Quantum dynamics with non-hermitian PT -symmetric operators: Models

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1 Hauptseminar Theoretische Physik 01 Quantum dynamics with non-hermitian PT -symmetric operators: Models Mario Schwartz Mario Schwartz PT -Symmetric Operators: Models 1 / 36

2 Overview Hauptseminar Theoretische Physik 01 1 Foundations Model one: PT -symmetric x Matrix 3 Model two: Delta peaks in an infinitely high potential well 4 Model three: Generalisation of the Harmonic Oscillator 5 Summary Mario Schwartz PT -Symmetric Operators: Models / 36

3 Hermitian Operators Foundations Hauptseminar Theoretische Physik 01 As yet The Hamiltonian H is Hermitian by definition. because to achieve but Expectation value of Energy E is real. E = ψ H ψ ψ ψ = ψ H ψ ψ ψ = ψ H ψ ψ ψ = E There are other, non-hermitian Hamiltonians H in physics which have real eigenvalues. Real eigenvalues are the foundation of these Hamiltonians. Mario Schwartz PT -Symmetric Operators: Models 3 / 36

4 Hermitian Operators Foundations Hauptseminar Theoretische Physik 01 As yet The Hamiltonian H is Hermitian by definition. because to achieve but Expectation value of Energy E is real. E = ψ H ψ ψ ψ = ψ H ψ ψ ψ = ψ H ψ ψ ψ = E There are other, non-hermitian Hamiltonians H in physics which have real eigenvalues. Real eigenvalues are the foundation of these Hamiltonians. Mario Schwartz PT -Symmetric Operators: Models 3 / 36

5 PT -symmetric operators Foundations Hauptseminar Theoretische Physik 01 New requirement for the Hamiltonian H: It needs to be symmetric with respect to the combination of spatial reflection P and time reversal T, which means [PT, H] = 0 Affect of these operators: P : ˆp ˆp, ˆx ˆx Pˆp = ˆpP, P ˆx = ˆxP T : ˆp ˆp, ˆx ˆx, i i T ˆp = ˆpT, T ˆx = ˆxT, T i = -it Mario Schwartz PT -Symmetric Operators: Models 4 / 36

6 PT -symmetric operators Foundations Hauptseminar Theoretische Physik 01 Partial proof: ( ) ( ) 1D=P 1. P ˆpψ(x) -i x ψ(x) = -i x ψ(x) x=-x ( ) ( ) 1D=-i x ˆp Pψ(x) Pψ(x) = -i x ψ( x) = i x ψ(x) x=-x ( ) ( ) P ˆpψ(x) = ˆp Pψ(x). T ( ) ( ) ( ) 1D= ˆpψ(x) T -i x ψ(x) = i x ψ(x) = -ˆp T ψ(x) Mario Schwartz PT -Symmetric Operators: Models 5 / 36

7 PT -symmetric operators Foundations Hauptseminar Theoretische Physik 01 But that is not enough for real energies T is an antilinear operator (So especially not linear: T (λψ) = λt (ψ)) Contrary to linear operators, [PT, H] = 0 is not sufficient to have for PT and H the same eigenvectors We also demand that PT and H have the same eigenvectors and claim that the eigenvalues are real then. From this it follows ( ) ( ( )! PT H ψ = PT E ψ = H PT ψ ) = E PT ψ [PT, H] = 0, and same eigenvectors: Only [PT, H] = 0 : unbroken PT -symmetry broken PT -symmetry Mario Schwartz PT -Symmetric Operators: Models 6 / 36

8 PT -symmetric operators Foundations Hauptseminar Theoretische Physik 01 But that is not enough for real energies T is an antilinear operator (So especially not linear: T (λψ) = λt (ψ)) Contrary to linear operators, [PT, H] = 0 is not sufficient to have for PT and H the same eigenvectors We also demand that PT and H have the same eigenvectors and claim that the eigenvalues are real then. From this it follows ( ) ( ( )! PT H ψ = PT E ψ = H PT ψ ) = E PT ψ [PT, H] = 0, and same eigenvectors: Only [PT, H] = 0 : unbroken PT -symmetry broken PT -symmetry Mario Schwartz PT -Symmetric Operators: Models 6 / 36

