H X n e γrx r ] e H +H X n n = 2 sinh 2H n/2 exp H H X0 γ r Xr H + H X n r= O-9a 9

10 V X r X r V O-9a O-9b X r X r ξ r X r χ λ V O-9b V S 3 S V S, χ = λχ 65 ln λ = n 2 n ln2 sinh 2H + H H ξ 0 γ r ξ r H + H ξ n 66 S = exp r= n 2 π ω r δr iz r r= [2] V λ max X r χ 0 r ξ r = 0r ξ r = n ω = nπ/n = π O-89a 67 cosh γ r = cosh 2H cosh 2H sinh H sinh H cos rπ n 68 cosh γ n = cosh 2H cosh 2H sinh H sinh H cos ω n = cosh 2H cosh 2H + sinh 2H sinh 2H = cosh 2H + H 69 γ n+r = γ n r 70 γ n = 2H + H 7 λ max n ln2 sinh 2H = 2 γ + γ γ 2m n = 2m γ + γ γ 2m + H + H = γ + γ γ 2m + 2 γ n n = 2m + O-93 3 V λ ψ V, ψ = λψ A O-9a O-9b X r X r V S X r X r χ S V S, χ = λχ B O-9b AS B S = e 2 π ωr δ r izr S, V, ψ = S, λψ S V SS, ψ = λs, ψ S V S, S, ψ = λs, ψ χ = S, ψ ψ = S, χ = e 2 π ωr δ r izr, χ ψ = S, χ 0

11 64 n n γ + γ γ 2n = 2γ + γ γ n 2 + γ n = 2γ + γ γ n 2 + 2H + H 72 γ + γ γ 2n = 2γ + γ γ n 73 n λ max 2 n ln2 sinh 2H = 2 n r= γ 2r γ 2r 62 λ max 2 n ln2 sinh 2H = 2 <2r<n cosh cosh 2H cosh 2H sinh H sinh H cos O-93-2r π O-93-2 n λ max B χ 0 ψ max = exp 2 π ω 2r δ2r iz 2r, χ 0 O-96a 4 <2r<n λ max A χ 0 n ψ max = exp, χ 0 r= 2 δ 2r iz 2r O-96b * * * * * 2 O-89acosh 2H / sinh 2H cosh 2H sinh 2H cos ω = cosh 2H cosh 2H cosh 2H cosh 2H cosh γ sinh 2H sinh 2H 2 cosh cosh 2H 46 cosh 2H sinh 2H cos ω = cosh 2H cosh 2H cosh 2H sinh 2H cosh 2H + sinh γ sinh 2H cos δ sinh 2H cosh 2H = sinh 2H cosh2 2H sinh γ cos δ cosh 2H = sinh 2H sinh 2 2H sinh γ cos δ A- = cosh 2H sinh 2H sinh γ cos δ 74 4 S

12 sinh γ cos δ = sinh 2H cosh 2H cosh 2H sinh 2H cos ω O-89b O-89c O-89b sinh 2H sin δ = sinh γ sin ω sinh γ sinh 2H cot δ = sinh 2H cosh 2H cosh 2H sinh 2H cos ω sinh γ sin ω cot δ = sinh 2H coth 2H cosh 2H cos ω sin ω O-89d [] 7 207//5 paper7.pdf [2] 8 207/2/2 paper4.pdf [3] 2 207/2/0 geo 2.pdf [4] 207/0/6 geo S.pdf 2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh