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MyEverettNews.com A Daily Website Devoted To Breaking News and Current Events in Everett, Washington
Former Kimberly-Clark Site Back To Creating Hundreds Of Jobs
Dignitaries cut the ribbon at the new terminal on Thursday. Photo credit: Port of Everett
In April 2012, the Kimberly-Clark plant closed on the Everett waterfront. Since then, the majority of jobs at the site have been in cleaning up the property. Soon hundreds of jobs will be created as the property comes back to life at the newest Cargo Terminal on the west coast. Here's the story from the Port of Everett.
The Port of Everett made history Thursday, officially opening its new Norton Terminal – the first all new cargo terminal to open on the U.S. West Coast in more than a decade.
The $40 million maritime infrastructure investment introduces 40 acres of new upland cargo capacity to the national freight network to further enhance movement of commerce throughout the global supply chain.
It also nearly doubles the Port of Everett Seaport's cargo handling footprint, fully realizing the Port's recent $57 million investment in its South Terminal Modernization that added another full-service berth in Everett in 2021 to support larger ships and heavier cargo.
"With most trade-compatible properties along our nation's waterways already spoken for, it's extremely rare to see an all-new cargo facility come online and add much-needed capacity to a strained logistics chain – which we have all had a front row seat to witness over the past year," Port of Everett CEO Lisa Lefeber said.
"The new Norton Terminal, combined with our upgraded South Terminal, are game changers for our working waterfront and region, allowing us to do our part to keep freight moving efficiently, restore thousands of meaningful family-wage jobs to this site, and support a thriving economy."
Completion of Norton Terminal caps off the Port's 5-year, $150 million Mills to Maritime initiative that set out to transform a former contaminated mill site into a new sustainable maritime hub. The two projects will add more than 1,000 jobs to the waterfront and generate $14.5 million annually in state and local taxes.
The Norton Terminal project, conducted in partnership with the Washington State Department of Ecology, integrated the property's cleanup requirements with redevelopment, completing the final physical cleanup of the uplands at a site that supported nearly 90 years of mill operations.
It was funded in part by a $17.75 million federal BUILD grant and $9.15 million in Washington State Department of Ecology MTCA grants.
The historic occasion was celebrated with a ribbon-cutting ceremony on December 1, alongside the Port's many project partners. Local, state and federal leaders shared remarks, including Sen. Patty Murray, Rep. Rick Larsen, Rep. Emily Wicks, Port and City leadership, the U.S. Department of Transportation Maritime Administration (MARAD) and the Washington State Department of Ecology.
Project Accomplishments
Realizes strategic acquisition of a former vacant mill site for maritime use at the working waterfront; situated at the heart of Everett's deep-water maritime complex, between two federally secured facilities – the Port of Everett Seaport and Naval Station Everett
Achieves environmental stewardship by completing final upland cleanup of legacy contamination at the site, work included:
Removal of more than 250,000 tons of contaminated material and rubble (about 14K dump truck loads)
Backfilling the site with approximately 14K dump truck loads of clean river sands from the Snohomish River
Pre-loading the site with 50,000 cubic yards of clean crushed material to compact the site, followed by grading the material to prepare for paving
Installation of a low permeability 9-inch-thick environmental pavement cap over a majority of the site to provide permanent environmental controls, which doubles as the cargo terminal
Installation of a state-of-the-art stormwater treatment system to protect the Puget Sound into the future – the first treatment system in the site's history
Enhances the movement of commerce by providing additional cargo laydown capacity
Better positions Everett's seaport facilities to aid in our regional economic recovery efforts, as well as cargo diversification
Provided necessary infrastructure for the Port of Everett Seaport to receive Strategic Seaport Designation from the U.S. Maritime Administration to support our armed forces – one of only 18 seaports nationwide
Restores family-wage jobs to the site and supports the 40,000 jobs already supported by seaport options
And, an added bonus, via the Port's voluntary 2% for public access policy, this project helped pay to dredge the public boat launch to ensure essential water access at the largest public boat launch in Washington state
The new Norton Terminal and South Terminal modernization projects combined, create major economic benefit, generating:
More than 1,050 direct jobs; 2,300 in total, including indirect and induced
$180 million in personal income
$46 million in local purchases
$14.5 million in state and local taxes
$46.5 million in federal taxes
The work also created benefit during construction, generating:
1,950 temporary construction jobs
$64 million in personal income
$6 million in state and local taxes
$20 million in federal taxes
Norton Terminal Features
40-acre paved, lit and secure cargo yard
Preserves the adaptive reuse of an existing 360,000 square foot warehouse
Power conduits to support future electric tugs
Provides the infrastructure to support short-sea shipping
Washdown area
Port Leadership Reaction
"This site has not been in operation for over a decade, but with the Port's determination, that changes now with the opening of the Norton Terminal. This terminal will restore jobs, support commerce, keep our region competitive, and provide an immediate economic boost to the City of Everett, Snohomish County, the neighboring Cascade Industrial Center in the cities of Arlington and Marysville, and the greater Puget Sound region. It also cleans up a big piece of the waterfront, which is a top priority to the Port."
– Port of Everett Commission President David Simpson
"With the opening of Norton Terminal and our modernized South Terminal, I'm confident there is no project cargo anywhere that we are not able to say, 'We can handle it.' These investments put Everett on the map and opens doors to new cargo opportunities. There is no stopping the success of Everett's working waterfront."
– Port of Everett Commission Vice President Glen Bachman
"The Port has been around for more than a century, and we have demonstrated time and time again our proven track record of putting former contaminated mill sites back into productive use to support trade and bring jobs to the waterfront. As a former longshoreman myself, this is a big win for maritime jobs, especially the skilled labor force we have here that is dedicated to keeping the supply chain moving – ILWU Locals 32, 52 and 98."
– Port of Everett Commission Secretary Tom Stiger
Local, State & Federal Partner Reaction
"Here in Washington state, our waterways are a cornerstone of our economy, our communities, and our way of life. Our ports create jobs, improve supply chain efficiency, and help move our state's world-class products and goods out to the rest of the world. The Norton Terminal is going to be a key part of that and serve as a major driver of jobs and economic activity in Everett and Snohomish County."
– Senator Patty Murray, United States Senate
"In the Pacific Northwest and across the country, you cannot have a big-league economy with little league maritime infrastructure. The opening of the Norton Terminal is great news for the hundreds of businesses and tens of thousands of jobs that depend on the Port of Everett to connect local goods and services with global markets."
– Rep. Rick Larsen (WA-02), a Senior Member of the Transportation and Infrastructure Committee
"The Norton Terminal Project is an impressive example of partnering to make a community thrive. Faced with an economy that could have easily stagnated and declined during such a tumultuous time in our history, the Port of Everett, instead, made opportunity and built resources out of environmental stewardship and a commitment to people and place. It is an honor to play a small role in securing the critical state support that helped deliver this project and essential jobs for our region in just two short years."
– Rep. Emily Wicks, 38th Legislative District, Washington State
"The new Norton Terminal is exactly the kind of forward thinking and strategic investment that will be critical to the recovery and growth of our local economy. Thank you to the Port of Everett for your vision and leadership."
– Mayor Cassie Franklin, City of Everett
"The Norton Terminal at the Port of Everett is important for businesses, workers, and our entire region. It isn't often that we get to celebrate not only the completion of a significant new part of our economic infrastructure but also the clean-up of the site. For generations, people have worked on the Everett waterfront at family-wage jobs. The mills have made way for new, diverse industries that will provide for thousands of families for many generations to come. The Norton Terminal will be another jewel on Puget Sound's waterfront."
– Executive Dave Somers, Snohomish County
"The U.S. DOT Maritime Administration congratulates the Port of Everett on the completion of Norton Terminal. We're proud to administer Federal investments that support our Nation's ports and waterways. This Mills to Maritime Project is a prime example of these investments strengthening our infrastructure, economy, and maritime industry."
– Catherine Simons, Gateway Director, Pacific Northwest & Alaska
U.S. Department of Transportation Maritime Administration (MARAD)
"We are pleased to help celebrate this important milestone. This is yet another great example of our fruitful partnership with the Port of Everett to clean up the environment and put once-contaminated properties back to productive use."
– Barry Rogowski, Program Manager, Washington State Department of Ecology's Toxics Cleanup Program
Port of Everett
About myeverettnews
My Everett News is a hyperlocal news website featuring breaking news and events in Everett, WA. We also cover City of Everett information and items of interest to those who live and work in Everett. It's written by Leland Dart a former Snohomish County based radio reporter born and raised in Everett.
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Over the past fifteen years, as I've spent time researching the life and career of the Yiddish author Jacob Dinezon, one piece of information has eluded me: the name of Dinezon's older sister with whom he lived in Warsaw from the mid-1880s until his death in 1919.
Yet there is no mention of Dinezon's sister's name in any of the Haynt's extensive coverage of Dinezon's final hours and elaborate funeral.
Having now entered the centennial year of Dinezon's yortsayt—the 100th anniversary of his death—I finally decided to reach out to the members of the JewishGen Discussion Group to see if anyone could help shed light on Dinezon's sister's name. JewishGen is a genealogical website offering the ability to search millions of records in a vast array of online databases. (Learn more on Wikipedia or watch a touching video about the organization.) Volunteers belong to a variety of special interest groups (SIGs) and provide answers, insights, and translations to people who are working on ancestry questions and family trees. I receive email digests from several SIGs and have marveled at the information that is shared between participants.
On January 17, 2019, I posted an inquiry to JewishGen seeking suggestions on ways to locate Jacob Dinezon's sister. I noted that Dinezon had lived with his sister and her family (including two daughters) at Dzielna No. 15 and Karmelicka No. 29 in Warsaw. I also mentioned the various transliterations I had found of Dinezon's name: Dineson, Dinesohn, Dienesohn, Dynezon, and Dynesohn.
First, Nicole and Simon used the search engine Genealogy Indexer created by genealogist Logan Kleinwaks to search for Jacob Dinezon in the 1908 Adresy Warszawy (Warsaw Address and Business Directory). What they found was a listing for "DYNCZON Eljasz literat, Dzielna 15." Dynczon, a new transliteration for Dinezon!
Then, using the database at JRI Poland (Jewish Records Indexing – Poland), Nicole discovered a memorial notice for "KAC, Fajga," with a notation that her brother was "Jakob Dineson, writer." Simon supported this with an image: the announcement of "Fajgi" Kac's headstone unveiling in the Nasz przeglad: Organ niezależny.
Here was information about Dinezon's sister Fajga (Feiga) Kac: her father's name was Binyamin, her husband's name was Chaim, her maiden name was Dinesohn, and she died on January 8, 1929. She was the sister of writer Yaakov Dinesohn and she lived to be 98 years old.
The database also showed a photograph of the "Mausoleum of the Three Writers" where Dinezon is buried with the Yiddish authors I. L. Peretz and S. Ansky in Warsaw's Okopowa Cemetery.
Now, if you look closely you will see Dinezon's name in the middle with the dates 1858-1919 etched in the stone.
This brings us to the other amazing discovery from our JewishGen researchers. For years scholars have been struggling to determine the year of Jacob Dinezon's birth. The most common date you see mentioned in biographical entries is 1856 (as shown in the image from the Warsaw cemetery database above). Some references cite 1852 and the Haynt at the time of Dinezon's death reported the year 1851.
Finally, the work of Simon Srebrny and Adam Teitelbaum may resolve this issue. As both researchers continued to investigate the Dinezon family tree, they discovered census information from Zagare, Lithuania for the years 1847, 1850, and 1851 in the Litvak-SIG database. There they uncovered a family named DENIZON that included a head of houshold named Benjamin Denizon, his wife Pera, and two daughters, Freida and Ginda.
Listing for the Benjamin Denizon family in the LitvakSIG Database, 1850, 1851.
Then the find! Again in the Litvak-SIG database, the 1858 census had a completely different spelling of the family name: DINERZUN. Included in this listing is a son: Yankel Eliash, seven years old! Again, the father is Benjamin, the mother is Pessa, the two daughters are Goda and Freyde, and there's even a new sister, Chane.
Now we have names to go with this wonderful photograph of Dinezon's parents that was sent to me by Dr. Agnieszka Żółkiewska of the Jewish Historical Institute in Warsaw, Poland.
So finally, through the miracle of modern technology, and all the volunteers who inputted volumes of data, we see that Yankel Eliash Dinerzun, son of Benjamin, was seven years old in 1858, and thus, was born in 1851! At last, confirmation of Jacob Dinezon's birth year, and verification of what appeared in the August 29, 1919 edition of the Warsaw Yiddish newspaper, Haynt.