9 Foundations Hauptseminar Theoretische Physik 01 Properties of H, PT and eigenfunctions H is unbroken PT -symmetric Real Energies, is only a conjecture and can t be shown rigorously. (PT ) = 1 Eigenvalues of PT : λ = e iω (PT ψ = PT λ ψ = λλ ψ! = ψ ) A new inner product is needed for orthogonal eigenvectors (ψ m, ψ n ) := C dx[pt ψ m(x)]ψ n (x) = ( 1) n δ nm }{{} For unbroken PT symmetry For orthonormal eigenvectors a new linear operator C must be constructed with the property to create a new inner product ψ m, ψ n := C dx[cpt ψ m(x)]ψ n (x) = δ }{{ nm } For unbroken PT symmetry Mario Schwartz PT -Symmetric Operators: Models 7 / 36

10 Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 Construct a PT -symmetric x Matrix [1, ] Consider a particle, in a space consisting of only two points Particle onto -1 In state -1 Particle onto 1 In state 1 1 x Mario Schwartz PT -Symmetric Operators: Models 8 / 36

11 Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 The shape of the Hamiltonian Given Hamiltonian H, eigenfunc. : ψ = c c, = 1 ( ) ( ) H 1 H 1 1 H a b = := H 1 H c d ( ) 0 1 P =, T : complex conjugation 1 0 Demands on H ( ) ( ) ( 0 1 a b c d PT H ψ = T ψ = 1 0 c d a b ( ) ( ) ( a b 0 1 HPT ψ = ψ b a = c d 1 0 d c d = a, c = b ) ψ! = ) ψ Mario Schwartz PT -Symmetric Operators: Models 9 / 36

12 Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 The shape of the Hamiltonian Given Hamiltonian H, eigenfunc. : ψ = c c, = 1 ( ) ( ) H 1 H 1 1 H a b = := H 1 H c d ( ) 0 1 P =, T : complex conjugation 1 0 Demands on H ( ) ( ) ( 0 1 a b c d PT H ψ = T ψ = 1 0 c d a b ( ) ( ) ( a b 0 1 HPT ψ = ψ b a = c d 1 0 d c d = a, c = b ) ψ! = ) ψ Mario Schwartz PT -Symmetric Operators: Models 9 / 36

13 Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 The shape of the Hamiltonian Generally the shape of the Hamiltonian must be ( ) a b H = (not necessarily Hermitian) b a Calculation of the eigenenergies E det(h E1) = 0 E = Re(a) ± b Im(a) For Hermitian Hamiltonian (Im(a) = 0) eigenenergies are real. For b > Im(a) eigenenergies are also real. Unbroken PT -symmetry For lim b Im(a) + (E) two real eigenvalues melt together to one. This is an exceptional point (proof later). For For b < Im(a) eigenenergies are complex. Broken PT -symmetry Mario Schwartz PT -Symmetric Operators: Models 10 / 36

14 Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 The shape of the Hamiltonian Generally the shape of the Hamiltonian must be ( ) a b H = (not necessarily Hermitian) b a Calculation of the eigenenergies E det(h E1) = 0 E = Re(a) ± b Im(a) For Hermitian Hamiltonian (Im(a) = 0) eigenenergies are real. For b > Im(a) eigenenergies are also real. Unbroken PT -symmetry For lim b Im(a) + (E) two real eigenvalues melt together to one. This is an exceptional point (proof later). For For b < Im(a) eigenenergies are complex. Broken PT -symmetry Mario Schwartz PT -Symmetric Operators: Models 10 / 36

15 Exampels Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 For example the unbroken ( > i ) PT -symmetric and non-hermitian Hamiltonian ( ) i H = i has the eigenenergies and eigenvectors E 1 = 3, E = ( ) ( ) i 3 i + 3 3, ψ 1 =, ψ = They are simultaneously eigenfunctions of PT. For ψ 1 : PT ( ) i 3 = ( ) -i = 3 ( 1 3 i + ) ( ) i 3 Mario Schwartz PT -Symmetric Operators: Models 11 / 36

16 Exampels Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 For example the unbroken ( > i ) PT -symmetric and non-hermitian Hamiltonian ( ) i H = i has the eigenenergies and eigenvectors E 1 = 3, E = ( ) ( ) i 3 i + 3 3, ψ 1 =, ψ = They are simultaneously eigenfunctions of PT. For ψ 1 : PT ( ) i 3 = ( ) -i = 3 ( 1 3 i + ) ( ) i 3 Mario Schwartz PT -Symmetric Operators: Models 11 / 36

