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Mathematical and Theoretical Physics

Short Communication Volume 2 Issue 1

Pseudo-hermiticity versus PT symmetry: spectral breakdown in nearly degenerate quantum energy eigenvalues

Biswanath Rath

Department of Physics, North Orissa University, India

Correspondence:

Received: July 26, 2018 | Published: January 14, 2019

Citation: 10.15406/oajmtp.2019.02.00046 DOI: 10.15406/oajmtp.2019.02.00046

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Abstract

It is interesting to note that pseudo-hermiticity condition fails to preserve spectral invariant nature in nearly degenerate quantum states of single and double well operators. In other words no spectral similarity exists between nearly degenerate energy eigenvalues of Hermitian operator and corresponding PT-symmetry operator.

Keywords: anti–PT–symmetry, pseudo-hermiticity condition, spectral variance, hermitian operator, nearly degenerate eigenvalues

Introduction

Quantum postulates are based on self-adjoint operators satisfying the condition

H= H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaacaWGibGaeyypa0 JaamisamaaCaaaleqabaGaaiiiGaaaaaa@3B61@ (1)

In terms of energy eigenvalues E n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadweapaWaaSbaaSqaa8qacaWGUbaapaqabaaaaa@3A07@  and corresponding eigenfunction one can write the above relation as1

H= n E n | ψ n >< ψ n | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeacqGH9aqpdaaeqbqaaiaadweapaWaaSbaaSqaa8qacaWGUbaa paqabaaapeqaaiaad6gaaeqaniabggHiLdGcdaabdaqaaiabeI8a5n aaBaaaleaacaWGUbaabeaakiabg6da+iabgYda8iabeI8a5naaBaaa leaacaWGUbaabeaaaOGaay5bSlaawIa7aaaa@4A1B@  (2)

The above relation assumes the well behaved nature of wave function

| ψ n >= ψ n 0(x±) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaadaabbaqaaiabeI 8a5naaBaaaleaacaWGUbaabeaakiabg6da+iabg2da9aGaay5bSdGa eqiYdK3aaSbaaSqaaiaad6gaaeqaaOGaeyOKH4QaaGimaiaacIcaca GG4bGaeyOKH4QaeyySaeRaeyOhIuQaaiykaaaa@4BA8@  (3)

As far as the author knows, almost all Hermitian operators satisfy this condition. Suppose the wave functions and corresponding operator can be transformed as

< ψ n | H | ψ n >=< ϕ n | h | ϕ n > MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaacqGH8aapcqaHip qEdaWgaaWcbaGaamOBaaqabaGcdaabdaqaaiaadIeaaiaawEa7caGL iWoacqaHipqEdaWgaaWcbaGaamOBaaqabaGccqGH+aGpcqGH9aqpcq GH8aapcqaHvpGzdaWgaaWcbaGaamOBaaqabaGcdaabdaqaaiaadIga aiaawEa7caGLiWoacqaHvpGzdaWgaaWcbaGaamOBaaqabaGccqGH+a Gpaaa@50BC@ (4)

then it is implied that | ϕ n >= S 1 | ψ n > MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaadaabbaqaaiabew 9aMnaaBaaaleaacaWGUbaabeaakiabg6da+aGaay5bSdGaeyypa0Ja am4uamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaeeaabaaeaaaaaa aaa8qacqaHipqEdaWgaaWcbaGaamOBaaqabaaak8aacaGLhWoacqGH +aGpaaa@46DC@         (5), < ϕ n |=< ψ n |S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaacqGH8aapdaabca qaaabaaaaaaaaapeGaeqy1dy2aaSbaaSqaaiaad6gaaeqaaaGcpaGa ayjcSdGaeyypa0JaeyipaWZdbmaaeiaabaGaeqiYdK3aaSbaaSqaai aad6gaaeqaaaGccaGLiWoacaWGtbaaaa@4509@   (6), h= S 1 HS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIgacqGH9aqpcaWGtbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGa amisaiaadofaaaa@3E3F@        (7)