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text:
Мъжки песни е шести студиен албум на българската рок група Епизод, издаден през 2004 година.

Песни
Албума се състои от 9 песни:

 Море сокол пие – 04:43
 Борба – 03:33
 Ако умра ил загина – 03:48
 На прощаване – 03:43
 Болен ми лежи Миле Попйорданов – 04:35
 Слушай как шумят шумите – 03:28
 В механата – 03:51
 Пристанала е – 03:06
 100 каба гайди – 04:12

Изпълнители
Изпълнители са:

 Емил Чендов
 Драгомир Драганов
 Симеон Христов
 Иво Георгиев
 Христо Гьошарков

Източници

Външни препратки
  Мъжки песни в Български рок архиви

Албуми на Епизод
Музикални албуми от 2004 година

redpajama_set_name: RedPajamaWikipedia


entry features and the articulated brick facade. The Flinders Street facades were cleaned to remove accumulated layers of paint and grime and re-tuckpointed. Significant restoration work was undertaken to the facade mouldings and pediment including the iconic Britannia figure mounted on the central plinth. Sections of the original terracotta plank floors, an early 20th century fireproofing system, were replaced to allow for contemporary services. The Empire Building redevelopment involved the removal of the rear section of the building and the construction of 14 new floors. The additional floors have been set back from the street and have a strong vertical rhythm sympathising with the verticality and heritage character of the original building. The new addition is characterised by an articulation of setbacks, massing and materials. A contrasting external colour scheme was selected for the Empire Building. For many years the two facades had been treated as one - now the individual character of each can be appreciated. Like the Mutual Stores, there are new retail tenancies at street level in the Empire Building to further develop the active edge promoted by the Council Planning Scheme. The Empire Apartments comprises 306 studio, one and two bedroom student apartments as well as communal facilities including lounge, laundry and shared terraces. The terraces are located behind the original facade and provide residents with an opportunity to experience Melbourne from a new perspective. All retail tenancies are fully let and the Mutual Store and Empire apartments are fully occupied.


text:
Octavian Morariu () este un fost jucător de rugby român, care a devenit conducător sportiv. Este membru al Comitetului Internațional Olimpic și președintele al Rugby Europe.

Carieră
Este fiul rugbistului Viorel Morariu și al vicecampioanei mondiale la volei Cornelia Timoșanu. S-a apucat de rugby la vârsta de 8 ani. Când avea 16 ani s-a legitimat la RC Grivița Roșie. Și-a făcut debutul la echipa națională a României într-o partidă de Campionatul FIRA în decembrie 1984. De-a lungul carierei a strâns cinci selecții cu "Stejarii". În anul 1986 a absolvit ca inginer constructor Facultatea de Căi Ferate, Drumuri și Poduri din Institutul de Construcții din București.

În 1987 a devenit primul român invitat să joace pentru echipa internațională Barbarians. După un turneu a ales să rămâne în Franța, unde a obținut azil politic. A fost angajat la o firmă de construcții. În paralel, a jucat pentru clubul ASPTT Paris Rugby, care evolua în prima divizie a campionatului național. S-a căsătorit cu o franțuzoaică de origine română, Dana Patrichi, și a dobândit cetățenia franceză. De la 1991 până la 1993 a fost antrenor la echipa de rugby a clubului Paris Saint-Germain.

S-a întors în România după Revoluția Română din 1989. În anul 2001 a fost ales președintele Federației Române de Rugby. În anul următor, a fost numit președintele  Agenției Naționale pentru Sport (ANS), cu rang de secretar de stat. În anul 2004 a preluat președinția al Comitetului Olimpic și Sportiv Roman, fiind reales în 2008 și în 2013. În anul respectiv, a devenit primul președinte al Rugby Europe, fosta Federația Internațională de Rugby Amator-Asociația Europeană de Rugby, și a fost ales membru Comitetului Internațional Olimpic.

Distincții
 cavaler al Ordinului Național de Merit (2015)

Referințe

Legături externe
  Prezentare la Comitetul Olimpic Internațional

Rugbiști români
Bucureșteni
Români reîntorși în România
Români expatriați în Franța
Oficialități sportive
Absolvenți ai Universității Tehnice de Construcții din București
Morariu
Cavaleri ai Ordre national du Mérite


text:
Dus Kahaaniyan: Bas… Stop stealing, Mr.Gupta
December 9, 2007 · by sudhishkamath
Director: Sanjay Gupta, Hansal Mehta, Rohit Roy, Meghna Gulzar, Apoorva Lakhia, Jasmeet Dhodi
Cast: Sanjay Dutt, Suniel Shetty, Arbaaz Khan, Shabana Azmi, Naseeruddin Shah
Storyline: Ten short stories – stolen, adapted and some credited – with a twist in the tale.
Bottomline: Film students make more original films.
Sanjay Gupta is a thief and an obsessive, compulsive kleptomaniac at that.
After recycling Reservoir Dogs (Kaante), modifying U-Turn (Musafir), plagiarising 'Old Boy' (Zinda), he turns to books this time. Thanks to Shilpa for letting me know about the source stories.
For the first short story, Sanjay Gupta rips off Roald Dahl's 'Mrs. Bixby and the Colonel's Coat,' calls it 'Matrimony' and takes the writing credits for the story about cheaters.
The eccentric cinematography, crisp editing and the grainy grading once again proves that Gupta is technically competent, so much that you are hooked to the story-telling than what his actors Mandira Bedi and Arbaaz Khan can do by way of histrionics.
Hansal Mehta's 'High on the Highway' too is stylistically shot to suit the mood, with a non-linear narrative that seemed intelligent until you learn that Jimmy Shergill is supposed to be passing out of college. Masumeh's presence more than makes up for the casting mistake and the silly plot.
Meghna Gulzar's 'Pooranmasi,' is also about sexual choices. Set in a rural milieu, this short has very little going for it and makes you understand why Minissha Lamba jumped into the well. I switched off halfway watching a middle-aged Amrita Singh wake up in the fields in the arms of her half-naked lover.
Sanjay Gupta then returns with 'Strangers in the night,' an excuse to Neha Dhupia strut her stuff, with suggestively phallic imagery. Erotic no doubt, by why these porn-movie metaphors if the story was not about lust but about nobility? Oh, okay, that's the twist.
His 'Zahir' that follows next, credited to Rajeev Gopalakrishnan, packed a nice twist towards the end – the only story to have actually caught the fancy of the audience. Manoj Bajpai acquits himself pretty well too. This short story made in Tamil, I'm told, played on Doordarshan many years ago, with what seems to be a much better twist.
After five stories woven around different excuses to set up sex scenes, Jasmeet Dhodi's 'Lovedale' post Interval, tries a supernatural spin bordering on incest. With Aftab doing the chunk of acting, this is as boring as it gets.
Apoorva Lakhia's 'Sex on the beach' is just a showcase for Tarina Patel's golden bikini. Dino Morea evokes a few laughs but this is seriously the kind of fare you don't mind from film students.
Rohit Roy's 'Rice Plate,' a reworking of Jeffrey Archer's 'Broken Routine' (again, uncredited) has Shabana struggle with a Tamil accent but this is clearly among the more watchable stories of the lot, especially with her facing off Naseeruddin Shah. A pretty decent debut for Rohit Roy.
'Gubbare,' written by Gulzaar is a beautiful tale, completely misdirected by Sanjay Gupta. What should have been a well-concealed twist that tugs at your heart-strings turns predictable half-way in spite of Nana Patekar's heart-breaking performance.
Sanjay Gupta's 'Rise and Fall' is spectacularly shot, highly stylised and inspired by Matrix Revolutions with Sanjay Dutt and Suniel Shetty doing what they do best: fight in slow-mos.
With no common thread running through the films (no, different excuses for sex and violence cannot qualify as a theme for an anthology film), nothing original about these stories, 'Dus Kahaaniyan' is not half as good as shorts made by film students with much lesser budgets.
Wait for a cheap DVD copy.
http://sudermovies.blogspot.com
Saying Sanjay Gupta copies from other movies is as redundant as saying like "those queers are a bunch of gay assholes." You get my point?
When I last searched, the definition for a Sanjay Gupta movie went somewhat like this – a movie which is mostly a scene-by-scene rip off of some Hollywood/Korean/Chinese/Iranian/Japanese/German/Polish/Dutch/Swahili/movies, with a motley group of actors who try so hard to catch your attention that you feel sad for them sometimes.

redpajama_set_name: RedPajamaCommonCrawl


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Stewart-Haas Racing Hires Ryan Preece As Reserve Driver For 2022
KANNAPOLIS, N.C. — Stewart-Haas Racing (SHR) has hired Ryan Preece as its reserve driver for the 2022 NASCAR season.
The 31-year-old racer from Berlin, Connecticut, will perform simulator work while running a mix of races across each of NASCAR's top three touring series — Cup, Xfinity and Camping World Trucks. Preece will be a Ford Performance driver in each series, and he will be the default driver if any of SHR's full-time drivers are unable to compete due to COVID-19 or any other unforeseen circumstance.
"Ryan is a versatile wheelman with a racer's mentality who fits extremely well within our culture at Stewart-Haas Racing," said Greg Zipadelli, vice president of competition, SHR. "Between the amount of testing and development work we're doing with the Next Gen car this year, his added insights and time in the simulator will make us better by allowing us to learn faster.
"Plus, we're still dealing with COVID. As much as we all want it to be over, it's not. We needed a more robust plan in the event COVID sidelines one of our drivers. We have that now with Ryan."
Preece will run a total of 12 races — two in Cup, three in Xfinity and seven in Truck. His Cup and Xfinity Series races will be in coordination with SHR and its technical alliance partners. His Truck Series starts will come with David Gilliland Racing.
"This is a unique setup, but I feel like it provides me with the best opportunity to win races and contribute to a championship-caliber team while expanding my racing experience," said Preece, who spent the last three seasons (2019-2021) competing full-time in Cup. "I'm a racer, and Stewart-Haas Racing is a team built by racers. They measure success by wins. Whether I'm in the simulator, in one of their cars, or in a Ford Mustang or Ford F-150 for another team, I'm here to help SHR and Ford win."
Preece's two Cup Series races will be May 1 at Dover (Del.) Motor Speedway and May 29 at Charlotte (N.C.) Motor Speedway. His three Xfinity Series races will be April 2 at Richmond (Va.) Raceway, May 28 at Charlotte and June 25 at Nashville (Tenn.) Superspeedway. His seven Truck Series races will be March 4 at Las Vegas Motor Speedway, March 19 at Atlanta Motor Speedway, May 6 at Darlington (S.C.) Raceway, May 20 at Texas Motor Speedway in Fort Worth, June 24 at Nashville, July 23 at Pocono Raceway in Pennsylvania and Sept. 9 at Kansas Speedway in Kansas City.
Preece made two Truck Series starts last year, winning in his series debut June 18 at Nashville. He is a two-time Xfinity Series victor, winning in July 2017 at Iowa Speedway in Newton and April 2018 at Bristol (Tenn.) Motor Speedway.
Preece is especially known for his exploits on the NASCAR Whelen Modified Tour. He won the series championship in 2013 and has amassed 25 victories, three of which came last year — July 17 at New Hampshire Motor Speedway in Loudon, Aug. 6 at Stafford (Conn.) Motor Speedway and Sept. 10 at Richmond.
CREDIT: SHR PR
PrevPreviousLandon Cassill Joins Kaulig Racing For 2022 XFINITY Season
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Web design projects are a natural continuation of User Experience Design projects. A great process starts with UX research, then moves to UX Design where you can prebuild a project and work out any issues that might create a problem for the people that will be interacting with your digital products. It is easier to adjust your work in the design phase of a project than to change what has been already built. A proper application of the UX process can also be a great way for a company of any size to discover opportunities that their consumers are looking for that may not be getting done.
I am currently researching different Web Design & development processes to determine the path I feel is a good one to take. I would be happy to sit down with you & your team to learn more about what your project needs are and how to develop a system that uses both a UX & Design process to build you a project to help you grow.