17 Exampels Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 But the eigenfunctions are not orthogonal with the ordinarily inner product ψ 1 ψ = ( i 3, ) ( ) i + 3 = 3i. The inner product (ψ 1, ψ ) = PT ( i 3, ) ( ) i + 3 = (, i 3 ) ( ) i + 3 = 0 has orthogonal eigenfunctions but it is indefinite because (ψ 1, ψ 1 ) = (, i 3 ) ( ) i 3 = 4 3 < 0 Mario Schwartz PT -Symmetric Operators: Models 1 / 36

18 Exampels Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 But the eigenfunctions are not orthogonal with the ordinarily inner product ψ 1 ψ = ( i 3, ) ( ) i + 3 = 3i. The inner product (ψ 1, ψ ) = PT ( i 3, ) ( ) i + 3 = (, i 3 ) ( ) i + 3 = 0 has orthogonal eigenfunctions but it is indefinite because (ψ 1, ψ 1 ) = (, i 3 ) ( ) i 3 = 4 3 < 0 Mario Schwartz PT -Symmetric Operators: Models 1 / 36

19 Exampels Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 So a new inner product must be constructed ψ 1, ψ = CPT (ψ 1 ) ψ := ψ 1 ψ it s close to the ordinarily inner product but without complex conjugation. We get ψ 1, ψ = ( i 3, ) ( ) i + 3 = 0 ψ 1, ψ 1 = ( i 3, ) ( ) i 3 = 6 3i ψ 1 = ( ) 1 i 3 6 3i ψ 1, ψ 1 = 1 Mario Schwartz PT -Symmetric Operators: Models 13 / 36

20 Exampels Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 So a new inner product must be constructed ψ 1, ψ = CPT (ψ 1 ) ψ := ψ 1 ψ it s close to the ordinarily inner product but without complex conjugation. We get ψ 1, ψ = ( i 3, ) ( ) i + 3 = 0 ψ 1, ψ 1 = ( i 3, ) ( ) i 3 = 6 3i ψ 1 = ( ) 1 i 3 6 3i ψ 1, ψ 1 = 1 Mario Schwartz PT -Symmetric Operators: Models 13 / 36

21 Exampels Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 For example the broken ( 1 < i ) PT -symmetric Hamiltonian ( ) i 1 H = 1 i has the eigenenergies and eigenvectors E 1 = 3i, E = ( ) ( ) ( + 3)i -(- + 3)i 3i, ψ 1 =, ψ 1 = 1 They are not simultaneously eigenfunctions of PT. For ψ 1 : PT ( ) ( ) ( ) ( + 3)i 1 = 1 ( + ( + 3)i c 3)i 1 Mario Schwartz PT -Symmetric Operators: Models 14 / 36

22 General case Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 For general H with real b the simultaneous eigenfunctions of PT and H are given by ( ) Im(a)i b ψ 1 = Im(a) b ( ) Im(a)i + b ψ = Im(a) b For Im(a) = b we have already seen that the two eigenenergies melt together. Here we see also ( ) ±bi ψ 1 = ψ =, ψ b 1, ψ 1 = ( ±bi, b ) ( ) ±bi = 0 b All conditions for an exceptional point are fulfilled. Mario Schwartz PT -Symmetric Operators: Models 15 / 36

23 General case Model one: PT -symmetric x Matrix Hauptseminar Theoretische Physik 01 For general H with real b the simultaneous eigenfunctions of PT and H are given by ( ) Im(a)i b ψ 1 = Im(a) b ( ) Im(a)i + b ψ = Im(a) b For Im(a) = b we have already seen that the two eigenenergies melt together. Here we see also ( ) ±bi ψ 1 = ψ =, ψ b 1, ψ 1 = ( ±bi, b ) ( ) ±bi = 0 b All conditions for an exceptional point are fulfilled. Mario Schwartz PT -Symmetric Operators: Models 15 / 36