Now question arises as to if (i)h = hermitian or h = non–hermitian.2 If h is hermitian then both < ϕ n |and| ϕ n > MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaacqGH8aapdaabca qaaabaaaaaaaaapeGaeqy1dy2aaSbaaSqaaiaad6gaaeqaaaGcpaGa ayjcSdGaamyyaiaad6gacaWGKbWaaqqaaeaapeGaeqy1dy2aaSbaaS qaaiaad6gaaeqaaOGaeyOpa4dapaGaay5bSdaaaa@45F8@ are well behaved like that of | ψ n >or< ψ n |. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaadaabbaqaaabaaa aaaaaapeGaeqiYdK3aaSbaaSqaaiaad6gaaeqaaaGcpaGaay5bSdGa eyOpa4Jaam4BaiaadkhacqGH8aapdaabcaqaa8qacqaHipqEdaWgaa WcbaGaamOBaaqabaaak8aacaGLiWoacaGGUaaaaa@45DF@ In other words spectral invariance exists between h and H. This may be a case of spectral invariance in non-degenerate quantum systems. In this context we would like to state that Mostafazadeh3 has explored some features relating to discrete non-degenerate levels. However till now no literature on similarity transformation is available involving nearly degenerate quantum levels pertaining to either single well or double well operators. In this context we would like to state that the recent paper of Rath4 has explored some interesting features on similarity transformation. However the aim of this paper is to explore eigenvalue relation between H and h, when H is having nearly degenerate eigenvalues.

Similarity Transformation operator(S)

Here we consider S as suggested earlier3 as

  S= e x 2 /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadofacqGH9aqpcaWGLbWaaWbaaSqabeaacqGHsislcaWG4bWaaWba aWqabeaacaaIYaaaaSGaai4laiaaikdaaaaaaa@3F33@ (8)

Above operator satisfies the following relations

S= S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadofacqGH9aqpcaWGtbWaaWbaaSqabeaacaGGGacaaaaa@3B97@  (9)

PS P 1 =S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadcfacaWGtbGaamiuamaaCaaaleqabaGaeyOeI0IaaGymaaaakiab g2da9iaadofaaaa@3E2F@  (10)

TS T 1 =S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadsfacaWGtbGaamivamaaCaaaleqabaGaeyOeI0IaaGymaaaakiab g2da9iaadofaaaa@3E37@  (11)

PTS (PT) 1 =S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadcfacaWGubGaam4uaiaacIcacaGGqbGaaiivaiaacMcadaahaaWc beqaaiabgkHiTiaaigdaaaGccqGH9aqpcaWGtbaaaa@4138@  (12)

In above P stands for space reflection P( x )( x ); MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadcfapaWaaeWaaeaapeGaamiEaaWdaiaawIcacaGLPaaapeGaeyOK H46damaabmaabaWdbiabgkHiTiaadIhaa8aacaGLOaGaayzkaaWdbi aacUdaaaa@41E6@ T stands for time reversal operator T( i )( i ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaaiaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadsfapaWaaeWaaeaapeGaamyAaaWdaiaawIcacaGLPaaapeGaeyOK H46damaabmaabaWdbiabgkHiTiaadMgaa8aacaGLOaGaayzkaaGaai Olaaaa@41AF@ 2 Further using this transformation one can see that the commutation relation remains invariant

S 1 [ x,p ]S=[ x,p+ix ]=[ x,p ]=i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaWbaaS qabeaacqGHsislcaaIXaaaaOWaamWaaeaacaWG4bGaaiilaiaadcha aiaawUfacaGLDbaacaWGtbGaeyypa0ZaamWaaeaacaWG4bGaaiilai aadchacqGHRaWkcaWGPbGaamiEaaGaay5waiaaw2faaiabg2da9maa dmaabaacbiGaa8hEaiaa=XcacaWFWbaacaGLBbGaayzxaaGaeyypa0 JaamyAaaaa@4FEA@ (13)

Or

S[ x,p ] S 1 =[ x,pix ]=[ x,p ]=i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbWaamWaae aacaWG4bGaaiilaiaadchaaiaawUfacaGLDbaacaWGtbWaaWbaaSqa beaacqGHsislcaaIXaaaaOGaeyypa0ZaamWaaeaacaWG4bGaaiilai aadchacqGHsislcaWGPbGaamiEaaGaay5waiaaw2faaiabg2da9maa dmaabaacbiGaa8hEaiaa=XcacaWFWbaacaGLBbGaayzxaaGaeyypa0 JaamyAaaaa@4FF5@ (14)

As reported earlier it is bounded3 in its behaviour.