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redpajama_set_name: RedPajamaGithub


Sun will be transiting into your 12th and 1st house indicating unfavorable position for entire this month. Jupiter and Rahu are in excellent position and very eagar to do positive things for you. Eventhough Saturn Moving onto your Janma Sthanam, you will feel wonderful results but only for this month. You will excellent progress and you will be very happy about your growth for entire this month. You will see improvements all aspects of your life including health, family, career and finance in this month. This month is looking excellent for you in this year 2014!
Saturn just moved onto your Janma Sthanam. But still you will feel positive energies in this month. You will recover from your ailing health. You will come out of the mental stress and you will feel relaxed. Your friends will be very much for supportive for you. You will get sound sleep which you wanted for the last couple of months. You will feel healthy and be happy with your progress. Gain as much energy as possible in this month with yoga and meditation. Since you will start facing the bad results again from next month onwards. The relief you get in this month would be significant but unfortunately can not continue into next month.
You will get excellent support in your family environment. Since exalted Jupiter gained full strength, it will make your life wonderful even with the presence of Janma Sani. You will strengthen the relationship with your spouse. Besides it is a great time for conjugal bliss. If eligible, Married couples will get blessed with a baby. Mars transit can also produce great happiness towards end of this month. You will also regain the social status and success in your life.
If you are single, you will find a suitable match in this month. Lovers will resolve their conflicts and can enjoy romance in this month. Subha Karyas happening in your family will make you feel very happy. Your time is looking excellent in this month but you will be placed under very severe testing period from next month onwards. So take advantage of entire this month to finish off the major tasks and take important decision on your life.
Students will make excellent progress on their studies. You will come out of unwanted tension and pressure you had in the past. You will also get very good friends in this month to support you.
If you are unemployed, your time is looking excellent to get a Job. But you would have to take a salary cut or work one level below from the past. Because your time is too good but only for this month. When Janma Sani is with you, you can not expect big salary package. Since Jupiter is in good position, you will get a Job and that can give you some happiness. You have no other option except to accept the offer with salary cut in this month. If you are currently employed, then it is better to avoid looking for a job change.
Business people will see sudden and surprising recovery in this month. Sudden windfalls are likely but only for people with strong maha dasa. Money will come into your business through multiple sources. It is an excellent time to refinance and pay off your higher APR loans. But do not plan on expanding your business since your time is looking excellent only for this month only.
If you apply for a visa, you will get it immediately with no delay in this month. If you miss it, then you have to wait for the next couple of months.
You will have excellent financial recovery for entire this month. Mars, Jupiter and Rahu are teaming up to support your financial growth and stop the malefic impact of Janma Sani completely in this month. Cash inflow from new sources will give you great relief in this month. Your expenses will also come down. It is a good time to apply for bank loan for refinance to lower your APR or consolidate your debt.
Stock market trading is very tricky. Your time is looking excellent but for this month. Hence it is a good time to come out of your open positions by end of this month. Your open positions on your portfolio will go up in this month. It may give you some profits or touch break even to close the position. If you have weak natal chart, then you will have to take some losses. In either case, your stock position will go up for sure from the current level. But it is not a good time to initiate any positions.
You will get excellent opportunities in this month to shine well. Movie artists, musicians, politicians, producers or VIPs will be happy with their progress on this month. Make sure you work on short term projects only because of Janma Sani.
You will have excellent time for entire this month even with the presence of Janma Sani. You can relax very much in this month and of course your bad time will start again from next month onwards.
1. Avoid taking non-veg food for entire this month to feel more positive energies.
2. Visit temple on Saturdays.


text:
VOL 12 No.03, August 17, 2018 (Bhadra 01, 2075) Online Register Number: DOI 584/074-75
Published on: Aug 18, 2018
JDS Fellows Will Contribute Greatly To Nepal's Development
By Masamichi Saigo | Aug 02, 2018
Nepal: A Country In Transition
The authors say "the country may have strong legal and institutional frameworks, but the image of the country has been growing from bad to worse from all indications " (P 301).
By Prof. Madhukar SJB Rana | Aug 12, 2018
Nepal: Five Months of Oli's Rule: A Disappointment?
By Dr. S. Chandrasekharan | Aug 12, 2018
Rains And Our Economy
It is a big disappointment that, contrary to the information provided by the Ministry of Finance, revenue collection suffered a setback, which had been exceeding the target each year in the last decade. It may be noted that the ministry had stated in a press conference that revenue collection last year had exceeded the target by Rs. 2 billion to reach Rs.732 billion but data made available by Nepal Rastra Bank and the Financial Comptroller General Office show that revenue collection in the year was only Rs. 726 billion, clearly showing that it fell sort of target for the first time in a decade. It would be unfair to put the entire blame for the debacle on the current dispensation because most parts of the last fiscal year, Oli and Khatiwada were not managing state affairs but they definitely owe public an apology for the misleading information formally provided in a press conference organized by the ministry soon after the annual closing.
By Dr. Tilak Rawal | Aug 18, 2018
Plogging For Plastic Wastes
In Europe there is awareness about recycling of plastics. The Guardian reported recently that the figure of recycling of plastic in Norway was as high as 95%. Norway recycles each used plastic bottle twelve times. Germany claims to do a lot of recycling too and claims that its efforts are more successful than that of the Scandinavians
By Hemang Dixit | Aug 18, 2018
Plastic Politicians, Polyester Parties
Traitors infest the Nepalese political class and voters think all politicians are corrupt. The voters' low ideological alignment with the parties, especially in municipal elections, is another reason that guarantees a politician with a few agitators when it comes to changing parties or even position.
By Deepak Raj Joshi | Aug 18, 2018
STABLE OLI GOVERNMENT On An Unstable Course
Although the government has come to power on promises to provide stability, such an indication is nowhere to be seen
By A Correspondent | Aug 18, 2018
The Myth Of Enlightened Despotism
It was introduced with the aim of nurturing a small group of clerical Indians who could serve the British's interests and act as interpreters. The British left India with a literacy rate of only 16 percent (female literacy rate was 8 percent) showing that they did not really do much to educate the masses. The case of Indian railways is even worse.
By Abijit Sharma | Aug 18, 2018
SINGHA DUBAR II Governance Drama
After three episodes, Singha Durbar II has gained popularity among audiences for showing the importance of good governance, transparency and accountability to the people
"Collaboration Is Key Message In Singha Durbar II"
Bhola Prasad Dahal, PhD, Country Director at Search for Common Ground Nepal, has extensive knowledge on good governance and development issues in Nepal. As Search for Common Ground recently released SINGHA DURBAR II, a public television show related to the contemporary issues of good governance, accountability, federal structure, civil society, service delivery, migration and SDGs, Dahal spoke to New Spotlight on related issues. Excerpts:
Page 1 of 3 Next chevronRight icon
COAS General Thapa Returns Completing A Week-Long Visit To United Nations
French Embassy Hosts A Reception On Bastille Day
19 hours, 3 minutes ago
COAS General Thapa Meets Joint Vice Chief Of Staff Of U.S Armed Forces General Selva
Son Of Venezuelan Leader Nicolas Maduro In Nepal To Build Solidarity With Ruling Nepal Communist Party
Ambassador Pyakuryal Resigns From Post
Nepal Is Ready To Increase The Numbers Of UN Peacekeepers: COAS General Thapa
The World Bank Provides Rs. 12 Billion To Nepal To Strengthen Fiscal Federalism
COAS General Thapa Attended Women Peace And Security Network Meeting
Supreme Court Issues Stay Order To Halt Testing Of Vegetables and Fruits
I Was Misinformed By The Officials On A Letter Of Indian Embassy: PM Oli


text:
Kennesaw State professor receives educator of the year award from Georgia Art Education Association
Sandra Bird' an associate professor of art education at Kennesaw State University' recently won the…
Georgia (Oct 18, 2007) — Kennesaw State professor receives educator of the year award from Georgia Art Education Association
Cheryl Anderson Brown
Director of University Relations
Frances Weyand Harrison
fharris4@kennesaw.edu
Contact: Cheryl Anderson Brown' Assistant Director of Public Relationsfor the College of the Arts' 770−499−3417 or cbrown@kennesaw.edu
KENNESAW' Ga. (Oct. 18' 2007) −— Sandra Bird' an associate professor of art education at Kennesaw State University' recently won the Higher Education Art Educator of the Year award from the Georgia Art Education Association. The award was announced at the GAEA's Fall Professional Learning Conference in Augusta' Ga.' earlier this month. Bird lives in Kennesaw.
"I'm excited to receive this award'" says Bird. "It's a truly a reflection on the people that I work with. It's always nice to be recognized for your hard work." Bird has been a member of the National Art Education Association since 1996 and has participated in several NAEA and GAEA projects' including presenting 15 presentations at various conferences since 2000.
Bird has been a KSU faculty member since 1999. She received her Ph. D. in art education from Florida State University and is particularly interested in non−Western art. Her latest project is working with Blackwell Elementary School in Marietta' Ga. to help students learn about China through interactive art activities. Bird is also active with several Islamic educational organizations' including the Islamic Speaker's Bureau of Atlanta and the Istanbul Center for Culture and Dialogue.
A member of the 35−unit University System of Georgia' Kennesaw State University is a comprehensive' residential institution with a growing student population of more than 20'000 from 132 countries. The third−largest university in Georgia' Kennesaw State offers more than 60 graduate and undergraduate degrees' including a new doctorate in education.


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From fairways to greens and everything in between, the IHCC team is keeping it green! In 2012, they introduced a new hybrid grass, called T-1, to the greens. This dark green grass stands up to Bowling Green's summer heat and is expected to perform as well or better than the current G-2 grasses. Using other types of warm season grasses, including the Midiron Bermudagrass on the fairways and Zoysia within the tee boxes (plus the right amount of irrigation) ensures a green and vibrant course every round throughout the spring and summer season.
Course Improvements began in 2014 and included installation of a new irrigation system, continuous cart paths on the back nine, improved and expanded practice areas, and the installation of new bunkers. The course also features the "Billy Bunker System" that provides consistent playing conditions and prevents washouts during heavy rains.


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text:
\section{Introduction}