24 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Delta peaks in an infinitely high potential well [3, 4] Consider two delta peaks in an infinitely high potential well with imaginary prefactors. The corresponding time-independent Schrödinger equation reads Hψ = ( iξδ(x + a) + iξδ(x a))ψ = Eψ iξ (L) (C) (R) -1 -a a 1 x -iξ Mario Schwartz PT -Symmetric Operators: Models 16 / 36

25 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Necessary condition of H to have PT -symmetry Given Hamiltonian with complex potential H = ˆp ˆp + V (x) = m m + V R(x) + iv I (x) Application of PT -operator ( ˆp ) PT H ψ = m + V R( x) ivi ( x) PT ψ (! ˆp ) = HPT ψ = m + V R(x) + iv I (x) PT ψ A PT -symmetric Hamiltonian requires V R (x) = V R ( x), V i (x) = V i ( x) Mario Schwartz PT -Symmetric Operators: Models 17 / 36

26 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Necessary condition of H to have PT -symmetry Given Hamiltonian with complex potential H = ˆp ˆp + V (x) = m m + V R(x) + iv I (x) Application of PT -operator ( ˆp ) PT H ψ = m + V R( x) ivi ( x) PT ψ (! ˆp ) = HPT ψ = m + V R(x) + iv I (x) PT ψ A PT -symmetric Hamiltonian requires V R (x) = V R ( x), V i (x) = V i ( x) Mario Schwartz PT -Symmetric Operators: Models 17 / 36

27 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 The Hamiltonian is PT -symmetric The Hamiltonian Hψ = ( iξδ(x + a) + iξδ(x a))ψ = Eψ is PT -symmetric because V R (x) = V R ( x), V i (x) = V i ( x). The wave function takes over the symmetry of the potential when the eigenenergies are real ψ(x) = ψ S (x) + iψ A (x), ψ S (x) = ψ S ( x), ψ A (x) = ψ A ( x) Mario Schwartz PT -Symmetric Operators: Models 18 / 36

28 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Proof The real and the imaginary part of the differential equation ( p ) m + V R(x) + iv i (x) can be solved separately. The real part is ( ) ( ) ψ S (x) + iψ A (x) = E ψ S (x) + iψ A (x) ( p ) m + V R(x) ψ S (x) V i (x)ψ A (x) = Eψ S (x). By changing the variable x to x we get with the symmetry of the potential ( p ) m + V R(x) ψ S ( x) V i (x) ( ) ψ A ( x) = Eψ S ( x) For given starting conditions the solution must be unique. ψ S (x) = ψ S ( x), ψ A (x) = ψ A ( x) Mario Schwartz PT -Symmetric Operators: Models 19 / 36

29 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Proof The real and the imaginary part of the differential equation ( p ) m + V R(x) + iv i (x) can be solved separately. The real part is ( ) ( ) ψ S (x) + iψ A (x) = E ψ S (x) + iψ A (x) ( p ) m + V R(x) ψ S (x) V i (x)ψ A (x) = Eψ S (x). By changing the variable x to x we get with the symmetry of the potential ( p ) m + V R(x) ψ S ( x) V i (x) ( ) ψ A ( x) = Eψ S ( x) For given starting conditions the solution must be unique. ψ S (x) = ψ S ( x), ψ A (x) = ψ A ( x) Mario Schwartz PT -Symmetric Operators: Models 19 / 36

30 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Ansatz for the wave function With E = κ and real κ (unbroken PT -symmetry) ψ L (x) = (α iβ) sin(κ(x + 1)) x ( 1, a) ψ(x) = ψ C (x) = γ cos(κx) + iδ sin(κx) x ( a, a) ψ R (x) = (α + iβ) sin(κ( x + 1)) x (a, 1) With the derivative ψ ψ L (x) = κ(α iβ) cos(κ(x + 1)) x ( 1, a) (x) = ψ C (x) = κγ sin(κx) + iκδ cos(κx) x ( a, a) (x) = κ(α + iβ) cos(κ( x + 1)) x (a, 1) ψ R Mario Schwartz PT -Symmetric Operators: Models 0 / 36

31 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Boundary conditions Continuity of the wave function ψ L ( a) = ψ C ( a), ψ C (a) = ψ R (a) Integrate within a small area interval (±a η, ±a + η) ±a+η ±a η d ±a+η ψ(x) + (-iξδ(x + a) + iξδ(x a))ψ(x) dx = Eψ(x) dx dx ±a η d dx ψ(±a + 0) d ψ(±a 0) = iξψ(±a) dx Mario Schwartz PT -Symmetric Operators: Models 1 / 36