Generation of space-time reversal model operators h PT (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaieqacaWFObWaa0 baaSqaaiaa=bfacaWFubaabaGaa8hkaiaa=LgacaWFPaaaaaaa@3CD5@  

As discussed earlier we can generate h as

SH S 1 =h S 1 hS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGtbGaamisai aadofadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcaWGObGa eyOKH4Qaam4uamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaadIgaca WGtbaaaa@448A@  (15)

Now consider the following cases

H= P 2 100 x 2 + x 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGibGaeyypa0 JaamiuamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaigdacaaIWaGa aGimaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG4bWaaW baaSqabeaacaaI0aaaaaaa@4343@  (16)

The above DWP (double well potential) was extensively studied in the past by many authors after the work of Balsa.5 However here we study the same after similarity transformation. The transformed operator is

h PT (1) = p 2 +i(xp+px)101 x 2 + x 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObWaa0baaS qaaiaadcfacaWGubaabaGaaiikaiaaigdacaGGPaaaaOGaeyypa0Ja amiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMgacaGGOaacbi Gaa8hEaiaa=bhacaWFRaGaa8hCaiaa=HhacaGGPaGaeyOeI0IaaGym aiaaicdacaaIXaGaa8hEamaaCaaaleqabaGaaGOmaaaakiabgUcaRi aadIhadaahaaWcbeqaaiaaisdaaaaaaa@4F2D@  (17)

Now consider another DWP as

H= p (2) 50 x 2 + x 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeacqGH9aqpcaWGWbWdamaaCaaaleqabaGaaiika8qacaaIYaWd aiaacMcaaaGcpeGaeyOeI0IaaGynaiaaicdacaWG4bWdamaaCaaale qabaWdbiaaikdaaaGcpaGaey4kaSYdbiaadIhapaWaaWbaaSqabeaa peGaaGOnaaaaaaa@44C3@  (18)

The transformed operator in this case is

h PT (2) = p 2 +i(xp+px)-51 x 2 + x 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObWaa0baaS qaaiaadcfacaWGubaabaGaaiikaiaaikdacaGGPaaaaOGaeyypa0Ja amiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMgacaGGOaacbi Gaa8hEaiaa=bhacaWFRaGaa8hCaiaa=HhacaWFPaGaa8xlaiaa=vda caWFXaGaa8hEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadIhada ahaaWcbeqaaiaaiAdaaaaaaa@4E26@ (19)

Let us consider another operator involving quartic and sextic term as

H= p 2 10 x 4 + x 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeacqGH9aqpcaWGWbWdamaaCaaaleqabaWdbiaaikdaaaGccqGH sislcaaIXaGaaGimaiaadIhapaWaaWbaaSqabeaapeGaaGinaaaaki abgUcaRiaadIhapaWaaWbaaSqabeaapeGaaGOnaaaaaaa@432A@  (20)

The transformed operator in this case is

h PT (3) = p 2 +i(xp+px) x 2 10 x 4 + x 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObWaa0baaS qaaiaadcfacaWGubaabaGaaiikaiaaiodacaGGPaaaaOGaeyypa0Ja amiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMgacaGGOaacbi Gaa8hEaiaa=bhacaWFRaGaa8hCaiaa=HhacaWFPaGaeyOeI0Iaa8hE amaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaigdacaaIWaGaamiEam aaCaaaleqabaGaaGinaaaakiabgUcaRiaadIhadaahaaWcbeqaaiaa iAdaaaaaaa@5152@ (21)