In \cite{Aizawa:2013vma} and \cite{Aizawa:2014uma} second-order PDEs, invariant under the ${\widehat{\mathfrak{cga}}}_\ell$ ($\ell=\textstyle{\frac{1}{2}}+\mathbb{N}_0$) centrally extended Conformal Galilei Algebra \cite{negro1997nonrelativistic}, were constructed. They were shown to possess a spectrum which is either continuous \cite{Aizawa:2013vma} or discrete (positive and bounded) \cite{Aizawa:2014uma}.
In \cite{Aizawa:2013vma} the invariant PDEs were obtained via Verma module representation, while in \cite{Aizawa:2014uma}  the so called on-shell condition was used (for the cases at hand the two approaches are proven to be equivalent). \par
In this paper we address several important issues that were not touched in these  two previous works. We name a few: the identification of the general class of invariant PDEs (which turns out to depend on real parameters  belonging  to a fundamental domain),  the existence of a contraction algebra, the reason for the cryptohermiticity (we use here the word adopted in \cite{smilga2009comments}) of the discrete spectrum, the construction of the {{Hilbert space connected with Pais-Uhlenbeck oscillators with unbounded spectrum, etc.}
\par
Specifically, the following list of results is derived in the present paper (we limit here in the Introduction to discuss the first non-trivial case obtained for ${\ell}={\textstyle{\frac{3}{2}}}$, the ${\ell}>{\textstyle{\frac{3}{2}}}$ cases are commented in Section {\bf 8}): two special differential realizations of ${\widehat{\mathfrak{cga}}}_\ell$ produce, as invariant PDEs,
Schr\"odinger equations with continuous and respectively discrete spectrum. Both realizations depend on a parameter $\gamma\neq0$. Unitarily inequivalent theories are recovered for $\gamma$ belonging to the fundamental domain $\gamma\in ]0,+\infty[$.\par
The $\gamma=0$ PDEs are decoupled equations. The continuum spectrum case corresponds to the free Schr\"odinger equation in $1+1$ dimensions, while the discrete spectrum case corresponds to a system of two decoupled {\em cryptohermitian oscillators} (namely, despite being non hermitian, possessing the same spectrum as two decoupled oscillators with the given frequencies). The parameter $\gamma$ can therefore be regarded as a deformation parameter and as a coupling constant.\par
Without loss of generality we can fix $\omega_1=1$ to be the energy mode of the first oscillator in the coupled PDE. Then, the ${\widehat{\mathfrak{cga}}}_{\textstyle{\frac{3}{2}}}$ invariance of the PDE is recovered if  the energy mode of the second oscillator possesses the critical values
$\omega_2=\pm {\frac{1}{3}}, \pm 3$ ($\omega_2=3$ is the solution given in \cite{Aizawa:2014uma}). The negative values correspond to an unbounded spectrum and, as explained later, are connected with special cases of the Pais-Uhlenbeck oscillators. At fixed $\omega_{1,2}$, the spectrum of the operators does not depend on the value of $\gamma$.\par
The $\gamma\rightarrow 0$ limit of the ${\widehat{\mathfrak{cga}}}_{\textstyle{\frac{3}{2}}}$ algebra produces a contraction algebra which is a symmetry subalgebra of the decoupled systems. For the decoupled oscillators (without loss of generality the analysis can be limited to the $\omega_1=1$, $\omega_2\geq 1$ domain), the PDE possesses a $9$-generator symmetry algebra at generic values, with enhanced symmetry at the critical values $\omega_2=1$ and $\omega_2=3$ (two different $12$-generator symmetry algebras are obtained at these special points). The $\gamma\rightarrow 0$ contraction algebra is a $8$-generator subalgebra of the $\omega_2=3$, $12$-generator decoupled symmetry.\par
In this paper we discuss the subtle connection with Pais-Uhlenbeck oscillators.
The Pais-Uhlenbeck model is a higher derivative system \cite{pais1950field,smilga2009comments,ketovMichiaki} which admits, via the Ostrogradski\u{\i} construction \cite{ostrogradski1850member},
a Hamiltonian formulation. The Ostrogradski\u{\i} Hamiltonian is canonically equivalent to a set of decoupled harmonic oscillators with alternating (positive and negative) energy modes.
In a series of papers \cite{galajinsky2013dynamical2,andrzejewski2014conformal,andrzejewski2014conformal2,andrzejewski2014hamiltonian,galajinsky2015dynamical}
the Pais-Uhlenbeck oscillators with energy modes given (up to a normalization factor) by the arithmetic progression $\omega_i = 2i-1$ were linked to the Conformal Galilei Algebras ${\widehat{\mathfrak{cga}}}_\ell$ (with $\ell = n-\textstyle{\frac{1}{2}}$).
As discussed in Section {\bf 8} the connection is rather subtle. To avoid any confusion, we should stress that the symmetry operators considered here are required to be realized as {\em first-order} differential operators. \par
The scheme of the paper is as follows: in Section {\bf 2} we present the ($\gamma\neq 0$-dependent) differential realization for the deformation of the free Schr\"odinger equation at ${\ell}={\textstyle\frac{3}{2}}$ and the differential realization for the coupled oscillator. The connection of the two differential realizations obtained by similarity transformations and change of the time coordinate is shown in Section {\bf 3}.  In Section {\bf 4} the most general solution of the ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$-invariant oscillator is given. The symmetry of the decoupled oscillator (with enhanced critical points at $\omega_2=1 $ and $\omega_2=3$)
is presented in Section {\bf 5}. The ${\ell}={\textstyle{\frac{3}{2}}}$ contraction algebra in the $\gamma\rightarrow 0$ limit is given in Section {\bf 6}. {{The Hilbert space for the oscillators and the relation with PT-symmetry
is discussed in Section {\bf 7}}}. In Section {\bf 8} the extension to the ${\ell}>{\textstyle\frac{3}{2}}$ cases and the relation to Pais-Uhlenbeck oscillators are commented. Generalizations of the present construction are discussed in the Conclusions.

\section{Differential realizations of the Conformal Galilei Algebra ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ }
The $d=1$ ${\ell}=\frac{3}{2}$ centrally extended Conformal Galilei algebra
${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ consists of eight generators ($z_0,z_\pm, w_{\pm 1}, w_{\pm 3}, c$), obeying the following
non-vanishing commutation relations
\bea\label{cga32}
\relax [z_0,z_\pm]&=& \pm 2i  z_\pm,\nonumber\\
\relax [z_+, z_-]&=& -4i z_0,\nonumber\\
\relax[ z_0,  w_k] &=& i k {w_k},\nonumber\\
\relax [ z_\pm,  w_k]&=& i(k\mp3) w_{k\pm2},\nonumber\\
\relax [ w_{|k|}, w_{-|k|}]&=&(3-2|k|){16} { c}.
\eea
We introduce in this Section two differential realizations of the above algebra.
As it will be explained in the following, the first differential realization corresponds to a symmetry of a deformed free Schr\"odinger equation in $1+1$ dimensions, while the second differential realization corresponds to a symmetry of another system
(possessing the same spectrum as two decoupled oscillators).\par
The first differential realization is a one-parameter ($\gamma\neq 0$) extension of the differential
realization obtained in \cite{Aizawa:2013vma} via standard left action on the space of functions defined on the coset (see also \cite{gomis2012schrodinger}). The presence of $\gamma$ can be understood by the fact that the combined rescalings of the space and time coordinates which preserve the conformal Galilei algebra structure lead to just one free parameter. In
this case we obtain first-order differential operators acting on functions of $\tau, x, y$. They are given by
\bea\label{freerep}
{\overline z}_+&=& \partial_\tau,\nonumber\\
{\overline z}_0&=&-2i\tau\partial_\tau -ix\partial_x-3iy\partial_y-2i,\nonumber\\
{\overline z}_-&=&-4\tau^2\partial_\tau-4(\tau x-\frac{3}{\gamma}y)\partial_x -12\tau y\partial_y -8(\tau-ix^2),\nonumber\\
{\overline w}_{+3}&=& \partial_y,\nonumber\\
{\overline w}_{+1}&=& -2i\tau\partial_y+\frac{2i}{\gamma}\partial_x,\nonumber\\
{\overline w}_{-1}&=& -4\tau^2 \partial_y+\frac{8}{\gamma}\tau\partial_x-\frac{8i}{\gamma}x,\nonumber\\
{\overline w}_{-3}&=& 8i\tau^3 \partial_y -\frac{24i}{\gamma}\tau^2\partial_x -48(\frac{1}{\gamma}\tau x +\frac{1}{\gamma^2}y),\nonumber\\
{\overline c}&=&\frac{1}{\gamma^2}.
\eea
The second differential realization is the $\gamma$ extension of the differential realization presented in \cite{Aizawa:2014uma}.
It is given by the first-order differential operators acting on functions of $t, x, y$,
\bea\label{oscrep}
{\widehat z}_0&=&\partial_t,\nonumber\\
{\widehat z}_+ &=& e^{2i t}(\partial_t+ ix\partial_x+ 3iy\partial_y+ix^2+2i),\nonumber\\
{\widehat z}_-&=&e^{-2it}(\partial_t-ix\partial_x-3iy\partial_y+\frac{12 }{\gamma}y\partial_x+7ix^2+
\frac{12 }{\gamma} xy-2i),\nonumber\\
{\widehat w}_{+3}&=& e^{3it}\partial_y,\nonumber\\
{\widehat w}_{+1}&=& e^{it}(\partial_y+\frac{2i}{\gamma}\partial_x+\frac{2i}{\gamma}x),\nonumber\\
{\widehat w}_{-1}&=&e^{-it}(\partial_y+\frac{4i}{\gamma}\partial_x-\frac{4i}{\gamma}x),\nonumber\\
{\widehat w}_{-3}&=& e^{-3it}(\partial_y+\frac{6i}{\gamma}\partial_x-\frac{18i}{\gamma}x-\frac{48}{\gamma^2}y),\nonumber\\
{\widehat c}&=& \frac{1}{\gamma^2}.
\eea
We consider now the three elements $\Omega_0, \Omega_{\pm 1}$ of the ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ enveloping algebra of degree $0,\pm 1$ (measured by the degree generator $-\frac{i}{2}z_0$), respectively:
\bea\label{omegafreerep}
{\Omega}_{+1}&=&  i {z}_++\frac{\gamma^2}{16} \left(\{{w}_{+3}, { w}_{-1}\}-\{{w}_{+1}, { w}_{+1} \}\right),\nonumber\\
{\Omega}_{0}&=&
i {\overline z}_0+\frac{\gamma^2}{32} \left(\{{w}_{+3}, { w}_{-3}\}-\{{ w}_{+1}, {w}_{-1} \}\right),\nonumber\\
{ \Omega}_{-1}&=&
i {\overline z}_{-}+\frac{\gamma^2}{16} \left(\{{w}_{+1}, { w}_{-3}\}-\{{w}_{-1}, { w}_{-1} \}\right),
\eea
where the curly brackets on the r.h.s. denote anticommutators.\par
The three operators ${\Omega}_{\pm 1,0}$ close the ${\mathfrak{sl}(2)}$ algebra, with ${\Omega}_0$ the Cartan element:
\bea
\relax [{\Omega}_0,{\Omega}_{\pm1}]&=& \mp 2{\Omega}_{\pm 1},\nonumber\\
\relax[{\Omega}_{+1},{\Omega}_{-1}]&=& 4{\Omega}_0.
\eea
It is important to note that for both differential realizations (\ref{freerep}) and (\ref{oscrep}), $\Omega_0, \Omega_{\pm 1}$ are presented
as second order differential operators.\par
For the differential realization (\ref{freerep}) we have
\bea\label{omegafreerep}
{\overline \Omega}_{+1}&=& i\partial_\tau -i\gamma x\partial_y + \frac{1}{2}\partial_x^2=i {\overline z}_+-\overline H_+,\nonumber\\
{\overline \Omega}_{0}&=& -2i\tau {\overline \Omega}_{+1}=i {\overline z}_0-\overline H_0,\nonumber\\
{\overline \Omega}_{-1}&=& -4\tau^2 {\overline \Omega}_{+1}=i {\overline z}_--\overline H_-.
\eea
The ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ on-shell invariant condition for ${\overline\Omega}_{\pm 1,0}$ (see \cite{Niederer:1972zz,Toppan:2015qla,Aizawa:2014uma} for a definition) is guaranteed by the fact that their only non-vanishing commutators with the ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ generators are expressed as
\bea
\relax [\overline z_0, {\overline{\Omega}}_{+1}]&=& 2i {\overline\Omega}_{+1} , \nonumber\\
\relax [\overline z_-, {\overline{\Omega}}_{+1}]&=& 4i{\overline \Omega}_0= 8\tau {\overline\Omega}_{+1} , \nonumber\\
\relax [\overline z_+, {\overline{\Omega}}_{0}]&=& -2i{\overline \Omega}_{+1}= \tau^{-1}{\overline\Omega}_0 , \nonumber\\
\relax [\overline z_-, {\overline{\Omega}}_{0}]&=& 2i{\overline \Omega}_{-1}= 4\tau{\overline\Omega}_0 , \nonumber\\
\relax [\overline z_+, {\overline{\Omega}}_{-1}]&=& -4i{\overline \Omega}_{0}=  2\tau^{-1}{\overline\Omega}_{-1}, \nonumber\\
\relax [\overline z_0, {\overline{\Omega}}_{-1}]&=& -2i{\overline \Omega}_{-1}.
\eea
The degree $1$ invariant equation
\bea\label{deforfree}
{\overline\Omega}_{1}\Psi(\tau,x,y)=0 \quad &\Rightarrow & i\partial_\tau\Psi =-\frac{1}{2}\partial_x^2\Psi
+i\gamma x\partial_y\Psi
\eea
is a Schr\"odinger equation with $\tau$ playing the role of a time coordinate. The parameter $\gamma$ is a coupling constant. This equation can be regarded as a $\gamma$-deformation of the free Schr\"odinger equation in $1+1$ dimensions.
\par
For the differential realization (\ref{oscrep}), $\Omega_0, \Omega_{\pm 1}$ are given by
\bea\label{omegaoscrep}
{\widehat{\Omega}}_{+1}&=& e^{2it}{\widehat \Omega}_0=i {{\widehat z}}_+- {\widehat H}_+,\nonumber\\
{ \widehat{\Omega}}_{0}&=& i\partial_t +\frac{1}{2}{\partial_x}^2-\frac{1}{2}x^2 -3y\partial_y-i\gamma x\partial_y-\frac{3}{2}=i { {\widehat z}}_0-{\widehat H}_0,\nonumber\\
{\widehat{\Omega}}_{-1}&=& e^{-2it}{\widehat \Omega}_0=i {{\widehat z}}_--{\widehat H}_-.
\eea