32 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Solve the model The boundary conditions lead to a 4 dimensional homogeneous system of eigenequations sin(κ(1 a)) 0 - cos(κa) 0 α 0 sin(κ(1 a)) 0 - sin(κa) β ξ - cos(κ(1 a)) κ sin(κ(1 a)) sin(κa) 0 γ = 0 sin(κ(1 a)) cos(κ(1 a)) 0 cos(κa) δ ξ κ Mario Schwartz PT -Symmetric Operators: Models / 36

33 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Solve the model Nontrivial solution only if secular determinant D(κ) vanishes D(κ) = 1 (sin(κ) ξ ) κ sin(κ) sin (κ(1 a)) = 0 κ ξ = sin(κ) sin(aκ) sin (κ (1 a)) This is a transcendental equation (Can t be solved for κ analytical). When ξ(κ) is plotted the solutions can be found. Mario Schwartz PT -Symmetric Operators: Models 3 / 36

34 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 First Plot Fig: The function ξ = ξ(κ) for a = /3. Robust and fragile types of spectral lines are present, ξ = 5t, κ = jπ. [3] Mario Schwartz PT -Symmetric Operators: Models 4 / 36

35 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 First Plot Mario Schwartz PT -Symmetric Operators: Models 5 / 36

36 Model two: Delta peaks in an infinitely high potential well Hauptseminar Theoretische Physik 01 Second Plot Fig: Overstepping complexification of energy levels for a = 1/11, ξ = 8t, κ = jπ. [3] Mario Schwartz PT -Symmetric Operators: Models 6 / 36

37 Model three: Generalisation of the Harmonic Oscillator Hauptseminar Theoretische Physik 01 Generalisation of the Harmonic Oscillator [5] The class of complex Hamiltonians H = p (ix) N (N real) is a generalisation of the harmonic oscillator with = 1 ω = 1 m = 1. With N = we get the normal harmonic oscillator potential H = p + x, E n = n + 1 In the general case for N the following Schrödinger equation must be solved ψ (x) (ix) N ψ(x) = Eψ(x) Solving can only be done numerically or by approximation. Mario Schwartz PT -Symmetric Operators: Models 7 / 36

38 Model three: Generalisation of the Harmonic Oscillator Hauptseminar Theoretische Physik 01 Generalisation of the Harmonic Oscillator [5] The class of complex Hamiltonians H = p (ix) N (N real) is a generalisation of the harmonic oscillator with = 1 ω = 1 m = 1. With N = we get the normal harmonic oscillator potential H = p + x, E n = n + 1 In the general case for N the following Schrödinger equation must be solved ψ (x) (ix) N ψ(x) = Eψ(x) Solving can only be done numerically or by approximation. Mario Schwartz PT -Symmetric Operators: Models 7 / 36

39 Model three: Generalisation of the Harmonic Oscillator Hauptseminar Theoretische Physik 01 General harmonic oscillator Fig: Energy levels of the Hamiltonian H = p (ix) N as a function of the parameter N. [5] There are three regions N : Spectrum is real and positiv. N = corresponds to the harmonic oscillator. 1 < N < : A finite number of real positiv eigenvalues and a infinite number of complex conjugate pairs of eigenvalues. N 1: There are no real eigenvalues. Mario Schwartz PT -Symmetric Operators: Models 8 / 36

40 Model three: Generalisation of the Harmonic Oscillator Hauptseminar Theoretische Physik 01 Importent remarks We can write the class of Hamiltonians also as H = p (ix) N = p + x (ix)ñ For Ñ 0 the PT -symmetry is spontaneously broken, which can also considered as equivalent to a phase transition. The boundery conditions that ψ(x) 0 as x only suffices when 1 < N < 4 (only then quantized energylevels E). The eigenvalue problem must be continued into the complex x plane Mario Schwartz PT -Symmetric Operators: Models 9 / 36