Let us consider model single well oscillator as

H= p 2 + x 2 +λ | x | (1+ x 2 ) 2.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGibGaeyypa0 JaamiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRGqaciaa=Hhadaah aaWcbeqaaiaaikdaaaGccqGHRaWkcqaH7oaBdaWcaaqaamaaemaaba GaamiEaaGaay5bSlaawIa7aaqaaiaacIcacaaIXaGaey4kaSIaa8hE amaaCaaaleqabaGaaGOmaaaakiaacMcadaahaaWcbeqaaiaaikdaca GGUaGaaGynaaaaaaaaaa@4C67@  (22)

h (4) = p 2 +i (xp+px) 2 +λ | x | (1+ x 2 ) 2.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObWaa0baaS qaaaqaaiaacIcacaaI0aGaaiykaaaakiabg2da9iaadchadaahaaWc beqaaiaaikdaaaGccqGHRaWkcaWGPbGaaiikaGqaciaa=HhacaWFWb Gaa83kaiaa=bhacaWF4bGaa8xkamaaCaaaleqabaGaaGOmaaaakiab gUcaRiabeU7aSnaalaaabaWaaqWaaeaacaWG4baacaGLhWUaayjcSd aabaGaaiikaiaaigdacqGHRaWkcaWF4bWaaWbaaSqabeaacaaIYaaa aOGaaiykamaaCaaaleqabaGaaGOmaiaac6cacaaI1aaaaaaaaaa@54A0@  (23)

Eigenvalues calculation

Here we solve the eigenvalue relation

H|Ψ>=E|Ψ> MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGibWaaqqaae aacqqHOoqwaiaawEa7aiabg6da+iabg2da9iaadweadaabbaqaaiab fI6azbGaay5bSdGaeyOpa4daaa@42C5@  (24)

With

|Ψ>= m A m±k |m> MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaabbaqaaiabfI 6azbGaay5bSdGaeyOpa4Jaeyypa0ZaaabuaeaacaWGbbWaaSbaaSqa aiaad2gacqGHXcqScaWGRbaabeaaaeaacaWGTbaabeqdcqGHris5aO WaaqqaaeaacaWGTbaacaGLhWoacqGH+aGpaaa@4867@ (25)

where |m> MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaabbaqaaiaad2 gaaiaawEa7aiabg6da+aaa@3B60@ satisfies the condition [ p 2 + x 2 ]|m>=(2m+1)|m> MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWadaqaaiaadc hadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG4bWaaWbaaSqabeaa caaIYaaaaaGccaGLBbGaayzxaaWaaqqaaeaacaWGTbaacaGLhWoacq GH+aGpcqGH9aqpcaGGOaGaaGOmaiaac2gacqGHRaWkcaaIXaGaaiyk amaaeeaabaGaamyBaaGaay5bSdGaeyOpa4daaa@4B43@  (26)

In this case one has to solve the recursion relation satisfied A m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadgeapaWaaSbaaSqaa8qacaWGTbaapaqabaaaaa@3A04@  considering different matrix size [15] and calculate the eigenvalues.

Results and discussion

In Table 1, we present convergent eigenvalues of the H ( i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaadaqadaqaaKqzadWdbiaadMgaaSWdaiaa wIcacaGLPaaaaaaaaa@3CCA@  and in Figures 1,3,5,7 reflect the same. Further we also notice that nearly degenerate eigenvalues reflected in Table 1 also remain the same on exchange of xp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIhacqGHsgIRcaWGWbaaaa@3BD1@  and px MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadchacqGHsgIRcaWG4baaaa@3BD1@  i.e

H (1) = x 2 100 p 2 + p 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaWdbiaaigdapaGaaiykaaaak8qa cqGH9aqpcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislca aIXaGaaGimaiaaicdacaWGWbWdamaaCaaaleqabaWdbiaaikdaaaGc cqGHRaWkcaWGWbWdamaaCaaaleqabaWdbiaaisdaaaaaaa@4661@ (27)