The on-shell invariant condition is guaranteed by the fact that their only non-vanishing commutators with the ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ generators are expressed as
\bea
\relax [ {\widehat z}_0, {{{\widehat \Omega}}}_{+1}]&=& 2i {{\widehat \Omega}}_{+1} , \nonumber\\
\relax [ {\widehat z}_-, {{{\widehat \Omega}}}_{+1}]&=& 4i{{\widehat \Omega}}_0=4i e^{-2it} {{\widehat \Omega}}_{+1} , \nonumber\\
\relax [{\widehat  z}_+, {{{\widehat \Omega}}}_{0}]&=& -2i{{\widehat \Omega}}_{+1}= -2ie^{2it}{{\widehat \Omega}}_0 , \nonumber\\
\relax [ {\widehat z}_-, {{ {\widehat \Omega}}}_{0}]&=& 2i{{\widehat \Omega}}_{-1}= 2i e^{-2it}{{\widehat \Omega}}_0 , \nonumber\\
\relax [ {\widehat z}_+, {{ {\widehat \Omega}}}_{-1}]&=& -4i{{\widehat \Omega}}_{0}=  -4ie^{2it}{{\widehat \Omega}}_{-1}, \nonumber\\
\relax [{\widehat z}_0, {{ {\widehat \Omega}}}_{-1}]&=& -2i{{\widehat \Omega}}_{-1}.
\eea

The degree $0$ invariant equation
\bea\label{cryptoosc}
{{\widehat \Omega}}_{0}\Psi(t,x,y)=0 \quad &\Rightarrow& i\partial_t\Psi =\left(
-\frac{1}{2}{\partial_x}^2+\frac{1}{2}x^2 +3y\partial_y+i\gamma x\partial_y+\frac{3}{2}\right)\Psi
\eea
is a Schr\"odinger equation with $t$ playing the role of the time coordinate. The non-hermitian operator in the right hand side possesses a discrete and positive spectrum. The parameter $\gamma$ is a coupling constant. The equation (\ref{cryptoosc}) can be regarded as a $\gamma$-deformation of a decoupled ``cryptohermitian oscillator" discussed in the following.

\section{Connection of the two differential realizations}
The two differential realizations (\ref{freerep}) and (\ref{oscrep}) of the ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ algebra introduced in Section {\bf 2}  induce Schr\"odinger equations from, respectively,
degree $1$ and degree $0$ invariant operators.\par
The two differential realizations are connected via a similarity transformation coupled with a redefinition of the time coordinate.\par
Let us denote as ${\widehat {g}}$ an operator entering (\ref{oscrep}) or (\ref{omegaoscrep}) and as ${\overline g}$ its  corresponding operator
entering (\ref{freerep}) or (\ref{omegafreerep}). For convenience we introduce the operator ${\widehat X}_+$ by setting, for ${\widehat z}_+$ in (\ref{oscrep}),
\bea
{\widehat z}_\pm = e^{\pm 2it}(\partial_t+{\widehat X}_\pm),\quad && {\widehat X}_+= ix\partial_x+ 3iy\partial_y+ix^2+2i.
\eea

The connection is explicitly realized by the similarity transformation
\bea
{\widehat g}\mapsto {\overline g} &= &e^S {\widehat g} e^{-S},\quad \quad (e^S= e^{S_2}e^{S_1}),\nonumber\\
S_1&=& t{\widehat X}_+,\nonumber\\
S_2&=&\frac{1}{2}x^2,
\eea
 supplemented by the redefinition of the time coordinate
\bea
t\mapsto\tau&=& \frac{i}{2}e^{-2it}.
\eea
The first similarity transformation (induced by $S_1$) allows to map
\bea
{\widehat z}_+\mapsto \widetilde{z}_+ &=&e^{S_1}{\widehat z}_+e^{-S_1}= e^{2it}\partial_t=\partial_\tau,
\eea
so that
\bea
{\widehat \Omega}_{+1}\mapsto \widetilde{ \Omega}_{+1}=e^{S_1}{\widehat \Omega}_{+1}e^{-S_1}= ie^{2it}\partial_t -\widetilde{H}_{+1},
\eea
with
\bea
{\widetilde H}_{+1}&=& e^{2it}\left( i{\widehat X}_++e^{t{\widehat X}_+}{\widehat H}_0e^{-t{\widehat X}_+}\right).
\eea
Due to the commutators
\bea
\relax [{\widehat X}_+, {\widehat H}_0] = 2i {\widehat K}_+, \quad &\quad&\quad [{\widehat X}_+,{\widehat  K}_+]= -2i{\widehat  K}_+,
\eea
where
\bea
{\widehat K}_+&=&\frac{1}{2}(\partial_x+x)^2-i\gamma x\partial_y,
\eea
we obtain
\bea
{\widetilde H}_{+1}&=& e^{2it} \left(i{\widehat X}_++{\widehat H}_0+{\widehat K}_+\right)-{\widehat K}_+.
\eea
The remarkable identity
\bea
i{\widehat X}_++{\widehat H}_0+{\widehat K}_+&=&0
\eea
implies that ${\widetilde H}_{+1}$ does not depend on the time coordinate (either $t$ or $\tau$).\par
The second similarity transformation (induced by $S_2$) allows to express
\bea
\widetilde{\Omega}_{+1}\mapsto {\overline \Omega}_{+1} &=& e^{S_2}\widetilde{\Omega}_{+1}e^{-S_2}= i\partial_\tau+\frac{1}{2}\partial_x^2-i\gamma x\partial_y
\eea
in the form which reduces, in the $\gamma\rightarrow 0$ limit, to the standard free Schr\"odinger equation in $1+1$ dimensions.\par
One should observe that the similarity transformation preserves the symmetry properties of the equations, mapping first-order invariant operators into first-order invariant operators.\par
The following commutative diagram is obtained:

\bea\label{commdiag}
&
\begin{array}{ccc}
~~~~coupled ~(\gamma\neq0):~~ \quad\quad {\bf{Free}}_\gamma ^{0,\pm1}(\tau)&
 \stackrel{{\bf S}}{\longleftrightarrow}&{\bf Osc}_\gamma^{0,\pm 1}(t)\\
 \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad~^{_{{\bf r}}}\downarrow\quad&&\downarrow^{_{{\bf r}}}\\
decoupled ~(\gamma=0):\quad~ ~~~{\bf Free}^{0,\pm 1}~ (\tau)&\stackrel{{\bf S}}{\longleftrightarrow}&{\bf Osc}^{0,\pm 1}(t)
\end{array}&
\eea
The left (right) part of the diagram  denotes the equations obtained from the  differential realizations (\ref{freerep}) and (\ref{oscrep}), respectively.  The horizontal arrows indicate the similarity transformation together with the change of the time coordinate, $\tau$ and $t$ respectively. \par
The three invariant PDEs (at degree $0, \pm 1$) are mapped into each other. \par
In the left part, the deformed Schr\"odinger invariant PDEs correspond to $deg~1$ and possess a continuous spectrum. \par
In the right part the deformed Schr\"odinger invariant PDEs correspond to $deg~0$. They possess a real, discrete spectrum which, as shown in Section {\bf 7}, coincides with the spectrum of two decoupled harmonic oscillators. We will call this system the {\em  $\ell=\frac{3}{2}$ oscillator}.
 \par
The vertical arrows denote the mapping to the decoupled systems.  This mapping can be reached in two ways:\par
{\em i}) the singular similarity transformation
\bea\label{notinv}
&g\mapsto R_1gR_1^{-1},  \quad { \textstyle with} \quad R_1= e^{\alpha y\partial_y}&
\eea
(such that $\gamma\rightarrow e^{-\alpha}\gamma) $ in the $\alpha\rightarrow\infty $ limit.\footnote{For $\alpha$ real the transformation (\ref{notinv}) is not unitary. The choice $\alpha=\ln \gamma$ allows to set $\gamma=1$. The free parameter $\gamma\in ]0,+\infty[$ labels different unitarily inequivalent theories with testable consequences. See, e.g., the probability decay to the vacuum state presented in (\ref{vacuumdecay}).}\par Despite the singularity of the limit, the invariant equations of the upper part of the diagram admits as non-singular limit the decoupled equations of the lower part of the diagram.  This similarity transformation preserves the symmetry of the equations, mapping first-order invariant operators into first-order invariant operators;\par
{\em ii}) the non-singular similarity transformation
\bea\label{nonsingular}
&g\mapsto R_2gR_2^{-1},  \quad R_2 =e^{\textstyle{({\textstyle\frac{3i}{8}} \gamma x+ {\textstyle{\frac{i}{8}}\gamma\partial_x-{\textstyle{\frac{1}{96}}}\gamma^2
\partial_y)\partial_y}}}.&
\eea
This non-singular transformation does not preserve the symmetry of the equation because some of the transformed generators are no longer first-order differential operators. \par
The four Schr\"odinger equations associated with the commutative diagram, starting from the upper right corner and proceeding clockwise, are:  (I) the $\ell=\frac{3}{2}$ oscillator (\ref{cryptoosc}),  (II) the decoupled (i.e. $\gamma=0$) $\ell=\frac{3}{2}$ oscillator, (III) the free Schr\"odinger equation in $1+1$ dimensions and, finally, (IV) the deformed free Schr\"odinger equation (\ref{deforfree}). \\
The three inequivalent (with constant, linear and quadratic potential, see \cite{Toppan:2015qla,Niederer:1972zz,Niederer:1973tz,Niederer:1974ba}) Schr\"odinger equations in $1+1$ dimensions invariant under the Schr\"odinger algebra are recovered as restrictions of the $\ell=\frac{3}{2}$ invariant PDEs.  Indeed, if we introduce the $x,y$ separation of variables, the equation of the harmonic oscillator and the free Schr\"odinger equation are recovered by setting $\partial_y\equiv 0$ from, respectively, equations (I)  and (IV). The linear Schr\"odinger equation is recovered from equation (IV)  after setting $\Psi(\tau,x,y) = \psi(\tau,x)\phi(y)$, with the restriction $\partial_y \phi(y)= k\phi(y)$.

\section{The general ${\ell} =\frac{3}{2}$ oscillator}

The ${\widehat{\mathfrak{cga}}}_{\frac{3}{2}}$ conformal Galilei invariance  requires the coupling parameter $\gamma\neq 0$. Since a unitary transformation changes its phase, we can assume
without loss of generality that $\gamma$ belongs to the fundamental domain $\gamma\in ]0,+\infty [$. \par
For $\gamma$ real, the invariant PDEs in the left part of the commutative diagram (\ref{commdiag}) are hermitian. This is not the case for the invariant PDEs in the right part of the diagram. Under hermitian conjugation, the $\ell=\frac{3}{2}$ oscillator equation is transformed into its conjugate
\bea
{\widehat \Omega}_0^\dagger(\gamma)\Psi(t,x,y) =0   &\equiv&( i\partial_t +\frac{1}{2}{\partial_x^2}-\frac{1}{2}x^2 +3 y\partial_y-i\gamma x\partial_y+\frac{3}{2})\Psi(t,x,y).
\eea
All operators
\bea\label{defosc}
\Theta &=& -\frac{1}{2}\partial_x^2 +\frac{1}{2}x^2 +\omega y\partial_y -i\gamma x\partial_y + C,
\eea
for any arbitrary constant $C$ and any $\gamma\neq 0$, induce a Schr\"odinger equation with $\ell=\frac{3}{2}$ Conformal Galilei symmetry, if $\omega$ is restricted to the values
\bea
\omega&=&\pm \frac{1}{3}, \pm 3.
\eea
The $\omega\leftrightarrow -\omega$ change of sign is explained by the hermitian conjugation. Understanding the $\omega\leftrightarrow
\frac{1}{\omega}$ transformation is subtler.  One should note at first that in the $\gamma=0$ decoupled case the role of the space coordinates $x,y$ can be exchanged by performing the canonical transformation
\bea
  y \ \leftrightarrow \ \frac{1}{\sqrt{2}} (x-\partial_x),&&\partial_y \ \leftrightarrow \ \frac{1}{\sqrt{2}} (x+\partial_x).  \nonumber
\eea
Next, the coupling term is introduced in terms of the non-singular similarity transformation
given by the inverse of equation (\ref{nonsingular}).  As it turns out, this procedure guarantees
the conformal Galilei invariance of the resulting PDE.\par
An explicit check of the symmetries of this class of PDEs proves that, in order to have the on-shell invariant equations $[{\widehat z}_\pm, {\widehat \Omega}_0]= f_\pm\cdot{\widehat \Omega}_0$, with $f_\pm$ arbitrary functions of the coordinates and symmetry generators of the form
${\widehat z}_\pm =e^{\pm i\lambda t}(\partial_t + {X}_\pm)$, (${X}_\pm$ time-independent operators and $\lambda\neq 0$), the following necessary and sufficient condition has to be satisfied:  the two equations
\bea
\lambda(\omega^2+1-\frac{5}{2}\lambda^2)&=&0,
\nonumber\\
-3\lambda^2+3\lambda^4+2\lambda\omega+
4\lambda^3\omega-\lambda^2\omega^2-2\lambda\omega^3&=&0,
\eea
must be simultaneously solved. The only non-vanishing solutions for $\lambda$ are encountered at  $\omega=\pm 3$ and $\omega=\pm\frac{1}{3}$. Therefore, the $\omega=\pm \frac{1}{3}, \pm 3$ critical values are special points of enhanced symmetry.

\section{Symmetry of the decoupled $\ell=\frac{3}{2}$ oscillator}

By applying the same considerations as in Section {\bf 4}, it is sufficient to analyze the symmetry of the decoupled ($\gamma=0$) $\ell=\frac{3}{2}$ operator
\bea
{\Omega} &=& i\partial_t +\frac{1}{2}\partial_x^2-\frac{1}{2}x^2 +\omega y\partial_y
\eea
in the range $\omega\in [1,\infty [$.\par
For a generic $\omega$ the following invariant operators can be encountered at degree $0, \pm\frac{1}{2}, \pm\frac{\omega}{2},\pm 1$:
\bea\label{generic}
{z}_\pm &=& e^{\pm2i t} (\partial_t\pm ix\partial_x +i\omega y\partial_y+ix^2\pm {\textstyle{\frac{i}{2}}}),\nonumber\\
{ z}_0&=& \partial_t +i\omega y\partial_y,\nonumber\\
{d}&=& -\frac{i}{2}\partial_t,\nonumber\\
{c}&=&1,\nonumber\\
{w}_{\omega}&=& e^{i\omega t} \partial_y,\nonumber\\
{w}_{1}&=& e^{it}(\partial_x +x),\nonumber\\
{w}_{-1}&=& e^{-it}(\partial_x -x),\nonumber\\
{w}_{-\omega}&=& e^{-i\omega t} y.
\eea
${ d}$ is the degree operator. Explicitly, the degree is
\bea
&\pm 1:~ {z}_\pm;\quad
0:~{ z}_0, { d}, { c};\quad
\frac{\pm \omega}{2}: ~{w}_{\pm \omega};\quad
\quad \frac{\pm 1}{2}: ~{w}_{\pm 1}.&
\eea
This $9$-generator symmetry algebra closes the ${\mathfrak{u}}(1){
\supset \hspace{-1em}\hspace{-1pt}+}
(\mathfrak{sch}(1)\oplus{\mathfrak{h}}(1))$ algebra, with non-vanishing commutation relations given by
\bea
\relax [{d}, { z}_\pm ] &=& \pm { z}_\pm,\nonumber\\
\relax [ {d}, {w}_k]&=& \frac{k}{2}{w}_k,\nonumber\\
\relax [{z}_0,{z}_\pm ] &=& \pm 2i { z}_\pm,\nonumber\\
\relax [{z}_+,{ z}_-] &=& - 4 i { z}_0,\nonumber\\
\relax [ {z}_0, {w}_{\pm 1}]&=& \pm i {w}_{\pm 1},\nonumber\\
\relax [{ z}_\pm, {w}_{\mp 1}]&=& \mp 2i {w}_{\pm 1},\nonumber\\
\relax [{w}_1,{w}_{-1}] &=& - 2 {c},\nonumber\\
\relax [w_\omega, w_{-\omega}]&=& {c}.
\eea
$ {d}$ is the generator of the ${\mathfrak{u}}(1)$ subalgebra, while $ {z}_0, {z}_{\pm}, {w}_{\pm 1}, {c} $ generate the Schr\"odinger algebra $ \mathfrak{sch}(1) $
and $ w_{\pm \omega}, {c} $ generate the Heisenberg algebra $ {\mathfrak{h}}(1). $ \\
The critical values $\omega=1$ and $\omega =3$ are points of enhanced symmetry for the decoupled system.\par
\subsection{The enhanced symmetry for the decoupled $\omega=1$ system}

At the critical value $\omega=1$ three extra generators are found at degree $0$ and $-1$:
\bea
q_1 &=& y(\partial_x+x),\nonumber\\
q_2 &=& e^{-2it}y^2,\nonumber\\
q_{3}&=& e^{-2it} y(\partial_x-x).
\eea
They have to be added to the previous set of (generic) symmetry generators
\bea
{ z}_\pm &=& e^{\pm2i t} (\partial_t\pm ix\partial_x +i y\partial_y+ix^2\pm {\textstyle{\frac{i}{2}}}),\nonumber\\
{z}_0&=& \partial_t +i y\partial_y,\nonumber\\
{d}&=& -\frac{i}{2}\partial_t,\nonumber\\
{c}&=&1,\nonumber\\
{w}_{1b}&=& e^{i t} \partial_y,\nonumber\\
{w}_{1a}&=& e^{it}(\partial_x +x),\nonumber\\
{w}_{-1a}&=& e^{-it}(\partial_x -x),\nonumber\\
{ w}_{-1b}&=& e^{-i t} y,
\eea
where we denoted, for $\omega =1$, $w_{\pm\omega}$ entering (\ref{generic}) as ``$w_{\pm 1b}$". \par
The extra non-vanishing commutation relations involving the $q_i$'s generators are
\bea
\relax [{ z}_0,q_1]&=& iq_1,\nonumber\\
\relax [ {d}, q_2] &=& - q_2,\nonumber\\
\relax[{d}, q_3] &=& -q_3,\nonumber\\
\relax[{z}_+, q_3]&=& -2i q_3,\nonumber\\
\relax [ {z}_-, q_1]&=& 2i q_3,\nonumber\\
\relax[ { w}_{1b}, q_1]&=& {w}_{1a},\nonumber\\
\relax[{ w}_{-1a}, q_1] &=& 2{ w}_{-1b},\nonumber\\
\relax[ { w}_{1b}, q_2]&=& 2{ w}_{-1b},\nonumber\\
\relax [ {w}_{1b}, q_3] &=& {w}_{-1a},\nonumber\\
\relax [ {w}_{1a}, q_3] &=& -2 { w}_{-1b},\nonumber\\
\relax [ q_1, q_3] &=& -2q_2.
\eea
The symmetry algebra closes as a non semi-simple, $12$-generator, Lie algebra.

\subsection{The enhanced symmetry for the decoupled $\omega=3$ system}

At the $\omega =3$ critical value the three extra generators
${ r}_{-j}$, $j=1,2,3$, of degree $-j$, are encountered.
We have, explicitly,
\bea\label{rgenerators}
{r}_{-1} &=& e^{-2it} y(\partial_x+x),\nonumber\\
{ r}_{-2} &=& e^{-4 it} y(\partial_x -x),\nonumber\\
{ r}_{-3} &=& e^{-6 i t} y^2.
\eea
At $\omega=3$ the symmetry algebra is a 12-generator algebra which differs from the $12$-generator symmetry algebra of the $\omega=1$ decoupled system.\par
 The extra non-vanishing commutation relations involving the ${r}_{-j}$ generators are given by
\bea
\relax [{d}, {r}_{-j}] &=& -j { r}_{-j},\nonumber\\
\relax [{z}_0, { r}_{-1}] &=& i { r}_{-1},\nonumber\\
\relax [{z}_0, { r}_{-2}]& =& -i {r}_{-2},\nonumber\\
\relax [{ z}_{-}, {r}_{-1}]& =& 2i {r}_{-2},\nonumber\\
\relax [{z}_{+}, {r}_{-2}]& =& -2i {r}_{-1},\nonumber\\
\relax [{ w}_{+3}, {r}_{-1}]& =& {w}_{+1},\nonumber\\