41 Model three: Generalisation of the Harmonic Oscillator Hauptseminar Theoretische Physik 01 Semiclassical approximation Classically we can write p(x) = m(e V (x)) Bohr-Sommerfeld quantisation for oscillations ( dx p(x) = π n + 1 ) x+ ( dx p(x) = π n + 1 ) x ± are the turning points where E + (ix) N = 0 x + = E 1/N e iπ(3/ 1/N), x = E 1/N e iπ(1/ 1/N) In this case we get ( π n + 1 ) = x+ x x dx E + (ix) N Also known as: Leading-order WKB phase-integral quantization condition. Mario Schwartz PT -Symmetric Operators: Models 30 / 36

42 Model three: Generalisation of the Harmonic Oscillator Hauptseminar Theoretische Physik 01 Semiclassical approximation Classically we can write p(x) = m(e V (x)) Bohr-Sommerfeld quantisation for oscillations ( dx p(x) = π n + 1 ) x+ ( dx p(x) = π n + 1 ) x ± are the turning points where E + (ix) N = 0 x + = E 1/N e iπ(3/ 1/N), x = E 1/N e iπ(1/ 1/N) In this case we get ( π n + 1 ) = x+ x x dx E + (ix) N Also known as: Leading-order WKB phase-integral quantization condition. Mario Schwartz PT -Symmetric Operators: Models 30 / 36

43 Model three: Generalisation of the Harmonic Oscillator Hauptseminar Theoretische Physik 01 Semiclassical approximation For N this is equivalent to the integral ( π n + 1 ) 1 = sin(π/n)e 1/N+1/ 0 ds 1 + s N with the solution (mathematica) [ ( E n = π n + 1 ) Γ(3/ + 1/N) π(n + 1/) sin(π/n)γ(1 + 1/N) ] N/N+ Fig: Comparison of the exact eigenvalues (obtained with the Runge-Kutta method) and the WKB result. [5] Mario Schwartz PT -Symmetric Operators: Models 31 / 36

44 Summary Summary Hauptseminar Theoretische Physik 01 New class of Hamiltonians: PT -symmetric Hamiltonians In general non-hermitian. [PT, H] = 0, and same eigenvectors means unbroken PT -symmetry Only real eigenvalues. Only [PT, H] = 0 means broken PT -symmetry Not only real eigenvalues. For orthonormal eigenvectors a new inner product is needed. A particle, in a space consisting of only two points General description with a PT -symmetric x Matrix ( ) a b H = (not necessarily Hermitian) b a All expected properties were proofed. Mario Schwartz PT -Symmetric Operators: Models 3 / 36

45 Summary Summary Hauptseminar Theoretische Physik 01 Delta peaks in an infinitely high potential well Necessary condition of H to have PT -symmetry: V R (x) = V R ( x), Vi (x) = Vi ( x) For real eigenenergies the wave function takes over the symmetry of the potential. Well-known solving method. Exact solvable except for a transcendental equation. Generalisation of the Harmonic Oscillator H = p (ix) N (N real) In general solving can only be done numerically or by approximation. Approximation with Bohr-Sommerfeld quantisation ( ) dx p(x) = π n + 1 Mario Schwartz PT -Symmetric Operators: Models 33 / 36

46 Thank you Hauptseminar Theoretische Physik 01 Danke für eure Aufmerksamkeit Mario Schwartz PT -Symmetric Operators: Models 34 / 36

47 References Hauptseminar Theoretische Physik 01 [1] Carl M. Bender, Dorje C. Brody, and Hugh F. Jones. Complex extension of quantum mechanics. Phys. Rev. Lett., 89:70401, Dec 00. [] Carl M Bender, M V Berry, and Aikaterini Mandilara. Generalized pt symmetry and real spectra. Journal of Physics A: Mathematical and General, 35(31):L467, 00. [3] Vit Jakubsky and Miloslav Znojil. An explicitly solvable model of the spontaneous PT -symmetry breaking. Czechoslovak Journal of Physics, 55: , /s x. [4] Miloslav Znojil. Solvable simulation of a double-well problem in PT -symmetric quantum mechanics. Journal of Physics A: Mathematical and General, 36(7):7639. Mario Schwartz PT -Symmetric Operators: Models 35 / 36

48 References Hauptseminar Theoretische Physik 01 [5] Carl M. Bender and Stefan Boettcher. Real spectra in non-hermitian hamiltonians having PT symmetry. Phys. Rev. Lett., 80: , Jun Mario Schwartz PT -Symmetric Operators: Models 36 / 36