H (2) = x 2 50 p 2 + p 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaGaaGOmaiaacMcaaaGcpeGaeyyp a0JaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGynai aaicdacaWGWbWdamaaCaaaleqabaWdbiaaikdaaaGccqGHRaWkcaWG WbWdamaaCaaaleqabaGaaGOnaaaaaaa@457F@  (28)

H (3) = x 2 10 p 4 + p 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaGaaG4maiaacMcaaaGcpeGaeyyp a0JaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGymai aaicdacaWGWbWdamaaCaaaleqabaGaaGinaaaak8qacqGHRaWkcaWG WbWdamaaCaaaleqabaGaaGOnaaaaaaa@457E@  (29)

H (4) = x 2 + p 2 +λ | p | (1+ p 2 ) 2.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaGaaGinaiaacMcaaaGcpeGaeyyp a0JaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOWdaiabgUcaR8qaca WGWbWdamaaCaaaleqabaGaaGOmaaaak8qacqGHRaWkpaGaeq4UdW2a aSaaaeaadaabdaqaaiaadchaaiaawEa7caGLiWoaaeaacaGGOaGaaG ymaiabgUcaRiaadchadaahaaWcbeqaaiaaikdaaaGccaGGPaWaaWba aSqabeaacaaIYaGaaiOlaiaaiwdaaaaaaaaa@4F4D@ (30)

Figure 1 Nearly degenerate eigenvalues.
Figure 2 Spectral breakdown.
Figure 3 Nearly degenerate eigenvalues.
Figure 4 Spectral breakdown.

Hamiltonian

First four eigenvalues

H ( 1 ) =  p 2   100 x 2  +  x 4 [ 5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeadaahaaWcbeqaa8aadaqadaqaa8qacaaIXaaapaGaayjkaiaa wMcaaaaak8qacqGH9aqpcaqGGaGaamiCamaaCaaaleqabaGaaGOmaa aakiaabccacqGHsislcaqGGaGaaGymaiaaicdacaaIWaGaamiEamaa CaaaleqabaGaaGOmaaaakiaabccacqGHRaWkcaqGGaGaamiEamaaCa aaleqabaGaaGinaaaak8aadaWadaqaa8qacaaI1aaapaGaay5waiaa w2faaaaa@4C54@

-2485.867 880

H (1) =  x 2 100 p 2 +  p 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaWdbiaaigdapaGaaiykaaaak8qa cqGH9aqpcaqGGaGaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaey OeI0IaaGymaiaaicdacaaIWaGaamiCa8aadaahaaWcbeqaa8qacaaI YaaaaOGaey4kaSIaaeiiaiaadchapaWaaWbaaSqabeaapeGaaGinaa aaaaa@47A7@

-2485.867 880

-2257.643 822

-2257.643 822

H ( 2 ) = p 2 50 x 2 + x 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaadaqadaqaa8qacaaIYaaapaGaayjkaiaa wMcaaaaak8qacqGH9aqpcaWGWbWdamaaCaaaleqabaWdbiaaikdaaa GccqGHsislcaaI1aGaaGimaiaadIhapaWaaWbaaSqabeaapeGaaGOm aaaakiabgUcaRiaadIhapaWaaWbaaSqabeaapeGaaGOnaaaaaaa@45E6@

-122. 182 076

H (2) = x 2 50 p 2 + p 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaWdbiaaikdapaGaaiykaaaak8qa cqGH9aqpcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislca aI1aGaaGimaiaadchapaWaaWbaaSqabeaapeGaaGOmaaaak8aacqGH RaWkpeGaamiCa8aadaahaaWcbeqaa8qacaaI2aaaaaaa@45CD@

-122. 182 076

-95.595 466

-95.595 466

H ( 3 ) = p 2 10 x 4 + x 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaadaqadaqaa8qacaaIZaaapaGaayjkaiaa wMcaaaaak8qacqGH9aqpcaWGWbWdamaaCaaaleqabaWdbiaaikdaaa GccqGHsislcaaIXaGaaGimaiaadIhapaWaaWbaaSqabeaapeGaaGin aaaak8aacqGHRaWkpeGaamiEa8aadaahaaWcbeqaa8qacaaI2aaaaa aa@4604@