\relax [{w}_{-1}, {r}_{-1}]& =& 2 { w}_{-3},\nonumber\\
\relax [{w}_{+3}, {r}_{-2}]& =& { w}_{-1},\nonumber\\
\relax [{w}_{+1}, {r}_{-2}]& =& -2 {w}_{-3},\nonumber\\
\relax [{w}_{+3}, { r}_{-3}]& =& 2 {w}_{-3},\nonumber\\
\relax [{r}_{-1}, { r}_{-2}]& =& -2 {r}_{-3}.
\eea

\section{The contraction algebra}

A contraction algebra is recovered from (\ref{oscrep})
by taking the $\gamma\rightarrow 0$ limit and by suitably rescaling the generators. The contraction requires the rescaling ${\widehat g}\mapsto {\widetilde g}= \gamma^s {\widehat g}$
(${\widehat g}$ is any generator entering (\ref{oscrep})), with the power $s$ given by
\bea
s=0 &:& {\widehat z}_0, {\widehat z}_+, {\widehat w}_3, {\widehat c},\nonumber\\
s=1 &:& {\widehat z}_-, {\widehat w}_1, {\widehat w}_{-1},\nonumber\\
s=2&:& {\widehat  w}_{-3}.
\eea
The contracted $8$-generator algebra expressed by ${\widetilde z}_\pm, {\widetilde z}_0, {\widetilde c}, {\widetilde w}_k$ ($k=\pm1, \pm 3$) is a subalgebra of the full $12$-generator symmetry algebra of the $\omega=3$ decoupled system with generators given by (\ref{generic}) and (\ref{rgenerators}). The identification goes as follows
\bea
{\widetilde z}_+ &=& e^{\widetilde S}{z}_+e^{-{\widetilde S}}= e^{2it}(\partial_t+ ix\partial_x +3iy\partial_y+ix^2+2i),\nonumber\\
{\widetilde z}_0 &=& e^{\widetilde S}(2i{d}-{\textstyle{\frac{3i}{2}}}{c})e^{-{\widetilde S}}= \partial_t,\nonumber\\
{\widetilde z}_- &=& e^{\widetilde S} (12 i{ r}_{-1})e^{-{\widetilde S}}= 12 i e^{-2it} y(\partial_x+x),\nonumber\\
{\widetilde w}_{+3} &=& e^{\widetilde S}{w}_{+3}e^{-{\widetilde S}}= e^{3it}\partial_y,\nonumber\\
{\widetilde w}_{+1} &=& e^{\widetilde S} (-2i{ w}_{+1})e^{-{\widetilde S}}= 2i e^{it} (\partial_x+x),\nonumber\\
{\widetilde w}_{-1} &=& e^{\widetilde S} (-4i{w}_{-1})e^{-{\widetilde S}}= 4i e^{-it} (\partial_x-x),\nonumber\\
{\widetilde w}_{-3} &=& e^{\widetilde S} (48{w}_{-3})e^{-{\widetilde S}}= -48 e^{-3it} y,\nonumber\\
{\widetilde c} &=& e^{\widetilde S} {c}e^{-{\widetilde S}}= 1,
\eea
with the similarity transformation  given by ${\widetilde S}= -\textstyle{\frac{3}{2}}it$.\par
The contraction algebra corresponds to the two-dimensional Euclidean algebra acting on {{two sets}} of creation/annihilation operators.
We have the
${\mathfrak{e}}(2){\supset \hspace{-1em}\hspace{-1pt}+}{\mathfrak{h}}(2)$ algebra, with non-vanishing commutators given by
\bea
\relax [{\widetilde z}_0,{\widetilde z}_{\pm}]&=& \pm 2i{\widetilde z}_\pm,\nonumber\\
\relax[ {\widetilde z}_0, {\widetilde w}_{k}]&=& k{\widetilde w}_k,\nonumber\\
\relax[{\widetilde z}_+,{\widetilde w}_{-1}]&=& -4i {\widetilde w}_{+1},\nonumber\\
\relax [ {\widetilde z}_-, {\widetilde w}_{+3}]&=& 6i {\widetilde w}_{+1},\nonumber\\
\relax[{\widetilde z}_-,{\widetilde w}_{-1}] &=& 2i {\widetilde w}_{-3},\nonumber\\
\relax[{\widetilde w}_{|k|}, {\widetilde w}_{-|k|}]&=& (3-2|k|) 16{\widetilde c}.
\eea

\section{Non-hermitian deformed oscillators with real eigenvalues}

The non-hermitian operator derived from the degree $0$ invariant equation (\ref{cryptoosc}) is
\bea\label{cryptoosc2}
H_0(\gamma)&=&
-\frac{1}{2}{\partial_x}^2+\frac{1}{2}x^2 +3y\partial_y+i\gamma x\partial_y+\frac{3}{2}.
\eea
It may be written (see (\ref{oscrep})) in the form
\begin{equation} \label{cryptHamil1}
  H_0(\gamma) = \frac{\gamma^2}{16} ({\widehat w}_{-1}{\widehat  w}_{+1} -{\widehat  w}_{-3}{\widehat  w}_{+3}) + 2.
\end{equation}
{{A discrete spectrum can be calculated algebraically from a lowest weight representation.}}
The ground state $\psi_{0,0} $ is defined by
\begin{equation} \label{Def:GS}
   {\widehat w}_{+1} \psi_{0,0} = {\widehat w}_{+3} \psi_{0,0} = 0.
\end{equation}
The solution of the above equations gives the explicit expression of the ground state (up to normalization)
\bea
 \psi_{0,0} &=& e^{-x^2/2}.
\eea
The excited states are given by
\begin{equation} \label{Def:ExS}
  \psi_{n,m} = {\widehat w}_{-1}^n {\widehat w}_{-3}^m \psi_{0,0}.
\end{equation}
The corresponding eigenvalue $E_{n,m}$ is computed by the commutation relations
\begin{equation} \label{SectGenAlg}
  [H_0(\gamma), {\widehat w}_{-1}] = {\widehat w}_{-1}, \qquad
  [H_0(\gamma),{\widehat w}_{-3}] = 3{\widehat w}_{-3}.
\end{equation}
One finds that the operator $ H_0(\gamma) $ has real discrete eigenvalues which are identical to the decoupled harmonic oscillator eigenvalues:
\bea
E_{n,m}&=& n+3m +2. \label{Eigen}
\eea
The explicit form of the eigenfunctions are readily obtained from (\ref{oscrep}). We list some of them.
\bea
   \psi_{1,0} &=& {\widehat w}_{-1} \psi_{0,0} = -\frac{8i}{\gamma} x e^{-x^2/2-it},
   \nonumber\\
   \psi_{2,0} &=& {\widehat w}_{-1}^2 \psi_{0,0} =  2\Big( \frac{4i}{\gamma} \Big)^2 (2x^2-1) e^{-x^2/2-2it},
  \nonumber \\
   \psi_{0,1} &=& {\widehat w}_{-3} \psi_{0,0} = -\frac{24}{\gamma^2} (i\gamma x+ 2y) e^{-x^2/2-3it},
  \nonumber \\
   \psi_{1,1} &=& {\widehat w}_{-1} {\widehat w}_{-3} \psi_{0,0} = \frac{48}{\gamma^2} \Big( 1 -4x^2 + \frac{4i}{\gamma}xy \Big) e^{-x^2/2-4it}.
\eea
In general, the excited state wavefunctions have the form of
\begin{equation} \label{ExStWaveF}
  \psi_{n,m} = {\cal P}_{n,m}(x,y) e^{-x^2/2} e^{-(n+3m)it},
\end{equation}
where $ {\cal P}_{n,m}(x,y) $ is a degree $n+m$ polynomial in $x, y. $
The phase factor $ e^{-(n+3m)it} $ is not essential to the eigenvalue problem. \par
Following
\cite{smilga2009comments}, we can call ``cryptohermitian" any non-hermitian operator (such as (\ref{cryptoosc2})) with a real spectrum. The operator (\ref{cryptoosc2}), being invariant under the transformations $x\mapsto -x, y\mapsto y, i\mapsto -i$, is PT-symmetric.  Therefore, it belongs to the class of non-hermitian PT-symmetric operators (see \cite{beboe} and \cite{beboe2} and \cite{ben} for a review) with real spectrum. It was pointed out in
\cite{mos1},
\cite{mos2} and
\cite{mos3} that PT-symmetric operators are often pseudo-hermitian. It turns out that this is not the case for the operator (\ref{cryptoosc2}), since the similarity transformation (\ref{nonsingular}) is not realized by a hermitian operator. For this reason the stronger notion of quasi-hermiticity, as well as the constructions based upon that (see \cite{sgh}, \cite{mobat} and \cite{bazno}),  is not applicable in our context. \par
In Section {\bf 8} we discuss the connection of the operator (\ref{cryptoosc2}) with a special class of Pais-Uhlenbeck oscillators, recovered at given algebraic frequencies. It is worth mentioning that PT-symmetry in the context of Pais-Uhlenbeck oscillators (at arbitrary, unequal, frequencies) was investigated in \cite{beman},
\cite{mann} and \cite{limi}.\par
In order to pave the way to discuss the connection between Conformal Galilei Algebras and Pais-Uhlenbeck
oscillators at special frequencies, we need to introduce the operator $K({\overline\gamma})$, obtained from
(\ref{cryptoosc2}) via a non-canonical transformation which preserves the canonical commutation relations, but does not preserve the hermitian conjugation property. Up to a vacuum energy constant, we can set
\bea\label{herosc}
K({\overline\gamma})&=& a^\dagger a+3 b^\dagger b +\frac{1}{2}+{\overline \gamma}(a+a^\dagger)b,
\eea
through the positions
\bea
&a=\frac{1}{\sqrt{2}}(x+\partial_x),\quad a^\dagger =\frac{1}{\sqrt{2}}(x-\partial_x),\quad b=
\frac{1}{\sqrt{2}}(z+\partial_z),\quad b^\dagger =
\frac{1}{\sqrt{2}}(z-\partial_z),\quad {\overline\gamma}=\frac{i}{\sqrt{2}}\gamma.&
\eea
The
two independent creation/annihilation operators ($a, a^\dagger$ and $b, b^\dagger$) have non vanishing commutators: $[a,a^\dagger]=[b,b^\dagger]=1$. \par
The similarity transformation (\ref{nonsingular}) maps $ K({\overline\gamma}) $ into the decoupled hermitian operator $K(0)$ through
\bea\label{Kgamma}
K({\overline\gamma}) &=& R K(0) R^{-1},  \quad  R =e^{-( {\frac{{\overline\gamma}}{2}}a^\dagger b+\frac{\overline\gamma}{4}ab+\frac{{\overline\gamma}^2}{48}b^2)}.
\eea
Since, as mentioned before, $R$ is not hermitian, $K({\overline{\gamma}})$, for ${\overline\gamma}\neq 0$, is not a pseudo-hermitian operator.\par
The operator $K({\overline\gamma})$ acts on the {{Hilbert space ${\mathcal L}^2({ {\mathbb R}^2})$}}, the space of square-integrable functions defined on the coordinates $x,z$. \par
\par
One can read from the commutators
\bea\label{mode}
[K(\overline\gamma), A_\lambda] &=& \lambda A_\lambda
\eea
which excited modes are created.\par
For any ${\overline \gamma}\neq 0$, the solutions of the (\ref{mode}) equation are obtained for $\lambda=\pm 3, \pm\frac{1}{3}$. The corresponding modes are
\bea\label{bogoljubovlike}
A_{-3}&=& b,\nonumber\\
A_{-1}&=& a+\frac{1}{2}{\overline\gamma}b,\nonumber\\
A_{+1}&=& a^\dagger-\frac{1}{4}{\overline\gamma}b,\nonumber\\
A_{+3}&=& b^\dagger-\frac{1}{2}\overline\gamma a^\dagger -\frac{1}{4}\overline\gamma a+\frac{1}{24}\overline\gamma^2b.
\eea
In this basis the non-vanishing commutators are
\bea
\relax [A_{-i},A_j]&=&\delta_{ij}, \quad\quad\quad (i,j=1,3).
\eea
The non-hermitian operator $K(\overline\gamma)$ commutes with the ``non-hermitian analog of the Number operator", $N(\overline\gamma)$. In terms of the $A_k$ modes, the operators are given by
\bea
K(\overline\gamma)&=& 3A_3A_{-3}+A_1A_{-1}+\frac{1}{2},\nonumber\\
N(\overline\gamma)&=& A_3A_{-3}+A_1A_{-1},\nonumber\\
\relax [K(\overline\gamma),N(\overline\gamma)]&=&0.
\eea
The Fock vacuum $|vac>$ satisfies
\bea
a|vac>=b|vac>=0&,& A_{-1}|vac>=A_{-3}|vac>=0.
\eea
The Hilbert space ${\mathcal L}^2({{\mathbb R}^2})$ can be spanned by both sets of (unnormalized) states,
\bea
|n,m> &=& (a^\dagger)^n(b^\dagger)^m|vac>,\nonumber\\
|\overline n,\overline m>&=& A_1^nA_3^m|vac>.
\eea
The invertibility of the Bogoliubov-type transformation (\ref{bogoljubovlike}) implies that the
$|\overline n,\overline m>$ eigenstates form a complete (albeit non-orthogonal) set which can be expressed in terms of the decoupled eigenvectors $|n,m>$ (the converse is also true).\par
We can therefore write
\bea
&|vac>=|0,0>=|\overline 0,\overline 0>.&
\eea
The spectrum of $K(\overline\gamma)$, $N(\overline\gamma)$ coincides with the spectrum of the Hamiltonian and Number operator of a decoupled harmonic oscillator.
$|\overline n,\overline m> $ is an eigenvector for $K(\overline\gamma)$, $N(\overline\gamma)$ with respective eigenvalues $n+3m+\frac{1}{2}$ and $n+m$. In increasing order of $K(\overline\gamma)$ eigenvalues, the first (unnormalized) common eigenvectors of $K(\overline\gamma)$, $N(\overline\gamma)$ are
\bea
(\frac{1}{2},0): && |\overline 0,\overline 0>=|0,0>=|vac>,\nonumber\\
(\frac{3}{2},1): && |\overline 1,\overline 0>=|1,0>,\nonumber\\
(\frac{5}{2},2): && |\overline 2,\overline 0>=|2,0>,\nonumber\\
(\frac{7}{2},1): && |\overline 0,\overline 1>=|0,1>-\frac{1}{2}\overline\gamma|1,0>,\nonumber\\
(\frac{7}{2},3): && |\overline 3,\overline 0>=|3,0>,\nonumber\\
(\frac{9}{2},2): && |\overline 1,\overline 1>=|1,1>-\frac{1}{2}\overline\gamma |2,0>-\frac{1}{4}\overline\gamma|0,0>,\nonumber\\
(\frac{9}{2},4): && |\overline 4,\overline 0>=|4,0>.
\eea
Since the operators are non-hermitian, their eigenvectors are non-orthogonal. This implies measurable physical consequences.
Let us suppose  that we are able to prepare the system in a given common eigenvector of $K(\overline\gamma)$, $N(\overline\gamma)$, let's say the state $|\overline 1,\overline 1>$. Following the standard rule of Quantum Mechanics we can compute the probability for this state to collapse, after a measurement operation, to the vacuum state. A simple computation shows that the probability is $p=|_N<\overline 1,\overline 1|\overline0,\overline0>_N|^2 $ ($|\overline 0,\overline 0>_N$, $|\overline 1,\overline 1>_N$ are the normalized states). In the case at hand we have
\bea\label{vacuumdecay}
p&=&\frac{|\overline \gamma|^2}{16+9|\overline\gamma|^2}.
\eea
This probability is restricted in the range $0\leq p<\frac{1}{9}<1$.
The deformation coupling parameter ${\overline \gamma}$, via its squared modulus,  has testable consequences.

\section{Comment on  Pais-Uhlenbeck oscillators and the $\ell\geq{\textstyle{\frac{5}{2}}} $ cases}

The same spectrum of eigenvalues is obtained for\\
{\em i}) the coupled ($\gamma\neq 0$) cryptohermitian operator (\ref{cryptoosc2}), \\
{\em ii}) the decoupled ($\gamma=0$) cryptohermitian operator and  (up to a vacuum energy shift) \\
{\em iii}) the hermitian
Hamiltonian (given by (\ref{herosc}) for ${\overline\gamma}=0$) of two decoupled oscillators.\par
The construction of Section {\bf 7} can be repeated by starting with the hermitian conjugate of the (\ref{cryptoosc2}) operator. In this case the spectrum of the three resulting operators is unbounded. It is given, up to the vacuum energy shift, by $E_{n,m}= n-3m$. The Hilbert space of the decoupled harmonic oscillators with energy modes $1,-3$ continues to be {{ ${\mathcal L}^2({ {\mathbb R}^2}$)}}, obtained by applying the creation operators $a^\dagger, b^\dagger$ to the Fock state $|0,0>$ ($a|0,0>=b|0,0>=0$).
Due to the unboundedness of the spectrum, $|0,0>$ can no longer be interpreted as the vacuum state.\par
The system with unbounded spectrum is related to the Pais-Uhlenbeck oscillators. We recall \cite{pais1950field,smilga2009comments} that the Pais-Uhlenbeck model is a higher derivative system. It admits, via the Ostrogradski\u{\i} construction \cite{ostrogradski1850member} (see \cite{Woodard:2015zca} for a review),
a Hamiltonian formulation. The resulting Ostrogradski\u{\i} Hamiltonian is canonically equivalent to a set of decoupled harmonic oscillators with alternating (positive and negative) energy modes. The $n$-oscillator Pais-Uhlenbeck system is canonically expressed as
\bea
H_n &=& \sum_{i=1}^{n} (-1)^{i+1} \omega_i a_i^\dagger a_i,
\eea
where $\omega_i\in {\mathbb R}$ and the constraint $\omega_i<\omega_{i+1}$ is satisfied.\par
The harmonic oscillator with energy modes $1,-3$ is a special case of the $2$-oscillator Pais-Uhlenbeck model.
In a series of papers \cite{galajinsky2013dynamical2,andrzejewski2014conformal,andrzejewski2014conformal2,andrzejewski2014hamiltonian,galajinsky2015dynamical}
the Pais-Uhlenbeck oscillators with energy modes given (up to a normalization factor) by the arithmetic progression $\omega_i = 2i-1$ were linked to the Conformal Galilei Algebras ${\widehat{\mathfrak{cga}}}_\ell$ (with $\ell = n-\textstyle{\frac{1}{2}}$).\par
The present analysis proves that this association is rather subtle.
We would like to stress that in this paper we consider (as it is standard in PDE's theory) the symmetry generators to be at most {\em first-order} differential operators. In this respect the Pais-Uhlenbeck PDE given by the decoupled harmonic oscillators does not possess any enhanced symmetry (not even at the special $1,-3$ energy modes).  The PDE, invariant under the Conformal Galilei Algebra, is obtained for the coupled operator only for $\gamma\neq 0$. The decoupled operator $(\gamma=0)$
possesses the 12-generator symmetry algebra (introduced in Section {\bf 6}) which does not contain the Conformal Galilei Algebra as a subalgebra. \par
Even so, these results are not in contradiction with the findings in previous works on Pais-Uhlenbeck oscillators.  For example,
in \cite{andrzejewski2014conformal} it was shown that on the Hamiltonian level the $\omega_1=1,\omega_2=-3$ Pais-Uhlenbeck oscillators possess the (centrally extended) $\frac{3}{2}$-conformal Galilei symmetry in terms of Poisson brackets among conserved charges. It can be shown, on the other hand, that at the quantum level, some of the Noether charges entering formula (39) in \cite{andrzejewski2014conformal} are {\em second-order} differential operators. \par
For general half-integer ${\ell}$, the invariant PDEs which possess the Conformal Galilei algebra ${\widehat{\mathfrak{cga}}}_\ell$ (for a definition, see \cite{negro1997nonrelativistic}) as a symmetry algebra, depend on ${\ell }+{\textstyle{\frac{3}{2}}}$ coordinates. The invariant PDEs are deformations of decoupled equations, depending
on ${\ell}-{\textstyle{\frac{1}{2}}}$ deformation parameters $\gamma_j\neq 0$ ($j=1,\ldots, {\ell}-{\textstyle{\frac{1}{2}}}$). The decoupled systems are recovered in the limit, for any $j$,$\gamma_j\rightarrow 0$.\par
The invariant PDEs with continuous spectrum are
\bea
i\partial_\tau \Psi(\tau, {\vec x})&=&\left(-{\textstyle{\frac{1}{2}}\partial_{x_1}^2+ i\sum_{j=1}^{\ell-{\textstyle{\frac{1}{2}}}}\gamma_jx_j\partial_{x_{j+1}}}\right)\Psi(\tau,\vec{ x}).\label{genfree}
\eea

The invariant PDEs with discrete spectrum are
\bea\label{genosc}
i\partial_t\Psi(t, {\vec x})&=&\left(-{\textstyle{\frac{1}{2}}\partial_{x_1}^2+
{\textstyle {\frac{1}{2}}}x_1^2+\sum_{i=2}^{\ell+{\textstyle{\frac{1}{2}}}}\omega_ix_i\partial_{x_i}+ i\sum_{j=1}^{\ell-{\textstyle{\frac{1}{2}}}}\gamma_jx_j\partial_{x_{j+1}}}\right)\Psi(t,\vec{ x}).
\eea
The energy modes $\omega_i$ are normalized so that $|\omega_i| = 2i-1$. The solution
$\omega_i = \epsilon_i |\omega _i|$ with all positive signs ($\forall i$, $\epsilon_i=+1$) corresponds to the bounded discrete spectrum discussed in \cite{Aizawa:2014uma}. By taking the hermitian conjugation, the solution with
flipped signs, $\epsilon_i =-1$ for all $i$, also leads to a ${\widehat{\mathfrak{cga}}}_\ell$-invariant PDE.\par
An explicit computation of the on-shell condition for $\ell={\textstyle{\frac{5}{2}}}$ (similar to the one presented in Section {\bf 4}), proves that  the ${\widehat{\mathfrak{cga}}}_{\frac{5}{2}}$ invariance is guaranteed by both choices of signs, $\epsilon_2=\pm 1$ and $\epsilon_3=\pm 1$. As explained above, the alternating choice ($\epsilon_2=-1, \epsilon_3=+1$) is related to a special case of three Pais Uhlenbeck oscillators (with $1,-3,5$ energy modes). \\
An open problem is finding a general proof, valid for all half-integer ${\ell}$, that every choice of $\epsilon
_i=\pm 1$ signs lead to the ${\widehat{\mathfrak{cga}}}_l$ symmetry algebra of the PDE equation (\ref{genosc}).
\section{Conclusions}

\par

We summarize, in the Conclusions, the list of new results obtained in this paper. \par
We constructed the most general class of second-order PDEs, invariant under the $d=1$ centrally extended Conformal Galilei Algebras ${\widehat{\mathfrak{cga}}}_\ell$ with half-integer $\ell$, proving that they are Schr\"odinger equations which are deformations of decoupled equations. For $\ell=\frac{3}{2}$ the unique deformation parameter is $\gamma\neq 0$ (the decoupled systems being recovered in the $\gamma\rightarrow 0$ limit).\par
At $\ell=\frac{3}{2}$ the invariant PDEs with discrete spectrum, besides $\gamma$, depend on two frequencies $\omega_1, \omega_2$ (entering the equation in non-symmetric form).  The invariance under ${\widehat{\mathfrak{cga}}}_{\ell=\frac{3}{2}}$ is only recovered at special critical values of the ratio
$r=\frac{\omega_2}{\omega_1}$ given by $r=\pm\frac{1}{3}, \pm 3$. The $r=3$ value reproduces the bounded spectrum presented in \cite{Aizawa:2014uma}, while the negative values $r<0$ produce an unbounded spectrum which coincides with the spectrum of two Pais-Uhlenbeck oscillators at the given ratio $r$. \par
We further investigated the symmetry algebra of the $\gamma=0$ decoupled systems for a generic value of  $r=\frac{\omega_2}{\omega_1}$, obtaining the following results.  Enhanced symmetries are encountered at the critical values $r=\pm\frac{1}{3},\pm 1, \pm 3$. Two inequivalent $12$-generator symmetry algebras are recovered at $r=\pm\frac{1}{3}, \pm 3$ and $r=\pm 1$, respectively. \par
The ${\widehat{\mathfrak{cga}}}_{\ell=\frac{3}{2}}$ Conformal Galilei Algebra  is not a subalgebra of the $12$-generator, $\gamma=0$ and $r=\pm\frac{1}{3},\pm 3$, decoupled symmetry. From ${\widehat{\mathfrak{cga}}}_{\ell=\frac{3}{2}}$, in the $\gamma\rightarrow 0$ limit, a contraction algebra is recovered.
The contraction algebra, see Section {\bf 6}, is an $8$-generator subalgebra of the full  $r=\pm\frac{1}{3}, \pm 3$ symmetry algebra of the $\gamma=0$ decoupled system. \par
As a corollary of this analysis we showed that the contraction algebra, rather than the Conformal Galilei algebra itself, is an enhanced symmetry of the
$r=-3$ decoupled Pais-Uhlenbeck oscillators.\par
Concerning the $\ell=\frac{3}{2}$ PDE with discrete spectrum, we showed that, besides the $r=3$ real bounded spectrum, a real unbounded spectrum
is obtained at $r=-\frac{1}{3}, -3$. For all admissible values $r$ the induced operators on the r.h.s. are not hermitian. They are, nevertheless, PT-symmetric
\cite{beboe}.
 They are not, on the other hand, pseudo-hermitian \cite{mos1}, since the non-singular similarity transformations (see equations (\ref{nonsingular}) and (\ref{Kgamma})), mapping the coupled into the decoupled system, are not realized by a hermitian operator. The non-hermitian operator (\ref{herosc}) acts on
the Hilbert space ${\mathcal L}^2({ {\mathbb R}^2})$. Its real discrete spectrum coincides with the spectrum of two decoupled harmonic oscillators. The Bogoljubov transformations relating the coupled and the decoupled systems are explicitly given.\par
For generic half-integer $\ell$, the number of non-vanishing deformation parameters $\gamma_j$ is $\ell-{\textstyle{\frac{1}{2}}}$. For $\ell=\frac{5}{2}$
the arithmetic progression of $\omega_i$ frequencies entering eq. (\ref{genosc}) and producing the ${\widehat{\mathfrak{cga}}}_{\ell=\frac{5}{2}}$ symmetry
algebra is given by $\pm 1, \pm 3, \pm 5$. The $1,3,5$ sequence corresponds to the bounded spectrum, while the $1,-3,5$ sequence corresponds to a
special case of three Pais-Uhlenbeck oscillators.
\par
The extension of this construction to the $\ell={\textstyle{\frac{1}{2}}}+{\mathbb N}_0$ centrally extended Conformal Galilei Algebras  with $d>1$ (see \cite{negro1997nonrelativistic}) is immediate. The invariant PDEs with discrete spectrum correspond to non-hermitian operators whose spectrum is real and given by $d$ copies of the energy modes created in the $d=1$ case.\par
In the class of systems here investigated, the oscillators turn out to have all different frequencies. Equal frequency oscillators are obtained from the different class of second-order PDEs invariant under the $d=2$ centrally extended Conformal Galilei Algebras
with integer $\ell$,
{{see \cite{aikutoexotic}.}}
\\ {~}~{~}
\par {\Large{\bf Acknowledgements}}
{}~\par{}~\par
We thank T. Yumibayashi for helpful discussions on Pais-Uhlenbeck oscillators.
Z.K. and F.T. are grateful to the Osaka Prefecture University, where this work was elaborated, for hospitality.
F.T. received support from CNPq (PQ Grant No. 306333/2013-9).
N. A. is supported by the  grants-in-aid from JSPS (Contract No. 26400209).
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