-132.132 256

H (3) = x 2 10 p 4 + p 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaWdbiaaiodapaGaaiykaaaak8qa cqGH9aqpcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislca aIXaGaaGimaiaadchapaWaaWbaaSqabeaapeGaaGinaaaakiabgUca RiaadchapaWaaWbaaSqabeaapeGaaGOnaaaaaaa@45AD@

-132.132 256

-101.636 451

-101.636 451

H (4) = p 2 x 2 + 1000| x | (1+ x 2 ) 2.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaGaaGinaiaacMcaaaGcpeGaeyyp a0JaamiCa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaamiEa8 aadaahaaWcbeqaaiaaikdaaaGcpeGaey4kaSYaaSaaaeaacaaIXaGa aGimaiaaicdacaaIWaWaaqWaaeaacaWG4baacaGLhWUaayjcSdaaba GaaiikaiaaigdacqGHRaWkieGacaWF4bWaaWbaaSqabeaacaaIYaaa aOGaaiykamaaCaaaleqabaGaaGOmaiaac6cacaaI1aaaaaaaaaa@5076@

 19.971 522

 19.971 522

H (4) = x 2 p 2 + 1000| p | (1+ p 2 ) 2.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaWbaaSqabeaacaGGOaGaaGinaiaacMcaaaGcpeGaeyyp a0JaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaamiCa8 aadaahaaWcbeqaaiaaikdaaaGcpeGaey4kaSYaaSaaaeaacaaIXaGa aGimaiaaicdacaaIWaWaaqWaaeaacaWGWbaacaGLhWUaayjcSdaaba GaaiikaiaaigdacqGHRaWkcaWGWbWaaWbaaSqabeaacaaIYaaaaOGa aiykamaaCaaaleqabaGaaGOmaiaac6cacaaI1aaaaaaaaaa@505F@          

 24.519 600

 

 24.519 600

Table 1 Nearly degenerate eigenvalues: 1-D quantum systems

In the case of PT-symmetry operators we reflect spectral nature in Figures 2,4,6,8. The main conclusion of the paper is that the spectral invariance nature of Hamiltonian is violated on similarity transformation only in nearly degenerate eigenvalues i.e

H E n n h P T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeacqGHsgIRcaWGfbWdamaaBaaaleaapeGaamOBaaWdaeqaaOWd biabgcMi5kabgIGio=aadaWgaaWcbaWdbiaad6gaa8aabeaak8qacq GHqgcRcaWGObWdamaaBaaaleaapeGaamiuaaWdaeqaaOWaaSbaaSqa a8qacaWGubaapaqabaaaaa@46C2@  (31)

However if the parameter in question is small there is no spectral variance. For example consider the case of non-degenerate eigenvalues of

H= p 2 x 2 + x 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeacqGH9aqpcaWGWbWdamaaCaaaleqabaWdbiaaikdaaaGccqGH sislcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHRaWkcaWG4b WdamaaCaaaleqabaWdbiaaisdaaaaaaa@41B1@ (32)

And h= p 2 2 x 2 +i(xp+px)+ x 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIgacqGH9aqpcaWGWbWdamaaCaaaleqabaWdbiaaikdaaaGccqGH sislcaaIYaGaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaey4kaS IaamyAa8aacaGGOaWdbiaadIhacaWGWbGaey4kaSIaamiCaiaadIha paGaaiyka8qacqGHRaWkcaWG4bWdamaaCaaaleqabaWdbiaaisdaaa aaaa@4ABA@  (33)

Figure 5 Nearly degenerate eigenvalues.
Figure 6 Spectral breakdown.
Figure 7 Nearly degenerate eigenvalues.
Figure 8 Spectral breakdown.

There is no spectral variance. Similar case exists for all operators cited above. However it is not known at present, why the spectral variance exists in large parameter reflecting nearly degenerate states?

Acknowledgements

Author is grateful to Referee for constructive remarks.

Conflict of interest

Authors declare that there is no conflicts of interest.

References

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©2019 10.15406/oajmtp.2019.02.00046. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.